Lecture Notes STA 2101 ALGEBRA FOR STATISTICS AND FINANCE · 2017-09-28 · ii) Recognize equations and graphs involving the conic sections. iii) Recognize graphs and properties of

Time is precious, but we do not know yet how precious it really is. We will only know when we are no longer able to take advantage of it…

Proverbs 21:5 The plans of the diligent lead to profit as surely as haste leads to poverty By J. K. Kiingati Page 1

Lecture Notes

STA 2101 ALGEBRA FOR STATISTICS AND FINANCE Pre-Requisites: None

Course Purpose By the end of this course, the student will have a good understanding of

algebraic notations, and gained knowledge on how to reason symbolically. Emphasis will be

placed on the study of functions, and their graphs, inequalities, and linear, quadratic, piece-wise defined,

rational, polynomial, exponential, and logarithmic functions.

Learning Objectives By the end of this course the student should be able to;

i) Solve linear, quadratic, and other nonlinear equations and be able to graph the solutions, using pencil and

paper as well as the graphics calculator.

ii) Recognize equations and graphs involving the conic sections.

iii) Recognize graphs and properties of functions and their inverses and interpret data involving graphs.

iv) Perform operations and procedures on polynomials, rational expressions, and numbers (real and

complex).

v) Use exponential and logarithmic functions in practical applications.

vi) Solve systems of equations using two and three variables by various methods, including the graphic

calculator.

vii) Use arithmetic and geometric sequences and series to find sums and products and to solve practical

application problems.

viii) Perform factorial notations and the binomial expansion.

ix) Apply permutations and combinations to fundamental principles of probability.

Course Description

Graphing parabolas, circles, ellipses, and hyperbolas; relations and functions; graphing functions; combining

functions; inverse functions; direct an inverse variations; solving problems whose mathematical models are

polynomial, rational, exponential and logarithmic functions; finding zeros of polynomial and rational

functions; solving systems of linear and nonlinear equations and inequalities with applications for each;

matrices and determinants; systems of nonlinear equations; binomial expan- sions; arithmetic and geometric

sequences and series; and counting techniques.

Course Text Books and Reference Text Books

1) Schay G., A Concise Introduction to Linear Algebra, Springer, ISBN: 9780817683252, 2012.

2) Strang G., Introduction to Linear Algebra, SIAM, ISBN: 9780961408893, 2003.

3) Hamilton A.G., Linear Algebra: An Introduction. Cambridge University Press, ISBN:

052132517X, 1989.

4) Andrilli S., Elementary Linear Algebra, Gulf Professional Publishing, ISBN: 9780120586219, 2003

5) Minc H., Introduction to Linear Algebra, Courier Dover Publications, ISBN: 9780486656953, 1988.

6) Lang S., Linear Algebra, Springer, ISBN: 9780387964126, 1987. 1573-8795.



VARIATIONS IN ALGEBRA Direct Variation If you know the retail price of one taifa laptop at JKUAT, then you can determine the total

revenue after selling 100, 1,000 or 15,000laptops since the total revenue is a constant multiple of

the retail price of one laptop. When two quantities y and x have a constant ratio k, they are said

to have direct variation and we write kxyxy then . where 0k

The constant k is called the constant of

Variation The equation kxy represents

direct variation between x and y, and y is

said to vary directly with x.

The graph of a direct variation equation

kxy is a line with slope k and y-intercept

0. The family of direct variation graphs

consists of lines through the origin, such as

those shown on the right.

Example 1 If y varies directly as x and y=20 when x=5 Write an equation that relates x and y

hence find the value of y when x=12.

Solution

Because x and y vary directly, then kxyxy Put x=5 and y=20 then

4 )5(20 kk Now when 48124, 12 yx

Example 2 Write and graph a direct

variation equation that has (-4, 8) as a

solution.

Solution

Use the given values of x and y to find the

constant of variation. aaxy 48

2a . Substituting -2 for a in axy

gives the direct variation equation xy 2 .

Its graph is shown.

Example 3 Hailstones form when strong updrafts support ice particles high in clouds, where

water droplets freeze onto the particles. The diagram shows a hailstone at two different times

during its formation.

a) Write an equation that gives the

hailstone’s diameter d (in inches) after t

minutes if you assume the diameter

varies directly with the time the

hailstone takes to form.

b) Using your equation from part (a),

predict the diameter of the hailstone after

20 minutes

c) Suppose that a hailstone forming in a

cloud has a radius of 0.6 inch. Predict

how long it has been forming.



Solution

a) Use the given values of t and d to find the constant of variation. aatd 1275.0

0625.0a . An equation that relates t and d is td 0625.0 .

b) After t=20 minutes, the predicted diameter of the hailstone is 25.1)20(0625.0 d inches.

c) Using the equation that relates t and d ie td 0625.0 put 6.0d to get

6.90625.06.0 tt minutes,

Because the direct variation equation kxy can be written as kx

y , a set of data pairs (x, y)

shows direct variation if the ratio of y to x is constant.

Example 4 Great white sharks have triangular teeth. The table below gives the length of a side

of a tooth and the body length for each of six great white sharks.

a) Does tooth length and body length show direct variation? If yes, write an equation that relates

the quantities.

Tooth length, t (cm) 1.8 2.4 2.9 3.6 4.7 5.8

Body length, b(cm) 215 290 350 430 565 695

b) The respective body masses m (in kilograms) of the great white sharks are; 80, 220, 375, 730,

1690, and 3195. Tell whether tooth length and body mass show direct variation. If so, write

an equation that relates the quantities

Solution

Find the ratio of the body length b to the tooth length t for each shark.

a) 4.1198.1

215 , 8.120

4.2

290 , 6.120

9.2

350 , 4.119

6.3

430 , 2.120

7.4

565 , 8.119

8.5

695

Because the ratios are approximately equal, the data show direct variation. An equation

relating tooth length and body length is tbt

b120120

b) Repeat the same procedure.

448.1

80 , 92

4.2

220 , 129

9.2

375 , 203

6.3

730 , 360

7.4

1690 , 551

8.5

3195

Because the ratios differ significantly, the data does not show direct variation.

Exercise

1) Write and graph a direct variation equation that has the given ordered pair as a solution.

(2, 6). (-3, 12). (6, -21). (4, 10). (-5, -1). (24, -8). (4

3, -4). (12.5, 5).

2) The variables x and y vary directly. Write an equation that relates x and y. Then find y

when x=12 given that; a) x=4 when y=8 b) x=-3 when y=-5 c) x=35 when y=-7

d) x=-18 when y=4 e) x=-4.8 when y=-1.6, f) x=2

3, when y=-10

3) Which equation is a direct variation equation that has (3,18) as a solution?

a) 𝑦 = 2𝑥2, b) 𝑦 = 1

6𝑥 c) 𝑦 = 6𝑥 , d) 𝑦 = 4𝑥 + 6

4) Does the given equation represents direct variation? If so, give the constant of variation?

𝑦 = 8𝑥, 𝑦 = 4 − 3𝑥, 3𝑦 − 7 = 10𝑥 , 2𝑦 − 5𝑥 = 0, 5𝑦 = −4𝑥 , 6𝑦 = 𝑥

5) The variables x and y vary directly. Write an equation that relates x and y. Then find x

when y=24 if; a) x=5 when y=15 b) x=-6 when y=8 c) x=-18 when y=-2

d) x=-12 when y=84 e) x=-20

3 when y=-15

8 f) x=-0.5 when y=3.6,

6) Does the data in the table show direct variation? If so, write an equation relating x and y.



X 3 6 9 12 15

Y -1 -2 -3 -4 -5

X 1 2 3 4 5

Y 7 9 11 13 15

X -8 -4 4 8 12

y 8 4 -4 -8 -12

X -5 -4 -3 -2 -1

y 20 16 12 8 4

7) Let (x1,y1) be a solution, other than (0, 0), of a direct variation equation. Write a second

direct variation equation whose graph is perpendicular to the graph of the first equation.

8) Let (x1,y1) and (x2 , y2) be any two distinct solutions of a direct variation equation. Show

that 𝑥2

𝑥1=

𝑦2

𝑦1

9) The amount A (in dollars) you pay for grapes varies directly with the amount P (in

pounds) that you buy. Suppose you buy 1.5 pounds for $2.25. Write a linear model that

gives A as a function of P.

10) The time t it takes a diver to ascend safely to the surface varies directly with the depth d.

It takes a minimum of 0.75 minute for a safe ascent from a depth of 45 feet. Write an

equation that relates d and t. Then predict the minimum time for a safe ascent from a

depth of 100 feet.

11) Hail 0.5 inch deep and weighing 1800 pounds covers a roof. The hail’s weight w varies

directly with its depth d. Write an equation that relates d and w. Then predict the weight

on the roof of hail that is 1.75 inches deep.

12) Your weight M on Mars varies directly with your weight E on Earth. If you weigh 116

pounds on Earth, you would weigh 44 pounds on Mars. Which equation relates E and M?

a) 𝑀 = 𝐸 − 72 b) 44𝑀 = 116𝐸 c) 𝑀 = 28

11𝐸 d) 𝑀 = 11

28𝐸

13) The ordered pairs (4.5, 23), (7.8, 40), and (16.0, 82) are in the form (s, t) where t

represents the time (in seconds) needed to download an Internet file of size s (in

megabytes). Tell whether the data show direct variation. If so, write an equation that

relates s and t.

14) Each year, gray whales migrate from Mexico’s Baja Peninsula to feeding grounds near

Alaska. A whale may travel 6000 miles at an average rate of 75 miles per day.

a) Write an equation that gives the distance d1 traveled in t days of migration.

b) Write an equation that gives the distance d2 that remains to be traveled after t days of

migration.

c) Do the equations from parts a) and b) represent direct variation. Explain your answers.

15) At a jewelry store, the price p of a gold necklace varies directly with its length l. Also, the

weight w of a necklace varies directly with its length. Show that the price of a necklace

varies directly with its weight.

Inverse Variation

In the previous section you learned that two variables x and y show direct variation if y = kx for

some nonzero constant k. Another type of variation is called inverse variation. Two variables x

and y show inverse variation if increase in one variable leads to decrease in the other variable

and we write x

k

xyy 1

. where 0k is called the constant of variation, and y is said to

vary inversely with x. The equation for inverse variation can be rewritten as xy = k. This tells

you that a set of data pairs (x, y) shows inverse variation if the products xy are constant or

approximately constant.



Example 1 The variables x and y vary inversely, and y = 8 when x = 3. Write an equation that

relates x and y hence find y when 𝑥 = −4.

Solution

Use the given values of x and y to find the constant of variation x

k

xyy 1

or xyk .

Thus 242483 xyk The inverse variation equation is xy 24 Now When 𝑥 = −4.,

the value of y is: 6424 y

Example 2 The speed of the current in a whirlpool varies inversely with the distance from the

whirlpool’s center. The Lofoten Maelstrom is a whirlpool located off the coast of Norway. At a

distance of 3 kilometers (3000 meters) from the center, the speed of the current is about 0.1

meter per second. Describe the change in the speed of the current as you move closer to the

whirlpool’s center.

Solution

First write an inverse variation model relating distance from center d and speed s

d

k

dss 1

or sdk . Thus 30030030001.0 sdk

The inverse variation equation is ds 300 . The table shows some speeds for different values of d.

.d 2000 1500 500 250 50

S 0.15 0.2 0.6 1.2 6

From the table you can see that the speed of the current increases as you move closer to the

whirlpool’s center.

Example 3 The table compares the wing

flapping rate r (in beats per second) to the

wing length l (in centimeters) for several

birds. Do these data show inverse variation?

If so, find a model for the relationship

between r and l.

Solution

Each product rl is approximately equal to 117. For instance, (3.6)(32.5) = 117 and (5.0)(23.5) =

117.5. So, the data do show inverse variation. A model for the relationship between wing

flapping rate and wing length is lr 117

Exercise

1) Tell whether x and y show direct variation, inverse variation, or neither

𝑥𝑦 = 0.25, 𝑥

𝑦= 5, 𝑦 = 𝑥 − 3 , 𝑥 = 7

𝑦, 𝑦

𝑥= 12 , 1

2𝑥𝑦 = 9, 2𝑥 + 𝑦 = 4

2) The variables x and y vary inversely. Use the given values to write an equation relating x and

y. Then find y when x = 2 a) X=5, y=-2 b) X=4, y=8 c) X=7, y=1 d) X=0.5, y=10

e) 𝑋 = 2

3-, y=-6, f) 𝑋 = 3

4-, y=-3

8,

3) If r varies inversely as the cube of s, and r = 17 when s = 3, find r when s = 2.

4) If g varies inversely as the square root of h, and g = 9 when h = 121, find g when h = 81.

5) Determine whether x and y show direct variation, inverse variation, or neither



x y b) x y © x y d) x y

1.5 20 31 217 3 36 1.6 40

2.5 12 20 140 5 50 4 16

4 7.5 17 119 7 105 5 12.8

5 6 12 84 16 48 20 3.2

6) The force F needed to loosen a bolt with a wrench varies inversely with the length l of the

handle. Write an equation relating F and l, given that 250 pounds of force must be exerted to

loosen a bolt when using a wrench with a handle 6 inches long. How much force must be

exerted when using a wrench with a handle 24 inches long?

7) The frequency of a vibrating guitar string varies inversely as its length. Suppose a guitar

string 0.65 meters long vibrates 4.3 times per second. What frequency would a string 0.5

meters long have?

8) On some tubes of caulking, the diameter of the circular nozzle opening can be adjusted to

produce lines of varying thickness. The table shows the length l of caulking obtained from a

tube when the nozzle opening has diameter d and cross-sectional area A.

D(in.) 81 4

1 83

21

A(in.2) 256

64

2569

16

I(in.) 1440 360 160 90

a) Determine whether l varies inversely with

d. If so, write an equation relating l and d.

b) Determine whether l varies inversely with

A. If so, write an equation relating l & A.

c) Find the length of caulking you get from a tube whose nozzle opening has a diameter of 3

4 inch.

9) The intensity I of a sound (in watts per square meter) varies inversely with the square of the

distance d (in meters) from the sound’s source. At a distance of 1 meter from the stage, the

intensity of the sound at a rock concert is about 10 watts per square meter. Write an equation

relating I and d. If you are sitting 15 meters back from the stage, what is the intensity of the

sound you hear?

10) The current in a simple electrical circuit is inversely proportional to the resistance. If the

current is 80 amps when the resistance is 50 ohms, find the current when the resistance is 22

ohms.

11) The intensity of light produced by a light source varies inversely as the square of the distance

from the source. If the intensity of light produced 3 feet from a light source is 750 foot-

candles, find the intensity of light produced 5 feet from the same source.

Joint variation

Joint Variation occurs when a quantity varies directly as the product of two or more other

quantities. For instance, if 𝑧 = 𝑘𝑥𝑦 where k ≠ 0, then z varies jointly with x and y. Other types

of variation are also possible, for instance

i) z varies directly as x and inversely with y.. So y

x

y

x kzz

ii) z varies directly as y and inversely with the square of x.. So 22 x

kyz

x

yz

iii) z varies directly as the square of x and inversely as the square root of y.. So

22

y

kxz

y

xz



Example 1 The law of universal gravitation states that the gravitational force F (in newtons)

between two objects varies jointly with their masses m1 and m2 (in kilograms) and inversely

with the square of the distance d (in meters)

between the two objects. The constant of

variation is denoted by G and is called the

universal gravitational constant. Write an

equation for the law of universal gravitation

hence estimate the universal gravitational

constant. Use the Earth and sun facts given

at the right

Solution

2

21

d

mGmF Substitute the given values and solve for G

)1067.6 G )1029.5(G )1050.1(

)1099.1)(1098.5(1053.3 1132

211

302422

G

The universal gravitational constant is about 2211 /1067.6 KgNM

Exercise

1) Write an equation for the given relationship.

a) x varies inversely with y and directly with z.

b) y varies jointly with z and the square root of x.

c) w varies inversely with x and jointly with y and z.

2) The variable z varies jointly with x and y. Use the given values to write an equation relating

x, y, and z. Then find z when x =-4 and y = 7. (a) X=3, y=8. Z=6 (b) X=-12, y=4, z=2

(c) X=1, y=1

3, z=5 (d) X=-6, y=3, z=2

3 (e) 𝑋 = 5

6-, y=0.3, z=8 (f) 𝑋 = 3

8,-, y=-16

17,, z=3

2,

3) If f varies jointly as g and the cube of h, and f = 200 when g = 5 and h = 4, find f when g = 3

and h = 6.

4) If a varies jointly as b and the square root of c, and a = 21 when b = 5 and c = 36, find a when

b = 9 and c = 225.

5) Wind resistance varies jointly as an object’s surface area and velocity. If an object traveling

at 40 mile per hour with a surface area of 25 square feet experiences a wind resistance of 225

Newtons, how fast must a car with 40 square feet of surface area travel in order to experience

a wind resistance of 270 Newtons?

6) For a given interest rate, simple interest varies jointly as principal and time. If $2000 left in

an account for 4 years earns interest of $320, how much interest would be earned in if you

deposit $5000 for 7 years?

7) The volume of a pyramid varies jointly as its height and the area of its base. A pyramid with

a height of 12 feet and a base with area of 23 square feet has a volume of 92 cubic feet. Find

the volume of a pyramid with a height of 17 feet and a base with an area of 27 square feet.

8) Tell whether x varies jointly with y and z. (a) 𝑥 = 15𝑦𝑧, (b) 𝑥

𝑧= 0.5𝑦, (c) 𝑥𝑦 = 4𝑧 ,

(d) 𝑥 = 𝑦𝑧

2, (e) x= 3𝑧

𝑦, (f) 2𝑦𝑧 = 7𝑥, (g) 𝑥

𝑦= 17𝑧, (h) 5𝑥 = 4𝑦𝑧

9) A star’s diameter D (as a multiple of the sun’s diameter) varies directly with the square root

of the star’s luminosity L (as a multiple of the sun’s luminosity) and inversely with the

square of the star’s temperature T (in kelvins).



a) Write an equation relating D, L, T, and a constant k.

b) The luminosity of Polaris is 10,000 times the luminosity of the sun. The surface

temperature of Polaris is about 5800 kelvins. Using k = 33,640,000, find how the

diameter of Polaris compares with the diameter of the sun.

c) The sun’s diameter is 1,390,000 kilometers. What is the diameter of Polaris?

10) The work W (in joules) done when lifting an object varies jointly with the mass m (in

kilograms) of the object and the height h (in meters) that the object is lifted. The work done

when a 120 kilogram object is lifted 1.8 meters is 2116.8 joules. Write an equation that

relates W, m, and h. How much work is done when lifting a 100 kilogram object 1.5 meters?

11) The heat loss h (in watts) through a single-pane glass window varies jointly with the

window’s area A (in square meters) and the difference between the inside and outside

temperatures d (in kelvins).

a) Write an equation relating h, A, d, and a constant k.

b) A single-pane window with an area of 1 square meter and a temperature difference of 1

kelvin has a heat loss of 5.7 watts. What is the heat loss through a single-pane window

with an area of 2.5 square meters and a temperature difference of 20 kelvins?

12) The area of a trapezoid varies jointly with the height and the sum of the lengths of the bases.

When the sum of the lengths of the bases is 18 inches and the height is 4 inches, the area is

36 square inches. Find a formula for the area of a trapezoid.

13) The load P (in pounds) that can be safely supported by a horizontal beam varies jointly with

the width W (in feet) of the beam and the square of its depth D (in feet), and inversely with

its length L (in feet). a. How does P change when the width and length of the beam are

doubled? b. How does P change when the width and depth of the beam are doubled? c. How

does P change when all three dimensions are doubled? Describe several ways a beam can be

modified if the safe load it is required to support is increased by a factor of 4.

14) Ohm’s law states that the resistance R (in ohms) of a conductor varies directly with the

potential difference V (in volts) between two points and inversely with the current I (in

amperes). The constant of variation is 1. What is the resistance of a light bulb if there is a

current of 0.80 ampere when the potential difference across the bulb is 120 volts?

LINEAR INEQUALITIES Solving Linear Inequalities with one Variable Inequalities have properties that are similar to those of equations, but the properties differ in

some important ways. Inequalities such as 𝑥 ≤ 1 and 2𝑛 − 3 > 9 are examples of linear

inequalities in one variable. A solution of an inequality in one variable is a value of the variable

that makes the inequality true. Eg,-2, 0, 0.872, and 1 are some of the many solutions of 𝑥 ≤ 1.

Transformations that Produce Equivalent Inequalities

When simplifying/transforming linear inequalities you may;

Add or subtract the same number to both sides.

Multiply or divide both sides by the same positive number.

Multiply or divide both sides by the same negative number and reverse the inequality.



The graph of an inequality in one variable consists of all points on a real number line that

correspond to solutions of the inequality. To graph an inequality in one variable, use an open dot

for < or > and a solid dot for ≤ or ≥. Eg the graphs of 𝑥 < 3 and 𝑥 ≥ −2 are shown below.

Example 1 Solve the inequality5𝑦 − 8 < 12 and graph the solution set.

Solution

5𝑦 − 8 < 12 ⇛ 5𝑦 < 20 ⇛ 𝑦 < 4 The solutions are all real numbers less than 4, as

shown in the graph below.

Example 2 Solve 2𝑥 + 1 ≤ 6𝑥 − 1.

Solution

Original inequality is 2𝑥 + 1 ≤ 6𝑥 − 1 Subtract 6x from each side −4𝑥 + 1 ≤ −1

Subtract 1 from each side. −4𝑥 ≤ −2 Divide each side by -4 to get . 𝑥 ≥ 0.5

The solutions are all real numbers greater than or equal to 0.5

Example 3

The weight w (in pounds) of an Icelandic saithe is given by 𝑤 = 10.4𝑡 − 2.2 where t is the age

of the fish in years. Describe the ages of a group of Icelandic saithe that weigh up to 29 pounds.

Solution

𝑤 ≤ 29 Weights are at most 29 pounds ⇛ 10.4𝑡 − 2.2 < 29 ie put 𝑤 = 10.4𝑡 − 2.2

⟹ 10.4𝑡 = 31.2 Add 2.2 to each side. ∴ 𝑡 ≤ 3 Divide both side by 10.4. The ages are less

than or equal to 3 years.

Example 4 If 𝑥 ∈ {−3, −2, −1, 0, 1, 2, 3}.find the solution set of each of the following

a) 𝑥 + 2 < 1 b) 2𝑥 − 1 < 4 c) 3 − 5𝑥 < −1 d) − 6 ≥ 2𝑥 − 4 e) 14 − 2𝑥 ≤ 6

Solution

a) 𝑥 + 2 < 1 ⟹ 𝑥 < −1 Subtracting 2 from both sides. Therefore the solution set is {−3, −2} b) 2𝑥 − 1 < 4 ⟹ 2𝑥 < 5 ⟹ 𝑥 < 2.5 Therefore the solution set is {−3, −2, −1, 0, 1, 2} c) 3 − 5𝑥 < −1 ⟹ −5𝑥 < −4 ⟹ 𝑥 > 0.8 Therefore the solution set is {1, 2, 3}. d) −6 ≥ 2𝑥 − 4 ⟹ 10 ≥ 2𝑥 ⟹ 5 ≥ 𝑥 or 𝑥 ≤ 5 Thus the solution set

is{−3, −2, −1, 0, 1, 2, 3} e) 14 − 2𝑥 ≤ 6 ⟹ −2𝑥 ≤ −8 or 𝑥 ≥ 4 Thus the solution set is{} or 𝜙

Solving Compound Inequalities

A Compound Inequality is two simple inequalities joined by “and” or “or.” Here are two

examples

−2 ≤ 𝑥 < 1

All real numbers that are greater than or

equal to -2 and less than 1.

𝑥 < −1 and 𝑥 ≥ 2

All real numbers that are less than -1 or

greater than or equal to 2.



Example 1 Solve −2 ≤ 3𝑡 − 8 ≤ 10.

Solution

To solve, you must isolate the variable between the two inequality signs.

−2 ≤ 3𝑡 − 8 ≤ 10 Write original inequality. ⟹ 6 ≤ 3𝑡 ≤ 18 Add 8 to each expression.

∴ 2 ≤ 𝑡 ≤ 6 Divide each expression by 3.

Because t is between 2 and 6, inclusive, the solutions are all real numbers greater than or equal to

2 and less than or equal to 6. The graph is shown below.

Example 2 Solve 2𝑥 + 3 < 5 or 4𝑥 − 7 > 9.

Solution

A solution of this compound inequality is a solution of either of its simple parts, so you should

solve each part separately. 2𝑥 + 3 < 5 ⟹ 𝑥 < 1 and 4𝑥 − 7 > 9 ⟹ 𝑥 > 4 thus the

solutions are all real numbers less than 1 or greater than 4. The graph is shown below.

Example 3 You have added enough antifreeze to your car’s cooling system to lower the

freezing point to º35°C and raise the boiling point to 125°C. The coolant will remain a liquid as

long as the temperature C (in degrees Celsius) satisfies the inequality −35 < 𝐶 < 125. Write

the inequality in degrees Fahrenheit.

Solution

Let F represent the temperature in degrees Fahrenheit, and use the formula 𝐶 =5

9(𝐹 − 32).

−35 <5

9(𝐹 − 32) < 125 ⟹ −63 < 𝐹 − 32 < 225 ⟹ −31 < 𝐹 < 257

The coolant will remain a liquid as long as the temperature stays between -31°F and 257°F.

Example 4 You are a state patrol officer who is assigned to work traffic enforcement on a

highway. The posted minimum speed on the highway is 45 miles per hour and the posted

maximum speed is 65 miles per hour. You need to detect vehicles that are traveling outside the

posted speed limits.

a) Write these conditions as a compound inequality.

b) Rewrite the conditions in kilometers per hour.

Solution

a) Let m represent the vehicle speeds in miles per hour. The speeds that you need to detect are

given by: 𝑚 < 45 or 𝑚 > 65

b) Let k be the vehicle speeds in kilometers per hour. The relationship between miles per hour

and kilometers per hour is given by the formula m≈ 0.621k. You can rewrite the conditions in

kilometers per hour by substituting 0.621k for m in each inequality and then solving for k. ie

0.621𝑘 < 45 or 0.621𝑘 > 65 ⟹ 𝑘 < 72.5 or 𝑘 > 105 You need to detect vehicles whose speeds are less than 72.5 kilometers per hour or greater than

105 kilometers per hour.

Linear Inequalities in Two Variables A Linear Inequality in two variables is an inequality that can be written in one of the following

forms: 𝐴𝑥 + 𝐵𝑦 < 𝐶, 𝐴𝑥 + 𝐵𝑦 ≤ 𝐶, 𝐴𝑥 + 𝐵𝑦 > 𝐶, 𝐴𝑥 + 𝐵𝑦 ≥ 𝐶. An ordered pair (x ,y)



is a solution of a linear inequality if the inequality is true when the values of x and y are

substituted into the inequality. Eg (-6, 2) is a solution of 𝑦 ≥ 3𝑥 − 9 because 2 ≥ 3(−6) − 9 is

a true statement.

Example 1 Is the given ordered pair is a solution of 2𝑥 + 3𝑦 ≥ 5? a) (0, 1) b (4, -1) c (2, 1)

Solution

Ordered Pair Substitute Conclusion

a. (0, 1) 2 × 0 + 3 × 1 = 3 ≱ 5 (0, 1) is not a solution.

b. (4, -1) 2 × 4 + 3 × −1 = 5 ≥ 5 (4, -1) is a solution.

c. (2, 1) 2( 2) + 3(1) = 7 ≥ 5 (2, 1) is a solution.

Graphing a Linear Inequality

The graph of a linear inequality in two variables is the graph of all solutions of the inequality.

The boundary line of the inequality divides the coordinate plane into two a shaded region which

contains the points that are solutions of the inequality, and an unshaded region which contains

the points that are not. To graph a linear inequality, follow these steps:

Graph the boundary line. Use a dotted line for < or > and a solid line for ≤ or ≥.

To decide which side of the boundary line to shade, test a point not on the boundary line to

see whether it is a solution of the inequality. Then shade the appropriate half-plane

Example 2 Graph a) ) 𝑦 < −2 and b) 𝑥 ≤ 1 in a coordinate plane

Solution

a) Graph the boundary line 𝑦 = −2. Use a dashed line because 𝑦 < −2. Test the point (0, 0).

Because (0, 0) is not a solution of the inequality, shade the half-plane below the line.

b) Graph the boundary line 𝑥 = 1. Use a solid line because 𝑥 ≤ 1. Test the point (0, 0). Because

(0, 0) is a solution of the inequality, shade the half-plane to the left of the line.

Example 3 Graph a) 𝑦 < 2𝑥 and b) 2𝑥 − 5𝑦 ≥ 10 in a coordinate plane

Solution

a) Graph the boundary line 𝑦 = 2𝑥 Use a dashed line because 𝑦 < 2𝑥 Test the point (1, 1).

Because (1, 1) is a solution of the inequality, shade the half-plane below the line.

b) Graph the boundary line 2𝑥 − 5𝑦 = 10 Use a solid line because 2𝑥 − 5𝑦 ≥ 10 Test the

point (0, 0). Since (0, 0) is not a solution of the inequality, shade the half-plane below the line



Example 4 You have relatives living in both the United States and Mexico. You are given a

prepaid phone card worth $50. Calls within the continental United States cost $0.16 per minute

and calls to Mexico cost $0.44 per minute.

a) Write a linear inequality in two variables to represent the number of minutes you can use for

calls within the United States and for calls to Mexico.

b) Graph the inequality & discuss 3 possible solutions in the context of the real-life situation.

Solution

a) Let x and y be the number of minutes you can use for calls within the United States and for

calls to Mexico respectively then 0.16𝑥 + 0.44𝑦 ≤ 50

b) Graph the boundary line 0.16𝑥 + 0.44𝑦 = 50. Use a solid line because 0.16𝑥 + 0.44𝑦 ≤ 50

Test the point (0, 0). Because (0, 0) is a solution of the inequality, shade the half-plane below

the line. Finally, because x and y cannot be negative, restrict the graph to points in the first

quadrant. Possible solutions are points within the shaded region shown.

Graphing and Solving Systems of Linear Inequalities The following is a Systems of Linear Inequalities in two variables.

𝑥 + 𝑦 ≤ 6 Inequality 1 2𝑥 − 𝑦 > 4 Inequality 2

A solution of a system of linear inequalities is an ordered pair that is a solution of each inequality

in the system. For example, (3, -1) is a solution of the system above. The graph of a system of

linear inequalities is the graph of all solutions of the system.

Investigating Graphs of Systems of Inequalities

The coordinate plane shows the four regions

determined by the lines 23 yx and

12 yx . Use the labeled points to help

you match each region with one of the

systems of inequalities

a) 12

23

yx

yx c)

12

23

yx

yx

b) 12

23

yx

yx d)

12

23

yx

yx



Conclusion; A system of linear inequalities defines a region in a plane. Here is a method for

graphing the region.

Graphing a System of Linear Inequalities

To graph a system of linear inequalities, do the following for each inequality in the system:

• Graph the line that corresponds to the inequality. Use a dashed line for an inequality with <

or > and a solid line for an inequality with ≤ or ≥.

• Lightly shade the half-plane that is the graph of the inequality. Colored pencils may help you

distinguish the different half-planes.

The graph of the system is the region common to all of the half-planes. If you used colored

pencils, it is the region that has been shaded with every color.

Example 1 Graph the system. 2 Inequality2 and 1 Inequality13 xyxy

Solution

Begin by graphing each linear inequality. Use a different color for each half-plane. For instance,

you can use red for Inequality 1 and blue for Inequality 2. The graph of the system is the region

that is shaded purple

Example 2 Graph the system. 2434and0,0 yxyx

Solution

Inequality 1 and Inequality 2 restrict the solutions to the first quadrant. Inequality 3 is the half-

plane that lies on and below the line 2434 yx . The graph of the system of inequalities is the

triangular region shown below



One can use a system of linear inequalities to describe a real-life situation, as shown in the

following example

Example 3 A person’s theoretical maximum heart rate is 220 − 𝑥 where x is the person’s age

in years (20 ≤ x≤ 65). When a person exercises, it is recommended that the person strive for a

heart rate that is at least 70% of the maximum and at most 85% of the maximum.

a) You are making a poster for health class. Write and graph a system of linear inequalities that

describes the information given above.

b) A 40-year-old person has a heart rate of 150 (heartbeats per minute) when exercising. Is the

person’s heart rate in the target zone?

Solution

a) Let y represent the person’s heart rate. From the given information, you can write the

following four inequalities

65and20 xx Person’s age must be at least 20 and at most 65

)220(8.0and)220(7.0 xyxy Target rate is at least 70% and at most 85% of

maximum rate. The graph of the system is shown below.

b) From the graph you can see that the target zone for a 40-year-old person is between 126 and

153, inclusive. That is, 126≤y≤153. A 40-year-old person who has a heart rate of 150 is

within the target zone

Exercise

1) Test whether the ordered pair is a solution of the system 22

1

xy

x (-1, 2), (0, 0), (1, 4) and (2, 7)

2) Test whether the ordered pair is a solution of the corresponding system of inequality graphed

below? (25, -5), (2, 3) and (2, 6)



3) Graph the system of linear inequalities a) 22

1

xy

x b)

1

3

y

yx c)

5

0

xy

x

4) Give an ordered pair that is a solution of the system;

a) 15

3

y

yx b)

2

6

x

yx c)

12

4

x

yx d)

yx

y

x

10

7

e)

132

5

3

yx

y

x

f)

xy

y

x

0

0

5) To be a flight attendant, you must be at least 18 years old and at most 55 years old, and you must

be between 60 and 74 inches tall, inclusive. Let X represent a person’s age (in years) and let y

represent a person’s height (in inches). Write and graph a system of linear inequalities showing

the possible ages and heights for flight attendants.

6) Match the system of linear inequalities with its graph

a) 4

2

y

x b)

4

2

y

x c)

xy

y

x

0

3

d)

xy

y

x

3

0

e)

xy

y

x

1

1

3

f)

xy

y

x

1

1

1

7) Write a system of linear inequalities for the region

8) Graph the system of linear inequalities

a) xyxy ,3,4

b) 52,6,1 xyxy c) 33,335,632 yxyxyx

d) 13,1,04 yxyxyx

e) 2,5,12 xyxyx

f) 3,8,435 yyxyx

g) 10,1,2 xyxyx

h) xyyxy ,24,0



i) xyyxyx 2,165,0

j) xyyxxy ,15,9,0

k) 22,41 yxyx

l) 3,12,56 xyyxy

9) You are a lifeguard at a community pool, and you are in charge of maintaining the proper pH

(amount of acidity) and chlorine levels. The water test-kit says that the pH level should be

between 7.4 and 7.6 pH units and the chlorine level should be between 1.0 and 1.5 PPM (parts per

million). Let p be the pH level and let c be the chlorine level (in PPM). Write and graph a system

of inequalities for the pH and chlorine levels the water should have.

10) For a healthy person who is 4 feet 10 inches tall, the recommended lower weight limit is about 91

pounds and increases by about 3.7 pounds for each additional inch of height. The recommended

upper weight limit is about 119 pounds and increases by about 4.9 pounds for each additional

inch of height. Source: Dietary Guidelines Advisory Committee

a) Let x be the number of inches by which a person’s height exceeds 4 feet 10 inches and let y

be the person’s weight in pounds. Write a system of inequalities describing the possible

values of X and y for a healthy person.

b) Use a graphing calculator to graph the system of inequalities from Exercise 52.

c) What is the recommended weight range for someone 6 feet tall?

11) A shoe store gives a discount of between 10 to 25% on all sales. Let x be the regular footwear

price and Y be the discount price

a) Write a system of inequalities for the regular footwear prices and possible sale prices.

b) Graph the system you have written in part a above. Use your graph to estimate the range of

possible sale prices for shoes that are regularly priced at $65

12) The men’s world weightlifting records for the 105-kg-and-over weight category are shown in the

table. The combined lift is the sum of the snatch lift and the clean and jerk lift. Let s be the weight

lifted in the snatch and let j be the weight lifted in the clean and jerk. Write and graph a system of

inequalities to describe the weights you could lift to break the records for both the snatch and

combined lifts, but not the clean and jerk lift.

13) Each day, an average adult moose can process about 32 kilograms of terrestrial vegetation (twigs

and leaves) and aquatic vegetation. From this food, it needs to obtain about 1.9 grams of sodium

and 11,000 Calories of energy. Aquatic vegetation has about 0.15 gram of sodium per kilogram

and about 193 Calories of energy per kilogram, while terrestrial vegetation has minimal sodium

and about four times more energy than aquatic vegetation. Write and graph a system of

inequalities describing the amounts t and a of terrestrial and aquatic vegetation, respectively, for

the daily diet of an average adult moose.

14) A potter has 70 pounds of clay and 40 hours to make soup bowls and dinner plates to sell at a

craft fair. A soup bowl uses 3 pounds of clay and a dinner plate uses 4 pounds of clay. It takes 3

hours to make a soup bowl and 1 hour to make a dinner plate. If the profit on a soup bowl is $25

and the profit on a dinner plate is $20, how many bowls and plates should the potter make in

order to maximize profit?



SEQUENCES AND SERIES Introduction Saying that a collection of objects is listed “in sequence” means that the collection is ordered so

that it has a first member, a second member, a third member, and so on. Below are two examples

of sequences of numbers. The numbers in the sequences are called terms

Sequence 1: 3 6 9 12 15 Sequence 2: 3 6 9 12 15 . . . . .

A sequence can be defined as an arrangement of numbers in a particular/ specific order. It can

also be defined as a function whose domain is a set of consecutive integers.

DOMAIN: 1 2 3 4 5 The domain gives the relative position of each term: 1st, 2nd, 3rd, and

so on. If a domain is not specified, it is understood that the domain starts with 1.

RANGE: 3 6 9 12 15 The range gives the terms of the sequence.

Sequence 1 above is a finite sequence because it has a last term. Sequence 2 is an infinite

sequence because it continues without stopping.

Each number in a sequence is called a term. 21,TT and n nT are used to denote the 1st, 2nd and

nth terms respectively.

Example 1 Determine the next 2 terms in each of the following sequences; a) ... 14, 11, 8, 5,

b) … 22,15,9,4, c) ,...,,,181

41

21 d) ,...,,,

54

43

32

21

Solution

a) The difference between the terms is 3 therefore the next 2 terms are 17 and 20.

b) The difference between the terms is increasing by 1 so the next 2 terms are 30 and 39.

c) Ratio between consecutive terms is 21 the next 2 terms are

161 and

321 .

d) Notice that the denominator exceeds the numerator by 1. Thus the next 2 terms are 65 and

76

Example 2 List the 1st 5 terms of the sequence whose nth term is; a) 32T nn b) 1)2(T n

n

Solution

a) 53)1(21 T 73)2(22 T 93)3(23 T 113)4(24 T 133)5(25 T

So the sequence is 5 7 9 11 13 . . . . .

b) 1)2( 11

1 T 2)2( 12

2 T 4)2( 13

3 T 8)2( 14

4 T 16)2( 15

5 T

Therefore the sequence is 1 2 4 8 16 . . . . .

Remarks If the terms of a sequence have a recognizable pattern, then you may be able to write a

rule for the nth term of the sequence.

Example 3 For each sequence, describe the pattern, write the next term, and write a formula for

the nth term a) . . . 81

1 ,

27

1- ,

9

1 ,

3

1 b) 2 6 12 20 . . .

Solution

a) We can write the terms as . . . )( )( )( )( 4

313

312

311

31

The next term is 24315

31

5 )(T . A formula for the nth term is n

n )(T31



b) We can write the terms as 1(2), 2(3), 3(4), 4(5), . . . .

The next term is 30)6(5T5 . A rule for the nth term is )1(T nnn .

When the terms of a sequence are added, the resulting expression is a series. A series can be

finite or infinite. Eg Finite series 1512963 infinite series .....1512963

We can use summation notation to write a series. For example, for the finite series shown above,

we can write

5

1

3i1512963i

where i is the index of summation,

The summation notation is read as “the sum from i equals 1 to 5 of 3i.” Summation notation is

also called sigma notation because it uses the uppercase Greek letter sigma, written .

Summation notation for an infinite series is similar to that for a finite series. For example, for the

infinite series shown above, we can write

1

3i.....1512963i

.

Example 4 Write each series with summation notation

a) 100....15105 b) ....54

43

32

21

Solution

a) Notice that the first term is 5(1), the second is 5(2), the third is 5(3), and the last is 5(20). So,

the terms of the series can be written as ii 5T where i = 1, 2, 3, . . . . 20 and the summation

notation for the series is

20

1

5ii

b) Notice that for each term the denominator of the fraction is 1 more than the numerator. So,

the terms of the series can be written as 1

T

i

ii where i = 1, 2, 3, 4, . . and the summation

notation for the series is

11i

i

i

Note The index of summation does not have to be i any letter can be used. Also, the index does

not have to begin at 1. For instance, in Example 5 part (b) below, the index begins at 3.

Example 5 Find the sum of the series a)

6

1

2i

i b)

6

3

2 )2(k

k

a) 4212108642)6(2)5(2)4(2)3(2)2(2)1(226

1

i

i

b) 9438271811)62()52()42()32()2( 22226

3

2 k

k

The sum of the terms of a finite sequence can be found by simply adding the terms. For

sequences with many terms, however, adding the terms can be tedious. Formulas for finding the

sum of the terms of four special types of sequences are given below.

nn

i

1

1 )1(21

1

nnin

i

)12)(1(61

1

2

nnnin

i

22122

41

1

3 )1()1(

nnnnin

i

In words, the 1st formula gives the sum of n 1’s. The 2nd gives the sum of the positive integers

from 1 to n. The 3rd formula gives the sum of the squares of the positive integers from 1 to n.



Exercise

1) Write the first six terms of the sequence a) 13T nn b) nn 2T c) 32T n

n d)

1T 2 nn e) n

nn

2

2T

f) 2)1(T nn g) 1

32)(3T n

n

2) Write the next term in the sequence. Then write a rule for the nth term.

a) .....7531

b) ....1000100101

c) ....1410842

d) .....2015105-

e) ...81

61

41

21

f) .....85

74

63

52

41

g) .....35

34

33

32

31

h) .....

504

403

302

201

i) ....1.53.45.37.29.1

3) Write the series with summation notation.

a) 1713951

b) 20161284

c) ....211512933-

d) ....-54-32-1

e) 11-10-9-8-7-

f) ....98

87

76

65

g) 001.001.01.01

h) 362516941

4) Find the sum of the series a)

7

1

3i

i b)

3

1

34n

n c)

5

1

2 )1(k

k d)

4

0

2 )12(n

n e)

10

2

2

nn

f)

4

1

)2(k

kk g)

12

21

1

nn

h)

5

11

nnn i)

6

2 1i i

i

5) The diagram shows part of a roof frame.

The length (in feet) of each vertical

support is given below the support.

These lengths form an arithmetic

sequence from each end to the middle.

a) Find the total length of the vertical

supports from one end to the middle.

b) Use your result from Exercise 65 to

find the total length of the vertical

supports from end to end.

6) Use one of the formulas for special series to find the sum: a)

42

1

1i

b)

10

1n

n c)

12

1

2

i

i d)

6

1

3

k

k

7) Suppose you are stacking tennis balls in a pyramid as a display at a sports store. If the base is

an equilateral triangle, then the number na of balls per layer would be )1(21 nnan where n

= 1 represents the top layer. How many balls are in the fifth layer? How many balls are in a

stack with 5 layers?

Arithmetic Progression (AP)

In an arithmetic progression the difference between consecutive terms is constant. The constant

difference is called the common difference and is usually denoted by d.

The nth term and the sum of the 1st n terms of an AP with first term a and common difference d,

are respectively given by dnaTn )1( and dnan

sn )1(22

Example 1 Decide whether each sequence is arithmetic.

a) -3, 1, 5, 9, 13, . . . b) 2, 5, 10, 17, 26, . . .



Solution

To decide whether a sequence is arithmetic, find the differences of consecutive terms.

a) ...4 45342312 TTTTTTTT Each difference is 4, so the sequence is arithmetic

b) 9753 5342312 TTTTTTTT The differences are not constant, so the

sequence is not arithmetic.

Example 2 Find the 10th term and the sum of the first 20 terms of the AP ...,5,2,1,4

Solution

4903)120()42[(2

20 and 233)110(434 2010 sTda

Example 3 A sequence is given by the formula 53 nTn for ...3,,2,1n Write down the

first 4 terms of this sequence hence find S12 and T15

Solution

50)3(148 and 294)]3(11)8(2[2

1231714118 15124321 TSdTTTT

Example 4 The 10th term of an AP is -15 and the 31st term is -57 find the 15th term.

Solution

25)2(143 thus242215730

15915

31

10

Tdd

daT

daT

Example 5 Which term of the AP ...17,11,5 is 119?

Solution

201206119)1(5565 nnnTda n

Thus 119 is the 20th term of the given arithmetic progression ...17,11,5 .

Example 6 The sum of the first n terms of a certain progression is given by nnSn 72 . What

kind of a sequence is the progression?

Solution

246219

46144671

333213

22212

2

11

TTTTTS

TTTTSTS

The sequence is ...,2,4,6 which is an AP with a common difference .2d

Example 7 The first row of a concert hall has 25 seats, and each row after the first has one

more seat than the row before it. There are 32 rows of seats.

a) Write a rule for the number of seats in the nth row.

b) 35 students from a class want to sit in the same row. How close to the front can they sit?

c) What is the total number of seats in the concert hall?

d) Suppose 12 more rows of seats are built (where each row has one more seat than the row

before it). How many additional seats will the concert hall have?



Solution

a) Use 25a and 1d to write a formula for the nth term

nndna 241)1(25)1(Tn

b) Using n 24Tn put 35Tn nad solve for n. ie 112435 nn . The class can sit in

the 11th row

c) Find the sum of an arithmetic series with 25a 1d and 32n . Ie

1296)3150(16]1)132()25(2[S])1(2[S232

322n dnan

There are 1296 seats in the concert hall.

d) The expanded concert hall has 441232 n rows of seats. Because 25a and 1d ,

the total number of seats in the expanded hall is

2046)4350(22]1)144()25(2[S244

44

The number of additional seats is 7501296-2046SS 3244 .

Exercise

1) Decide whether the sequence is arithmetic. Explain why or why not.

a) ....2581114

b) ....8127931

c) ...15-13-11-7-5-

d) ....2.521.510.5

e) ....5

1658

54

52

51

f) ....1131

31

35

2) Which term of the sequence ....,,0,,3,-23

23

29 is 27?

3) Find the sum of the first 10 terms of the arithmetic series;

a) ....18141062

b) ....54.543.53

c) ....(-6)(-3)036

d) ....5.54.33.11.90.7

4) The nth term of a sequence is given by the formula below. Write down the first 4 terms of this

sequence hence find S12 and T15

a) nTn 27

b) 35 nTn

c) nTn 25

d) nTn 312

e) nTn 214

f) 25.045.0 nTn

5) Suppose a movie theater has 42 rows of seats and there are 29 seats in the first row. Each row

after the first has two more seats than the row before it. How many seats are in the theater?

6) Write a rule for the nth term of the arithmetic sequence. Then find the 25th term

a) ....97531

b) ....3022146

c) ....5137239

d) ....32101

e) ...83221

211

21

f) ....5214

g) ....23

617

625

211

h) ....21

67

611

25

i) ....8.84.646.1

7) Write a formula for the nth term of the arithmetic sequence

a) 46T and 4 14 d

b) 80 and 12 ad

c) 24T and 835 d

d) 77T and 17T 155

e) 4T and 6 12 d

f) 52T and 28T 202

g) 61

9T and 2 a

h) 122T and 34T 187

i) 2.48T and 1.4 16 d

8) For part (i), find the sum of the first n terms of the arithmetic series. For part (ii), find n for

the given sum nS

a) 366S ii) 20 i) ....23181383 nn

b) 182S ii) 40 i) .. ..1826344250 nn



c) 375S ii) 19 i) ....1050)5(10- nn

d) -12S ii) 23 i) ....2225283134 nn

e) 6611S ii) 68 i) ....30231692 nn

f) 2178S ii) 24 i) ....584430162 nn

9) Which term of the sequence ....75.542.5 is 31? Find also the 12th term and the

sum of the first 15 terms.

10) The sum of the first n terms of a certain progression is given by; a) nnSn 23 b) 28 nnSn c) nnSn 72 what kind of a progression is this?

11) Three numbers are in an AP. The difference between the 1st and 3rd number is 8. If the product of

these 2 numbers is 20, find the 3 numbers.

12) The distance covered by a rolling object at intervals of 1 second was recorded as 4cm, 16cm,

28cm, 40cm and so on. How long will the object be 22.6m from the starting point?

13) Evaluate; a)

20

1

5i)(3i

b)

15

1

3i)-(-10i

c)

22

1

43 i)-(6

i

d)

43

1

i)4(11i

e)

18

1

i)4.4(8.1i

14) Domestic bees make their honeycomb

by starting with a single hexagonal cell,

then forming ring after ring of hexagonal

cells around the initial cell, as shown.

The numbers of cells in successive rings

form an arithmetic sequence.

a) Write a rule for the number of cells in the nth ring.

b) What is the total number of cells in the honeycomb after the 9th ring is formed? (Hint:

Do not forget to count the initial cell.)

15) Logs are stacked in a pile, as shown at

the right. The bottom row has 21 logs

and the top row has 15 logs. Each row

has one less log than the row below it.

How many logs are in the pile? 2

16) Suppose each seat in rows 1 through 11 of the concert hall in Example 7 costs $24, each seat

in rows 12 through 22 costs $18, and each seat in rows 23 through 32 costs $12. How much

money does the concert hall take in for a sold-out event?

17) A quilt is made up of strips of cloth,

starting with an inner square surrounded

by rectangles to form successively larger

squares. The inner square and all

rectangles have a width of 1 foot. Write

an expression using summation notation

that gives the sum of the areas of all the

strips of cloth used to make the quilt

shown. Then evaluate the ex pression

18) A paper manufacturer sells paper rolled onto cardboard dowels. The thickness of the paper is

0.004 inch. The diameter of a dowel is 3 inches, and the total diameter of a roll is 7 inches as

shown.



a) Let n be the number of times the

paper is wrapped around the dowel,

let Tn be the diameter of the roll just

before the nth wrap, and let ln be the

length of paper added in the nth

wrap. Copy and complete the table

a) What can you say about the sequence ....,,,, 4321 llll ? Write a formula for the nth

term of the sequence.

b) Find the number of times the paper must be wrapped around the dowel to create a roll

with a 7 inch diameter. Use your answer and the formula from part (b) to find the length

of paper in a roll with a 7 inch diameter.

c) Suppose a roll with a 7 inch diameter costs $15. How much would you expect to pay for

a roll with an 11 inch diameter whose dowel also has a diameter of 3 inches? Explain

your reasoning and any assumptions you make.

19) One of the major sources of our knowledge of Egyptian mathematics is the Ahmes papyrus

(also known as the Rhind papyrus), which is a scroll copied in 1650 B.C. by an Egyptian

scribe. The following problem is from the Ahmes papyrus. Divide 10 hekats of barley among

10 men so that the common difference is81 of a hekat of barley. Use what you know about

arithmetic sequences and series to solve the problem.

Geometric Progression (GP)

It is a sequence in which the ratio between any two consecutive terms is a constant. This ratio is

called the common ratio and is denoted by r. Thus ...3

4

2

3

1

2

term

term

term

term

term

termr

rd

th

nd

rd

st

nd

Examples of G.P are a) ...,8,4,2,1 b) ...,,1,391

31 c) ...,27,9,3,1 d) ...,,,,, 432 xxxx

Generally a G.P will be of the form ...,,,, 32 ararara . The nth term and the sum of the first n

terms of a geometric sequence with first term a and common ratio r are respectively given by;

1 n

n arT and r

raS

n

n

1

)1(

However if 0a and 1|| r then r

aS

1 called the sum to infinity/limiting sum of the G.P.

Example 1 Decide whether each sequence is geometric.

a) 1 , 2 , 6 , 24 , 120, . . . . b) 81 , 27 , 9 , 3 , 1, . . .

Solution

To decide whether a sequence is geometric, find the ratios of consecutive terms.

a) 54

5...4

3

43

2

32

1

2

term

term

term

term

term

term

term

termth

th

rd

th

nd

rd

st

nd

The ratios are different, so the sequence is not geometric.

b) 3

1

4

5...

3

1

9

3

3

4

3

1

27

9

2

3

3

1

81

27

1

2

term

term

term

term

term

term

term

termth

th

rd

th

nd

rd

st

nd

The ratios are the same, so the sequence is geometric



Example 2 Find the 10th term and the sum of the first 10 terms of the G.P ...,,,181

41

21 Also

find the sum to infinity.

Solution

3

2

)(1

1 and

512

341

)(1

])(1[1,)(1 and 1

21

21

10

21

1051219

21

1021

SSTra

Example 3 The 4th and 9th terms of a G.P are 8 and 256 respectively. Determine the 6th term and

the sum of the first 7 terms.

Solution

1 so 2328

256T and 8T 5

3

88

9

3

4 arrar

ararar

12721

)21(1S and 32)2(1T

7

7

5

6

Example 4 For the geometric sequence . . . 40- 20 10- 5 find the terms whose value is 320.

Solution

761)2(64)2(320)2(5T2 and 5 611

n nnra nn

Example 5 The ratio between the 7th and 5th terms of a geometric sequence is 4. If the 3rd term

is 8, find the 15th term and the sum of the first 8 terms.

Solution

14

15

2

3

2

4

6

5

7 )2(2T284Tbut 24T

Taaarrr

ar

ar

17021

])2(1[2S ,2 when and 510

21

])2(1[2S ,2 when Now

8

8

8

8

rr

Example 6 In 1990 the average monthly bill for cellular telephone service in the United States

was $80.90. From 1990 to 1997, the average monthly bill decreased by about 8.6% per year.

a) Write a rule for the average monthly cellular telephone bill an (in dollars) in terms of the

year. Let n = 1 represent 1990.

b) What was the average monthly cellular telephone bill in 1993?

c) When did the average monthly cellular telephone bill fall to $50?

d) On average, what did a person pay for cellular telephone service during 1990–1997?

Solution

a) Because the average monthly bill decreased by the same percent each year, the average

monthly bills from year to year form a geometric sequence. Use a = 80.9 and

914.0086.01 r , a rule for the average monthly bill is 1)914.0(9.80T n

n

b) In 1993, n = 4. So, the average monthly bill was 77.61$)914.0(9.80T 3 n .

c) We want to find n such that 50T n . Therefore

65914.0ln

618.0ln10.618

9.80

50)914.0()914.0(9.8050 11 nnnn

The average monthly cellular telephone bill reached $50 in 1995 (when n = 6).



d) Because the model 1)914.0(9.80T n

n gives the average monthly bill, the model

11 )914.0(8.970)914.0(9.8012Y nn

n gives the average annual bill. Using a = 970.8 and

914.0r , we can estimate a person’s total cost for cellular telephone service during the 8

year period 1990–1997 to be

5790914.01

)914.0(18.970S

8

n

A person paid about $5790 for cellular telephone service during 1990–1997.

Exercise

1) Decide whether the sequence is arithmetic, geometric, or neither. Explain your answer

a) .. . . 384 96 24 6 d) . . . . 9- 5- 1- 3 g) ....134

32

31

b) . . . . 13 7 3 1 e) . . . 1 3- 7- 11- h) ...323

161

81

43

c) . . . . 31 22 13 4 f) ....227

29

23

21 i) ...

6256

1255

254

53

2) te a rule for the nth term of the geometric sequence below then find 6T .

a) . . . . 64- 16 4- 1 c) . . . . 686 98 14 2 e) . . . . - - 5275

95

35

b) . . . . 40 20 10 5 d) . . . . 750- 150 50- 6 f) . . . . 22716

98

34

3) Write a rule for the nth term of the geometric sequence.

a) 4 and 3 ar

b) 45 and 31 ar

c) 72T and 6 3 r

d) 4 and 81 ar

e) 2 and 8 ar

f) 16T and 421 a

g) 300T & 10T 63

h) 5T & 20T 42

i) 3750T& 30T 52

4) For part (i), find the sum of the first n terms of the geometric series. For part (ii), find n for

the given sum Sn

a) 341S ii) 14 i) . . . . 641641 nn

b) 208S ii) 10 i) . . . . 7291891 nn

c) 3829S ii) 18 i) . . . . (-189)63)21(7 nn

d) 67.66S ii) 16 i) . . . . 10)(3090-3

10 nn

5) Find the sum of the series; a)

10

1

16(2)i

i b)

8

1

15(4)i

i c)

9

121 )12(-

i

i d)

10

1

18(0.75)i

i

e)

6

123 )4(

i

i f)

12

1

1(-2)i

i

6) The ratio between the 5th and 2nd terms of a GP is 827 if the 3rd term is 4.5 find the 10th term

and the sum of the first 10 terms.

7) The men’s U.S. Open tennis tournament is held annually in Flushing Meadow in New York

City. In the first round of the tournament, 64 matches are played. In each successive round,

the number of matches played decreases by one half.

a) Find a rule for the number of matches played in the nth round.

b) For what values of n does your rule make sense?

c) Find the total number of matches played in the men’s U.S. Open tennis tournament.

8) Which term of the GP . . . . 313

131 is 81? Also find the sum upto this term and the

limiting sum of the GP.



9) The sum to infinity of a certain GP is 8 and the sum of the 1st 2 terms is38 , find the 1st term and

the sum of the fist 4 terms.

10) When a computer must find an item in an ordered list of data (such as an alphabetical list of

names), it may be programmed to perform a binary search. This search technique involves

jumping to the middle of the list and deciding whether the item is there. If not, the computer

decides whether the item comes before or after the middle. Half of the list is then ignored on

the next pass through the list, and the computer jumps to the middle of the remaining list.

This is repeated until the item is found.

a) An ordered list contains 1024 items. Find a rule for the number of items remaining after

the nth pass through the list.

b) In the worst case, the item to be found is the only one left in the list after n passes through

the list. What is the worst-case value of n for a binary search of a list with 1024 items?

11) In 1990 factory sales of pagers in the United States totaled $118 million. From 1990 through

1996, the sales increased by about 20% per year.

a) Write a rule for pager sales Tn (in $”000,000) in terms of the year. Let n = 1 rep 1990.

b) What did factory sales of pagers total in 1992?

c) When did factory sales of pagers reach $300 million?

d) What was the total of factory sales of pagers for the period 1990–1996?

12) The Sierpinski triangle is a design using equilateral triangles. The process involves removing

smaller triangles from larger triangles by joining the midpoints of the sides of the larger

triangles as shown below. Assume that the initial triangle is equilateral with sides 1 unit long.

a) Let Tn be the number of triangles removed at the nth stage. Find a rule for Tn. Then find

the total number of triangles removed through the 10th stage.

b) Let An be the remaining area of the original triangle at the nth stage. Find a rule for An.

Then find the remaining area of the original triangle at the 15th stage.

13) Suppose two computer companies, Company A and Company B, opened in 1991. The

revenues of Company A increased arithmetically through 2000, while the revenues of

Company B increased geometrically through 2000. In 1996 the revenue of Company A was

$523.7 million. In 1996 the revenue of Company B was $65.6 million.

a) The revenues of Company A have a common difference of 55.5. The revenues of

Company B have a common ratio of 2. Find a rule for the revenues in the nth year of each

company. Let a represent 1991.

b) Find the sum of the revenues from 1991 through 2000 for each company.

c) Find the year when the revenue of Company B exceeds the revenue of Company A. Write

a brief paragraph explaining which company you would rather own.

14) Using the rule for the sum of the first n terms of a geometric series, write the polynomial as a

rational expression a) 4321 xxxx

15) b) 753 241263 xxxx



MATRICES A matrix (in plural matrices) is a rectangular arrangement of numbers in rows and columns. For

instance, matrix A below has two rows and three columns.

The dimension of this matrix are 32 (read

“2 by 3”). The numbers in a matrix are its

entries. In matrix A, the entry in the second

row and third column is 5.

rows 2502

126

column 3

A

Special Matrices Some matrices have special names because of their dimensions or entries

i) Row matrix: a matrix with only one row. Example 4023

ii) Column matrix: a matrix with only one column. Example

1

2

iii) Square matrix: a matrix whose number

of rows equals the number of columns.

Example

233

163

121

iv) Zero matrix: a matrix whose entries are

all zeros. Example

00

00

Equality of matrices

Two matrices are equal if their dimensions are the same and the entries in corresponding

positions are equal.

Example Given

42

33

42

83

x

x

y

x, find the values of x and y

Solution The two matrices are equal corresponding entries are equal. So

3622 and 48238 yxyxxxx

Addition and Subtraction of Matrices To add or subtract matrices, you simply add or subtract corresponding entries.

You can add or subtract matrices only if they have the same dimensions

Example Perform the indicated operation, if possible.

a)

3

0

1

7

4

3

b)

16

72

04

38 c)

5

1

43

02

S0lution

a) Since the matrices have the same dimensions, you can add them.

5

1

4

32

54

13

3

5

1

2

4

3

b) Since the matrices have the same dimensions, you can subtract them.



12

66

)1(064

)3(328

16

32

04

38

c) Since the order of

43

02is 22 and the order of

5

1is 12 ,we cannot add the matrices.

Remark Addition of matrices is both commutative and associative that is, if A, B, and C are

matrices with the same dimensions When adding matrices, you can regroup them and change

their order without affecting the result.

A + B = B + A commutative property

(A + B) + C = A + (B + C) Associative property

Scalar multiplication

In matrix algebra, a real number is often called a scalar. To multiply a matrix by

a scalar, you multiply each entry in the matrix by the scalar. This process is called scalar

multiplication

Note Multiplication of a sum or difference of matrices by a scalar obeys the distributive

property. That is B +A )B +A ( kkk and B -A )B -A ( kkk

Example 1 Perform the indicated operation(s) a)

24

02

2

3 b)

62

86

54

54

30

21

2

S0lution

a)

36

03

24

02

24

02

2

3

2

3

2

3

2

3

2

3

b)

46

146

96

62

86

54

108

60

42

62

86

54

54

30

21

2

We can use what we know about matrix operations and matrix equality to solve a matrix

equation.

Example 2 Solve the matrix equation for x and y:

812

026

2

14

58

132

y

x

S0lution

812

026

21012

086

56

0432

2

14

58

132

y

x

y

x

y

x

Equate corresponding entries and solve the two resulting equations.

1 and38210 and2686 yxyx

Example 3 Use matrices to organize the following information about health care plans.

This Year For individuals, Comprehensive, HMO Standard, and HMO Plus cost

$694.32, $451.80, and $489.48, respectively. For families, the Comprehensive, HMO

Standard, and HMO Plus plans cost $1725.36, $1187.76, and $1248.12.



Next Year For individuals, Comprehensive, HMO Standard, and HMO Plus will cost

$683.91, $463.10, and $499.27, respectively. For families, the Comprehensive, HMO

Standard, and HMO Plus plans will cost $1699.48, $1217.45, and $1273.08.

S0lution

One way to organize the data is to use 23 matrices, as shown.

This Year (A) Next Year (B)

Plus HNO

Standard HNO

iveComprehens

$1248.12 48.489$

$1187.76 80.451$

$1725.36 32.694$

Family Individual

$1273.08 27.499$

$1217.45 10.463$

$1699.48 91.683$

Family Individual

Note We can also organize the data using 32 matrices where the row labels are levels of

coverage (individual and family) and the column labels are the types of plans(Comprehensive,

HMO Standard, and HMO Plus).

Example 4 A company offers the health care plans in Example 3 above to its employees. The

employees receive monthly paychecks from which health care payments are deducted. Use the

matrices in Example 3 to write a matrix that shows the monthly changes in health care payments

from this year to next year.

S0lution

Begin by subtracting matrix A from matrix B to determine the yearly changes in health care

payments. Then multiply the result by 12

1 and round answers to the nearest cent to find the

monthly changes

$2.08 82.0$

$2.47 94.0$

$2.16- 87.0$

$24.96 79.9$

$29.69 30.11$

$25.88- 41.10$

12

1

$1248.12 48.489$

$1187.76 80.451$

$1725.36 32.694$

$1273.08 27.499$

$1217.45 10.463$

$1699.48 91.683$

12

1)A - B(

12

1

The monthly deductions for the Comprehensive plan will decrease, but the monthly deductions

for the other two plans will increase.

Exercise

1) Perform the indicated operation(s), if possible

6-

10-

11-

9

22-

20

a-) b)

9-129

85-1-

1-04-

47-6- c)

54-

024- d)

3-4

01-5

02

1-5-6

2) In Example 3, suppose the annual health care costs given in matrix B increase by 4% the

following year. Write a matrix that shows the new monthly payment.

3) Perform the indicated operation, if possible. If not possible, state the reason.

11

54

66

28 a) b)

94

72

10

53 c)

3.56.2

7.81.4

1.52.0

5,32.1



5110

363

2911

417 d) e)

5

2

832

1

4

3

4

1

2

1

f)

2111

972

937

82

61

61

4) Perform the indicated operation(s), if possible

604-

73-1- 4 -a) b)

1-3

6-1-5 c)

11-5-3-

1555-

931

4 d)

94

31

2-1-

00

9- e)

10-

42-2-

2

1

116

41

8-10

4.48.4-

1.5-2.1

4.36.8-

5.2 f) g)

54

23

01

4

30

50

812

h)

1963

1351

474

21062

65

424

31

772 i) j)

553

7142

268

0173

5) Solve the matrix equation for x and y.

a)

910

6

910

82 yx

b)

0

216

87

04

81

23

y

x

c)

20

1612

511

432

yx

d)

75

0

81

94

77

5

21

78

34

y

x

6) Use the following information about three Major League Baseball teams’ wins and losses in

1998 before and after the All-Star Game. Before The Atlanta Braves had 59 wins and 29

losses, the Seattle Mariners had 37 wins and 51 losses, and the Chicago Cubs had 48 wins

and 39 losses. After The Atlanta Braves had 47 wins and 27 losses, the Seattle Mariners had

39 wins and 34 losses, and the Chicago Cubs had 42 wins and 34 losses.

a) Use matrices to organize the information.

b) Using your matrices from Exercise 37, write a matrix that shows the total numbers of

wins and losses for the three teams in 1998.

7) The matrices below show the average yearly cost (in dollars) of tuition and room and board

at colleges in the United States from 1995 through 1997. Use matrix addition to write a

matrix showing the totals of these costs. _Source: U.S. Department of Education TUITION ROOM AND BOARD

Collegeyear - 4 Private

Collegeyear - 2 Private

Collegeyear - 4 Public

Collegeyear - 2 Public

920,12243,12481,11

190,7094,7914,6

986,2848,2681,2

283,1239,1192,1

199719961995

555,5368,5121,5

699,4469,4256,4

345,4166,4990,3

128,3978,2944,2

199719961995

8) The figures below give the number (in millions) of Hispanic CD, cassette, and music video

units shipped to all market channels and the dollar value (in millions) of those shipments (at

suggested list prices).



1996 Number of units—CDs:

20,779; cassettes: 15,299; and music

videos: 45.

Dollar value—CDs: 268,441;

cassettes: 122,329; and music

videos: 916.

1997 Number of units—CDs:

26,277; cassettes: 17,799; and music

videos: 70.

Dollar value—CDs: 344,697;

cassettes: 144,645; and music

videos: 1,260.

a) Use matrices to organize the

information.

b) Write a matrix that gives the total

numbers of units shipped and total

values for both years.

c) Write a matrix that gives the change

in units shipped and dollar value

from 1996 to 1997.

9) Eligibility for a National Merit Scholarship is based on a student’s PSAT score. Through

1996, this total score was found by doubling a student’s verbal score and adding this value to

the student’s mathematics score. Let V represent the average verbal scores and let M

represent the average mathematics scores earned by sophomores and juniors at Central High

for tests taken in 1993 through 1996.

VERBAL SCORES (V) MATHEMATIS SCORES (M)

1996

1995

1994

1993

6.482.48

8.487.48

9.489.48

0.499.48

Juniors Sophonore

9.508.49

8.504.49

0.503.48

4.500.49

Juniors Sophonore

a) Write an expression in terms of V and M that you could use to determine the average total

PSAT scores for sophomores and juniors at Central High from 1993 through 1996. Then

evaluate the expression.

b) Use the matrix from Exercise 43 to determine the average total PSAT score for juniors at

Central High in 1996.

10) A triangle has vertices (2, 2), (8, 2), and (5, 6). Assign a letter to each vertex and organize the

triangle’s vertices in a matrix. When you multiply the matrix by 4, what does the “new”

triangle look like? How are the two triangles related? Use a graph to help you.

11) The matrices show the number of people (in thousands) who lived in each region of the

United States in 1991 and the number of people (in thousands) projected to live in each

region in 2010. The regional populations are separated into three age categories



1991 2010

Pacific

Moutains

South

Midwest

Northwest

331.4001.25603.10

580.1461,8903.3

942.10474.53504.22

857.7554.36814.15

043.7391.31142.12

65Over 6518170

551.5125.31655.13

707.2420,1294.5

832.14557.67128.25

980.8095.41840.15

377.7822.38493.12

65Over 6518170

a) The total population in 1991 was 252,177,000 and the projected total population in 2010

is 297,716,000. Rewrite the matrices to give the information as percents of the total

population. (Hint: Multiply each matrix by the reciprocal of the total population (in

thousands), and then multiply by 100.)

b) Write a matrix that gives the projected change in the percent of the population in each

region and age group from 1991 to 2010.

c) Based on the result of Exercise 46, which region(s) and age group(s) are projected to

show relative growth from 1991 to 2010?

12) The matrices show the number of hardcover volumes sold and the average price per volume

(in dollars) for different subject areas.

_ 1995 (A) 1996 (B)

per volume sold

pric Average volume

per volume sold

pric Average volume

Travel

M usic

Law

Art

30.38000,199

27.43000,251

09.73000,716

23.41000,116,1

92.33000,179

21.39000,253

51.88000,827

40.53000,070,1

b) Calculate B o A. How many more (or fewer) law volumes were sold in 1996 than in

1995? How much more (or less) did the average music book cost in 1996 than in 1995?

c) Calculate B + A. Does the “volumes sold” column in B + A give you meaningful

information? Does the “average price per volume” column in B + A give you meaningful

information? Explain.

d) What conclusions can you make about the number of volumes sold and the average price

per volume of these books from 1995 to 1996?

Multiplying Matrices The product of two matrices A and B is

defined provided the number of columns in

A is equal to the number of rows in B. If A

is an nm matrix and B is an pn matrix,

then the product AB is an pm matrix.

Example 1 State whether the product AB is defined. If so, give the dimensions of AB.

a) A: 32 , B: 43 b) A: 23 , B: 43



Solution

a) Because A is a 32 matrix and B is a 43 matrix, the product AB is defined and is a 42

matrix.

b) Because the number of columns in A (two) is not equal to the number of rows in B (three),

the product AB is not defined.

Example 2 Given matrices

06

4-1

32-

A and

42-

31-B , find the product AB if possible

Solution

Because A is a 23 matrix and B is a 22 matrix, the product AB is defined and is a 23

matrix. To write the entry in the first row and first column of AB, multiply corresponding entries

in the first row of A and the first column of B. Then add. Use a similar procedure to write the

other entries of the product.

186

13-7

64-

(4)03))(6((-2)0(6)(-1)

(-4)(4)3))(1((-2))4(1(-1)

3(4)3))(2(3(-2)(-2)(-1)

42-

31-

06

4-1

32-

AB

Example 3 If

01

23A and

12

41B , find the product AB and BA if possible

Solution

41

107

12

41

01

23AB

45

27

01

23

12

41BA

Notice that AB ≠ BA. Matrix multiplication is not, in general, commutative.

Example 4 Given

31

12A ,

24

02-B and

23

11C , obtain A(B+C) and AB+AC

Solution

1122

65

47

11-

31

12

23

11

24

02-

31

12C)A(B

1122

65

58

45

614

20

23

11

31

12

24

02-

31

12CAB A

Notice that A(B + C) = AB + AC, which is true in general. This and other properties of matrix

multiplication are summarized below

Remark Matrix multiplication is both associative and distributive. That is if A, B and C are

matrices then;

A(BC)=(AB)C :- associative property of matrix multiplication

A(B+C)=AB+AC :- left distributive property

(A+B)C=AC+BC :- right distributive property



Matrix multiplication is useful in business applications because an inventory matrix, when

multiplied by a cost per item matrix, results in a total cost matrix.

pmpnnm

matrix

Cost Total

matrix

unitper Cost

matrix

Inventory

For the total cost matrix to be meaningful, the column labels for the inventory matrix must match

the row labels for the cost per item matrix.

Example 5 Two softball teams submit equipment lists for the season.

Women’s team 12 bates 45 balls 15 uniforms

Men’s team 15 bates 35 balls 17 uniforms

Each bat costs $21, each ball costs $4, and each uniform costs $30. Use matrix multiplication to

find the total cost of equipment for each team

Solution

To begin, write the equipment lists and the costs per item in matrix form. Because you want to

use matrix multiplication to find the total cost, set up the matrices so that the columns of the

equipment matrix match the rows of the cost matrix.

Equipment cost

teamsMen;

teamsWomen;

17 35 15

15 45 12

uniforms Balls Bates

Uniform

Ball

Bat

30$

4$

21$

The total cost of equipment for each team can now be obtained by multiplying the equipment

matrix by the cost per item matrix. The equipment matrix is 32 and the cost per item matrix is

13 , so their product is a 32 matrix.

977

882

)30(17 )4(35 )21(15

)30(15 )4(45 )21(12

30

4

21

17 35 15

15 45 12

The labels for the product matrix are as follows

teamsMen;

teamsWomen;

977$

882$

Cost Total

The total cost of equipment for the women’s

team is $882, and the total cost of equipment

for the men’s team is $977.

Exercise

1) State whether the product AB is defined. If so, give the dimensions of AB.

a) A: 23 , B: 32

b) A: 33 , B: 33

c) A: 23 , B: 23

d) A: 21 , B: 23

e) A: 42 , B: 34

f) A: 24 , B: 53

g) A: 55 , B: 45

h) A: 43 , B: 14

i) A: 33 , B: 42

2) Find the product.; a)

3

2

5- 3

4- 4 b)

3 1

0 2

1- 2-

0 1 c)

1- 2-

0 1

1- 0

2- 3

3 3-



3) Use matrix multiplication to find the total cost of equipment in Example 5 if the women’s

team needs 16 bats, 42 balls, and 16 uniforms and the men’s team needs 14 bats, 43 balls,

and 15 uniforms.

4) Find the product. If it is not defined, state the reason.

a)

21-

0

12

- -3

1

2

1

6

1

b)

3.2 2.0

0 1.8

1.5 37.

-8.7- 2.6 4.2-

c)

3- 0

1- 4

2- 3

4- 1

d)

3 5-

4 1-

3 0

2- 6-

e)

5 2- 8

2- 1 0

2 5- 0

1 8- 2

f)

0.5- 1.5

0 1

0.3 2.9

0.2 0.2-

0 6.0

g)

0

0.2

1.2

0.25- 1.5- 1

1.25 0.5- 1-

h)

4 3 1-

4 2- 7-

3 1- 0

1 7 0.1

8 3 2-

1 1 6

i)

4 3 1-

4 2- 7-

3 1- 0

1 7 0.1

8 3 2-

1 1 6-

j)

1- 6- 4

5 2- 4-

5 0

4 1

2- 6

k)

12- 5- 4-

6 12 3

3 7- 5

3 5 2-

18- 3- 6

0 1 0

5) Using the given matrices, simplify the expression.

4 1 3

2 4 1-

6 5 2-

E

3 3- 2-

4 2 1-

1 2- 3

D1 2-

3 1-C

4 2-

0 1 B

1- 6

2- 4A a) AB

b) AB + AC c) D(D + E) d) (E + D)E e) -3AC f) 0.5AB+2AC

6) Solve for x and y; a)

y

x 19

6

3

1

4 2- 0

4 2 3

2 1 2-

b)

11 13-

5y

4 1-

1 2

2- 9

1 x 2-

3 1 4

7) The percents of the total 1997 world production of wheat, rice, and maize are shown in the

matrix for the four countries that grow the most grain: China, India, the Commonwealth of

Independent States (formerly the Soviet Union), and the United States. The total 1997 world

production (in thousands of metric tons) of wheat, rice, and maize is 608,846, 570,906, and

586,923 respectively.

a) Rewrite the matrix to give the

percents as decimals.

b) Show how matrix multiplication can

be used to determine how many

metric tons of all three grains were

produced in each of the four

countries.

GRAIN PRODUCTION

U.S

C.I.S

India

China

40.5 4.1 3.11

0.5 1.0 5.7

7.1 5.21 22

18 8.34 1.20

Maize Ricewheat



8) In Exercises 37º39, use the following information. Three teams participated in a debating

competition. The final score for each team is based on how many students ranked first,

second, and third in a debate. The results of 12 debates are shown in matrix A.

MATRIX A

3 Team

12 Team

1 Team

2 6 4

5 2 5

4 5 3

3rd 2nd1st

a) Teams earn 6 points for each first place,

5 points for each second place, and 4

points for each third place. Organize this

information into a matrix B.

b) Find the product AB.

9) Which team won the competition? How

many points did the winning team score?

10) The numbers of calories burned by people of different weights doing different activities for

20 minutes are shown in the matrix.

Show how matrix multiplication can be

used to write the total number of calories

burned by a 120 pound person and a 150

pound person who each bicycled for 40

minutes, jogged for 10 minutes, and then

walked for 60 minutes.

CALORIES BURNED

person person

150lb 120lb

Walking

Jogging

Cycling

79 64

159 127

136 109

11) Matrix A is a 90° rotational matrix. Matrix B contains the coordinates of the triangle’s

vertices shown in the graph.

0 1

1- 0A

2 8 4

4- 4- 7B

a) Calculate AB. Graph the

coordinates of the vertices given

by AB. What rotation does AB

represent in the graph?

b) Find the 180° and 270° rotations

of the original triangle by using

repeated multiplication of the 90°

rotational matrix. What are the

coordinates of the vertices of the

rotated triangles?

Determinants and Cramer’s Rule Associated with each square matrix is a real number called its determinant The determinant of a

matrix A is denoted by det A or by |A|.

Determinant of a 22 Matrix

The determinant of a 22 matrix is the difference of the products of the entries on the diagonals



Determinant of a 33 Matrix

Repeat the first two columns to the right of the determinant.

Subtract the sum of the products in red from the sum of the products in blue.

2

1 Example 1 Evaluate the determinant of the matrix. a)

52

31 b)

421

102

312

Solution

You can use a determinant to find the area of a triangle whose vertices are points in a coordinate

plane. The area of a triangle with vertices (x1, y1), (x2, y2), and (x3, y3) is given by

1

1

1

2

1 Area

33

22

11

yx

yx

yx

where indicates

the absolute value for the determinant.

Example 2 The area of the triangle shown

is:

126

104

121

2

1 Area

Example 3 The Bermuda Triangle is a large triangular region in the Atlantic Ocean. Many

ships and airplanes have been lost in this region. The triangle is formed by imaginary lines

connecting Bermuda, Puerto Rico, and Miami, Florida. Use a determinant to estimate the area of

the Bermuda Triangle.



Solution

The approximate coordinates of the Bermuda Triangle’s three vertices are (938, 454), (900,

o518), and (0, 0). So, the area of the region is as follows:

242,447)600,40800()00884,485(2

1

100

1518900

1454938

2

1 Area

The area of the Bermuda Triangle is about 447,000 square miles.

Cramer’s Rule

We can use determinants to solve a system of linear equations. The method, called Cramer’s

rule and named after the Swiss mathematician Gabriel Cramer (1704o1752), uses the coefficient

matrix of the linear system.

Linear system fdycx

ebyax

coefficient matrix

dc

ba

Cramer’s Rule for a 22 System

Let A be the coefficient matrix of this linear system: fdycx

ebyax

. If 0A , then the system has

exactly one solution. The solution is: df

bex

A

1 and

fc

eay

A

1

In Cramer’s rule, notice that the denominator for x and y is the determinant of the coefficient

matrix of the system. The numerators for x and y are the determinants of the matrices formed by

using the column of constants as replacements for the coefficients of x and y, respectively.

Example 4 Use Cramer’s rule to solve this system: 1042

258

yx

yx

Solution

Evaluate the determinant of the coefficient matrix

Coefficient matrix A is

42

58A and it’s determinant is 42

42

58A

Apply Cramer’s rule since the determinant is not 0.

1410

52

42

1

x and 2

102

28

42

1

y The solution is (o1, 2).

Check this solution in the original equations.



Cramer’s Rule for a 33 System

Let A be the coefficient matrix of this linear system:

lizhygx

kfzeydx

jczbyax

If 0A , then the system has exactly one solution. The solution is:

ihl

fek

cbj

xA

1 ,

ilg

fkd

cja

yA

1 and

lhg

ked

jba

zA

1

Example 5 The atomic weights of three compounds are shown. Use a linear system and

Cramer’s rule to find the atomic weights of carbon (C), hydrogen (H), and oxygen (O)

Compound Formular Atomic weight

Methane 4CH 16

Glycerol 383 OHC 92

Water OH2 18

Solution

Write a linear system using the formula for each compound. Let C, H, and O represent the

atomic weights of carbon, hydrogen, and oxygen.

18O2H

92O3H83C

164HC

Evaluate the determinant of the coefficient matrix.

10

120

383

041

Apply Cramer’s rule since the determinant is not 0.

12

1218

3892

0416

10

1C

, 1

1180

3923

0161

10

1H

and 16

1820

9283

1641

10

1O

The weights of carbon, hydrogen, and oxygen are 12, 1, and 16, respectively.

Exercise

1) Evaluate the determinant of the matrix.

a) i)

26

10 ii)

15

41 iii)

42

28 iv)

25

24 v)

31

08 vi)

12

39

vii)

27

117 viii)

43

04 ix)

93

56 x)

108

30 xi)

85

212



b) i)

185

232-

1-412

ii)

110

124

495

iii)

145

41310

250

iv)

417

2420

2161

v)

250

980

104

vi)

4133

9312

928

Vii)

265

0910

1123

viii)

121511

18910

2023

ix)

1428

6010

10415

2) Find the area of the triangle with the given vertices

a) A(0, 1), B(2, 7), C(5, 5)

b) A(3, 6), B(3, 0), C(1, 3)

c) A(6, -1), B(2, 2), C(4, 8)

d) A(-4, 2), B(3, -1), C(-2, -2)

e) A(2, -6), B(-1, -4), C(0, 2)

f) A(1, 3), B(-2, 6), C(-1, 1)

3) Use Cramer’s rule to solve the linear system.

a) 454

486

yx

yx

b) 2383

372

yx

yx

c) 511114

2212

yx

yx

d) 465

32

yx

yx

e) 56103

1157

yx

yx

f) 4234

729

yx

yx

g) 1753

37

yx

yx

h) 511512

4412

yx

yx

i) 3478

1834

yx

yx

j) 2472

1354

yx

yx

k) 4075

3298

yx

yx

l) 641512

50103

yx

yx

4) Use Cramer’s rule to solve the linear system.

a)

4443

1

232

zyx

zyx

zyx

b)

4253

362

13

zyx

zyx

zyx

c)

23

86

10523

zy

zx

zyx

d)

125

3

92

zx

zyx

zyx

e)

9

17233

764

zyx

zyx

zyx

f)

2233

1522

74

zyx

zyx

zyx

g)

1656

924

52

yx

zyx

zyx

h)

14253

222

1372

zyx

zyx

zyx

i)

6458

829

1423

zyx

zyx

zyx

5) You are making a large pennant for your

school football team. A diagram of the

pennant is shown at the right. The

coordinates given are measured in

inches. How many square inches of

material will you need to make the

pennant?



6) Black-necked stilts are birds that live

throughout Florida and surrounding

areas but breed mostly in the triangular

region shown on the map. Estimate the

area of this region. The coordinates

given are measured in miles.

7) On a Marconi-rigged sloop, there are two triangular sails, a mainsail and a jib. These sails are

shown in a coordinate plane at the right. The coordinates in the plane are measured in feet

a) Find the area of the mainsail shown.

b) Find the area of the jib shown.

c) Suppose you are making a scale model

of the sailboat with the sails shown using

a scale of l in. = 6 ft. What is the area of

the model’s mainsail?

8) You fill up your car with 10 gallons of premium gasoline and fill a small gas can with 2

gallons of regular gasoline for your lawn mower. You pay the cashier $13.56. The price of

premium gasoline is 12 cents more per gallon than the price of regular gasoline. Use a linear

system and Cramer’s rule to find the price per gallon for regular and premium gasoline

9) The Golden Triangle refers to a large

triangular region in India. The Taj Mahal

is one of the many wonders that lie

within the boundaries of this triangle.

The triangle is formed by imaginary

lines that connect the cities of New

Delhi, Jaipur, and Agra. Use the

coordinates on the map and a

determinant to estimate the area of the

Golden Triangle. The coordinates given

are measured in miles

10) The atomic weights of three compounds are shown



Compound Formular Atomic weight

Tetrasulphur tetranitrate 44NS 184

Sulphur hexafluoride 6SF 146

Dinitrogen tetrafloride 42FN 104

Use a linear system and Cramer’s rule to find the atomic weights of sulphur (S), nitrogen

(N), and flourine (F).

11) Explain what happens to the determinant of a matrix when you switch two rows or two

columns.

Identity and Inverse Matrices The number 1 is the multiplicative identity for real numbers because aaa 11 . For

matrices, the nn identity matrix is the matrix that has 1’s on the main diagonal and 0’s

elsewhere.

Eg

10

01 and

100

010

001

are the 22 and 33 identity matrix respectively.

If A is any nn matrix and I is the nn identity matrix, then IA=AI=A.

Two nn matrices are inverses of each other if their product (in both orders) is the nn

identity matrix. For example, matrices A and B below are inverses of each other.

I10

01

33

12

23

13AB

and I

10

01

23

13

33

12BA

The symbol used for the inverse of A is -1A .

The Inverse of a 22 Matrix

The inverse of the matrix

dc

baA is

ac-

bd1

ac-

bd

A

1A 1-

bcadprovided

0bcad

Example 1 Find the inverse of

24

13A

Solution

5.12-

5.01

34-

12

46

1A 1-

check the inverse by showing that AAIAA -1-1

10

01

24

13

5.12-

5.01AA and

10

01

5.12-

5.01

24

13AA 1-1-

Example 2 Given the matrices

13-

14A and

36-

58B Solve the matrix equation AX

= B for the 22 matrix X where

Solution

Begin by finding the inverse of A.

43

11

43

11

34

1A 1-



To solve the equation for X, multiply both sides of the equation by -1A on the left.

30

22X

30

22X

10

011

36-

58

43

11X

13-

14

43

11BAAXA 1-1-

You can check the solution by multiplying A and X to see if you get B.

Some matrices do not have an inverse. You can tell whether a matrix has an inverse by

evaluating its determinant. If det A = 0, then A does not have an inverse. If det A ≠ 0, then A has

an inverse.

Exercise

1) Find the inverse of the matrix.

a)

23

34

b)

12

33

c)

46

01

d)

4

1

21

2

4

e)

45.2

35.0

f)

42.3

26.1

g)

43

54

h)

38

26

i)

171

181

j)

31

176

k)

13

27

l)

14_

27

m)

22

76

n)

44

45

o)

39

311

p)

12

2

1

2

3

q)

108

5.22.2

r)

2

5

4

3

5

4

1

2) Solve the matrix equation.

a)

04

13

50

135X

b)

2026

2017

28

15X

c)

513

604

10

42X

d)

1334

18512

14

35X

e)

92

37

151

58

41

73X

f)

66

91

34

43

54

97X

g)

11

23

03

12

64

21X

h)

28

64

75

11

26

34X

3) Tell whether the matrices are inverses of each other.

a)

103

31 and

13

310

b)

10121

5711

8102

and

437

325

120



c)

354

120

243

and

618

314

8211

d)

221

052

543

and

2339

1014

25210

4) Use the matrices shown. The columns of matrix T give the coordinates of the vertices of a

triangle. Matrix A is a transformation matrix.

241

321T and

01

10A

a) Find AT and AAT. Then draw the original triangle and the two transformed triangles.

What transformation does A represent?

b) Suppose you start with the triangle determined by AAT and want to reverse the

transformation process to produce the triangle determined by AT and then the triangle

determined by T. Describe how you can do this.

Solving Systems Using Inverse Matrices

In session.3 you learned how to solve a system of linear equations using Cramer’s rule. Here you

will learn to solve a system using inverse matrices

To solve a system of linear equations using inverse we must first write the system as a matrix

equation AX = B, where the matrix A is the coefficient matrix of the system, X is the matrix of

variables, and B is the matrix of constants. For instance the system f

dycx

ebyax can be written

in matrix form as

f

e

y

x

dc

ba.

Once we have written a linear system as AX = B, the next step is to pre-multiply both sides of

the matrix equation by the inverse of the coefficient matrix and then simplify both sides. Ie

B A=X AXAB = AX 11 For instance

ce-af

f1

f

11

f

bde

bcady

xe

ac

bd

bcady

x

dc

ba

ac

bd

bcad

e

y

x

dc

ba

Finally obtain the solution by equating the corresponding entries of the resulting matrices in step

2 above. Ie bcad

ceay

bcad

bdex

-f and

f

Example 1 Use the inverse matrices to solve the linear system 102

543

yx

yx

Solution

Begin by writing the linear system in matrix form, ie

10

5

12

43

y

x

Then pre-multiply both sides by the inverse of the coefficient matrix to get

4

7

20

35

5

1

10

01

10

5

32

41

83

1

12

43

32

41

83

1

y

x

y

x

y

x



The solution of the system is (-7, -4). Check this solution in the original equations

Example 2 Show that matrices A and B are inverses of each other. Hence solve the system of

linear equations

242

133

132

zyx

zyx

zyx

where

142

133

132

A and

326

101

011

B

Solution

Start by showing that AB = BA = I

142

133

132

326

101

011

100

010

001

326

101

011

142

133

132

write the linear system in matrix form then pre-multiply both sides by the inverse to get

2

1

2

2

1

1

326

101

011

142

133

132

326

101

011

2

1

1

142

133

132

z

y

x

z

y

x

z

y

x

Thus the solution of the system is (2, -1, -2).

Exercise

1) Write the linear system as a matrix equation.

a) 62

8

yx

yx

b) 724

93

yx

yx

c) 843

5

yx

yx

d) 54

62

yx

yx

e) 1024

935

yx

yx

f) 1573

1152

yx

yx

g) 1154

48

yx

yx

h) 13

452

yx

yx

i)

843

15

510

zyx

yx

zyx

j)

38254

2372

454

zyx

zyx

zyx

k)

313

1042

1643

zyx

zyx

zyx

l)

8.43.48.43.0

2.27.05.22.1

9.52.01.35.0

zyx

zyx

zyx

m)

472

62

9

zyx

zyx

zx

n)

059

14126

23108

zx

zy

zy

o)

2

12

0

zy

zx

zyx

2) Use an inverse matrix to solve the linear system.

a) 2187

2

yx

yx

b) 182

3

yx

yx

c) 1026

634

yx

yx

d) 1125

823

yx

yx

e) 81211

1

yx

yx

f) 223

5372

yx

yx

g) 434

857

yx

yx

h) 3042

5475

yx

yx



i) 332

975

yx

yx

j) 1432

92

yx

yx

k) 3152

2642

yx

yx

l) 2222

4359

yx

yx

3) Use the given inverse of the coefficient matrix to solve the linear system.

a)

5437

4325

22

zyx

zyx

zy

10141

571

8111

A 1-

b)

43

3025

93

yx

zyx

zyx

741

1693

531

A 1-

4) You are planning a birthday party for your younger brother at a skating rink. The cost of

admission is $3.50 per adult and $2.25 per child, and there is a limit of 20 people. You have

$50 to spend. Use an inverse matrix to determine how many adults and how many children

you can invite.

5) The price of flatware varies depending on the number of place settings you buy as well as

other items included in the set. Suppose a set with 4 place settings costs $142 and a set with 8

place settings and a serving set costs $351. Find the cost of a place setting and a serving set.

Assume that the cost of each item is the same for each flatware set.

6) Solve the linear system using the given inverse of the coefficient matrix.

6326

3335

5272

2336

zyx

zyxw

zyxw

zyxw

520224

833339

0101

933340

A 1-

7) Solve the system of linear equations using any algebraic method.

a) 823

64

yx

yx

b) 453

92

yx

yx

c) 223

159

yx

yx

d) 1

822

yx

yx

e) 726

143

yx

yx

f) 4083

1025

yx

yx

g) 3252

1137

yx

yx

h) 1092

145

yx

yx

i) 2412

497

yx

yx

932

1323

1323

zyx

zyx

yx

j)

7522

14823

43

zyx

zyx

zyx

k)

16652

1584

21553

zyx

zyx

zyx

l)

022

55

22

zyx

zyx

zx

m)

21353

96

1434

zyx

yx

zyx

n)

1127

122

173

zyx

zx

zyx

8) Dentists use various amalgams for silver fillings. The matrix shows the percents (expressed

as decimals) of powdered alloys used in preparing three different amalgams. Suppose a



dentist has 5483 grams of silver, 2009 grams of tin, and 129 grams of copper. How much of

each amalgam can be made?

PERCENT ALLOY BY WEIGHT

Amalgam

copper

Tin

Silver

00.003.004.0

27.025.026.0

73.072.070.0

C BA

9) You are making mosaic tiles from three types of stained glass. You need 6 square feet of

glass for the project and you want there to be as much iridescent glass as red and blue glass

combined. The cost of a sheet of glass having an area of 0.75 square foot is $6.50 for

iridescent, $4.50 for red, and $5.50 for blue. How many sheets of each type should you

purchase if you plan to spend $45 on the project?

10) You are an accountant for a construction business and are planning next year’s budget. You

have $200,000 to spend on salaries, equipment maintenance, and other general expenses.

Based on previous financial records of the business, you expect to spend five times as much

on salaries as on equipment maintenance, and you expect general expenses to be 10% of the

amount spent on the other two categories combined. Write and solve a system of equations to

find the amount you should budget for each category.

11) A company sells different sizes of gift baskets with a varying assortment of meat and cheese.

A basic basket with 2 cheeses and 3 meats costs $15, a big basket with 3 cheeses and 5 meats

costs $24, and a super basket with 7 cheeses and 10 meats costs $50.

a) Write and solve a system of equations using the information about the basic and big

baskets.

b) Write and solve a system of equations using the information about the big and super

baskets.

c) Compare the results from parts (a) and (b) and make a conjecture about why there is a

discrepancy.

12) A walkway lighting package includes a transformer, a certain length of wire, and a certain

number of lights on the wire. The price of each lighting package depends on the length of

wire and the number of lights on the wire.

• A package that contains a transformer, 25 feet of wire, and 5 lights costs $20.



Write and solve a system of equations to find the cost of a transformer, the cost per foot of

wire, and the cost of a light. Assume the cost of each item is the same in each lighting

package.



Permutations, Combinations and Binomial Expansion

The Fundamental Counting Principle and Permutations

In many real-life problems you want to count the number of possibilities. For instance, suppose

you own a small deli. You offer 4 types of meat (ham, turkey, roast beef, and pastrami) and 3

types of bread (white, wheat, and rye). How many choices do your customers have for a meat

sandwich?

To count the number of possible sandwiches we can use the fundamental counting principle.

Because you have 4 choices for meat and 3 choices for bread, the total number of choices is

1234 .

The Fundamental Counting Principle

Two Events If one event can occur in m ways and another event can occur in n ways, then the

number of ways that both events can occur is nm . For instance, if one event can occur in 12

ways and another event can occur in 5 ways, then both events can occur in 60512 ways.

Three or More Events. The fundamental counting principle can be extended to three or more

events. For example, if three events can occur in m, n, and p ways, then the number of ways that

all three events can occur is pnm . For instance, if three events can occur in 3, 5, and 4 ways,

then all three events can occur in 60453 ways.

Example 1 Police use photographs of various facial features to help witnesses identify suspects.

One basic identification kit contains 195 hairlines, 99 eyes and eyebrows, 89 noses, 105 mouths,

and 74 chins and cheeks.

a) The developer of the identification kit claims that it can produce billions of different faces. Is

this claim correct?

b) A witness can clearly remember the hairline and the eyes and eyebrows of a suspect. How

many different faces can be produced with this information?

Solution

a) You can use the fundamental counting principle to find the total number of different faces.

Number of faces = 195 × 99 × 89 × 105 × 74 = 13,349,986,650 The developer’s claim

is correct since the kit can produce over 13 billion faces

b) Because the witness clearly remembers the hairline and the eyes and eyebrows, there is only

1 choice for each of these features. You can use the fundamental counting principle to find

the number of different faces. Number of faces = 1 × 1 × 89 × 105 × 74 = 691,530 The

number of faces that can be produced has been reduced to 691,530.

Example 2 The standard configuration for a New York license plate is 3 digits followed by 3

letters.

a) How many different license plates are possible if digits and letters can be repeated?

b) How many different license plates are possible if digits and letters cannot be repeated?

Solution

a) There are 10 choices for each digit and 26 choices for each letter. You can use the

fundamental counting principle to find the number of different plates.



Number of plates = 10 × 10 × 10 × 26 × 26 × 26 = 17,576,000

The number of different license plates is 17,576,000.

b) If you cannot repeat digits there are still 10 choices for the first digit, but then only 9

remaining choices for the second digit and only 8 remaining choices for the third digit.

Similarly, there are 26 choices for the first letter, 25 choices for the second letter, and 24

choices for the third letter. You can use the fundamental counting principle to find the

number of different plates. Number of plates = 10 • 9 • 8 • 26 • 25 • 24 = 11,232,000 The

number of different license plates is 11,232,000.

Permutations

Definition (n factorial) We define n!, pronounced n factorial as 123...)2)(1(! nnnn

Eg 6123!3 241234!4 12012345!5 720123456!6

Permutation; number of different arrangements of a group of items where order matters

When order matters BAAB .

An ordering of n objects is a Permutations of the objects. For instance, there are six

permutations of the letters A, B, and C that is ABC, ACB, BAC, BCA, CAB, CBA. The

fundamental counting principle can be used to determine the number of permutations of n

objects. For instance, you can find the number of ways you can arrange the letters A, B, and C by

multiplying. There are 3 choices for the first letter, 2 choices for the second letter, and 1 choice

for the third letter, so there are 6123 ways to arrange the letters.

In general, the number of permutations of n distinct objects is: 123...)2)(1(! nnnn

and the number of permutations of n objects taken r at a time is: )!(

!

rn

nrn p

Example 3 Twelve skiers are competing in the final round of the Olympic freestyle skiing

aerial competition.

a) In how many different ways can the skiers finish the competition? (Assume there are no ties.)

b) In how many different ways can 3 of the skiers finish first, second, and third to win the gold,

silver, and bronze medals?

Solution

a) There are 12! different ways that the skiers can finish the competition. 12! = 479,001,600

b) Any of the 12 skiers can finish first, then any of the remaining 11 skiers can finish second,

and finally any of the remaining 10 skiers can finish third. So, the number of ways that the

skiers can win the medals is: 12 × 11 × 10 = 1320 This is the number of permutations of 12

objects taken 3 at a time, is denoted by 312 p , and is given by )!512(

!12

Example 4 You are considering 10 different colleges. Before you decide to apply to the

colleges, you want to visit some or all of them. In how many orders can you visit (a) 6 of the

colleges and (b) all 10 colleges?

Solution

a) The number of permutations of 10 objects taken 6 at a time is 200,151610 p



b) The number of permutations of 10 objects taken 10 at a time is: 800,628,31010 p

So far you have been finding permutations of distinct objects (ie we dealt with selections without

repetitions). If some of the objects are repeated, then some of the permutations are not

distinguishable. For instance, of the six ways to order the letters M, O, and M

M O M M O M O M M O M M M M O M M O

Only three are distinguishable without color: MOM, OMM, and MMO. In this case, the number

of permutations is 3!2

!3 , not 3!=6

The number of distinguishable permutations of n objects where 1r are identical objects of type

1, 2r are identical objects of type 2,. . . . kr are identical objects of type k. is given by

!.....!!

!

21 krrr

n

Example 4 Find the number of distinguishable permutations of the letters in (a) OHIO and (b)

MISSISSIPPI.

Solution

a) OHIO has 4 letters of which O is repeated 2 times. So, the number of distinguishable

permutations is 12!2

!4 .

b) MISSISSIPPI has 11 letters of which I is repeated 4 times, S is repeated 4 times, and P is

repeated 2 times. So, the number of distinguishable permutations is 650,34!2!4!4

!11

Exercise

1. Find the number of possible outcomes if you toss a coin and roll a dice.

2. Find the number of distinguishable permutations of the letters in the word a) FORMAT b)

WYOMING c) THURSDAY d) SEPTEMBER e) MATHEMATICS

3. How many 3-digit numbers are possible using the digits 0, 1, 4, 5, 7 and 9?

4. How many arrangements of three types of flowers are there if there are 6 types to choose

from?

5. How many different ways can you have 1st 2nd 3rd and 4th in a race with 10 runners?

6. How many ways can 7 people be arranged around a roundtable?

7. A license plate has 3 letters and 3 digits in that order. A witness to a hit and run accident saw

the first 2 letters and the last digit. If the letters and digits can be repeated, how many license

plates must be checked by the police to find the culprit?

8. You are going to set up a stereo system by purchasing separate components. In your price

range you find 5 different receivers, 8 different compact disc players, and 12 different

speaker systems. If you want one of each of these components, how many different stereo

systems are possible?

9. A pizza shop runs a special where you can buy a large pizza with one cheese, one vegetable,

and one meat for $9.00. You have a choice of 7 cheeses, 11 vegetables, and 6 meats.

Additionally, you have a choice of 3 crusts and 2 sauces. How many different variations of

the pizza special are possible?



10. To keep computer files secure, many programs require the user to enter a password. The

shortest allowable passwords are typically six characters long and can contain both numbers

and letters. How many six-character passwords are possible if (a) characters can be repeated

and (b) characters cannot be repeated?

11. Simplify the formula for nPr when r = 0. Explain why this result makes sense.

12. A particular classroom has 24 seats and 24 students. Assuming the seats are not moved, how many

different seating arrangements are possible? Write your answer in scientific notation.

13. “Ringing the changes” is a process where the bells in a tower are rung in all possible permutations.

Westminster Abbey has 10 bells in its tower. In how many ways can its bells be rung?

14. A music store wants to display 3 identical keyboards, 2 identical trumpets, and 2 identical guitars in

its store window. How many distinguishable displays are possible?

15. In a dog show how many ways can 3 Chihuahuas, 5 Labradors, 4 poodles, and 3 beagles line up in

front of the judges if the dogs of the same breed are considered identical?

16. Find the number of permutations of n objects taken n º 1 at a time for any positive integer n. Compare

this answer with the number of permutations of all n objects. Does this make sense? Explain

17. You have learned that n! represents the number of ways that n objects can be placed in a linear order,

where it matters which object is placed first. Now consider circular permutations, where

objects are placed in a circle so it does not matter which object is placed first. Find a formula

for the number of permutations of n objects placed in clockwise order around a circle when

only the relative order of the objects matters. Explain how you derived your formula.

Combinations and the Binomial Theorem

Combination; different arrangements of a group of items where order does not matter

When order is not important BAAB .

The number of combinations of a group of n distinct objects taken r at a time is: )!(!

!

rnr

nrnC

for

instance, How many ways are there to select a subcommittee of 7 members from among a

committee of 17? Ans 448,19717 C

Example 1 A standard deck of 52 playing

cards has 4 suits with 13 different cards in

each suit as shown. a. If the order in which

the cards are dealt is not important, how

many different 5-card hands are possible? b.

In how many of these hands are all five

cards of the same suit?

Solution

a) The number of ways to choose 5 cards

from a deck of 52 cards is:

960,598,2552 C

b) For all five cards to be the same suit, you

need to choose 1 of the 4 suits and then 5

of the 13 cards in the suit. So, the

number of possible hands is:

148,551314 CC



When finding the number of ways both an event A and an event B can occur, you need to

multiply (as you did in part (b) of Example 1). When finding the number of ways that an event A

or an event B can occur, you add instead.

Example 2 A restaurant serves omelets that can be ordered with any of the ingredients shown.

a) Suppose you want exactly 2 vegetarian

ingredients and 1 meat ingredient in your

omelet. How many different types of

omelets can you order?

b) Suppose you can afford at most 3

ingredients in your omelet. How many

different types of omelets can you order?

Solution

a) You can choose 2 of 6 vegetarian ingredients and 1 of 4 meat ingredients. So, the number of

possible omelets is: 601426 CC

b) You can order an omelet with 0, 1, 2, or 3 ingredients. Because there are 10 items to choose

from, the number of possible omelets is: 17612045101310210110010 CCCC

Remark Counting problems that involve phrases like “at least” or “at most” are sometimes

easier to solve by subtracting possibilities you do not want from the total number of possibilities.

Example 3 A theater is staging a series of 12 different plays. You want to attend at least 3 of

the plays. How many different combinations of plays can you attend?

Solution

You want to attend 3 plays, or 4 plays, or 5 plays, and so on. So, the number of combinations of

plays you can attend is 1212512412312 ... CCCC . Instead of adding these combinations, it is

easier to use the following reasoning. For each of the 12 plays, you can choose to attend or not

attend the play, so there are 122 total combinations. If you attend at least 3 plays you do not

attend only 0, 1, or 2 plays. So, the number of ways you can attend at least 3 plays is:

017,4)66121(086,4)(2 212112012

12 CCC

Binomial Theorem

If you arrange the values of nCr in a triangular pattern in which each row corresponds to a value

of n, you get what is called Pascal’s triangle. It is named after the famous French mathematician

Blaise Pascal (1623–1662)



Pascal’s triangle has many interesting patterns and properties. For instance, each number other

than 1 is the sum of the two numbers directly above it.

The binomial expansion of nqp )( for any positive integer n is:

n

r

rrn

rn

n

nn

n

n

n

n

n

n

n qpCqpCqpCqpCqpCqp0

022

2

11

1

0

0 .....)(

To expand a power of a binomial difference, you can rewrite the binomial as a sum. The

resulting expansion will have terms whose signs alternate between + and º.

Example 1 Expand a) 4)2( x b)

32 )( yx c) 5)( yx d)

4)25( x

Solution

1632248

22222)2(

234

40

44

31

34

22

24

13

14

04

04

4

xxxx

xCxCxCxCxCx

64223

320

33

221

23

122

13

023

03

32

33

)()()()()(

yxyyxx

yxCyxCyxCyxCyx

54322345

50

55

41

45

32

35

23

25

14

15

05

05

5

510105

)()()()()()()(

yxyyxyxyxx

yxCyxCyxCyxCyxCyxCyx

432

40

44

31

34

22

24

13

14

04

04

4

161606001000625

)2(5)2(5)2(5)2(5)2(5)25(

xxxx

xCxCxCxCxCx

Example 2 Find the coefficient of 4x in the expansion of 12)32( x

Solution

From the binomial theorem you know the following:

12

0

12

12

12 )3()2()32(r

rr

r xCx

The term that has 4x occurs when r=8 and is is 4484

812 120,963,51656116495)3()2( xxxC

The coefficient is 51,963,120.

Exercise

1) Your English teacher has asked you to select 3 novels from a list of 10 to read as an

independent project. In how many ways can you choose which books to read?

2) Your friend is having a party and has 15 games to choose from. There is enough time to play

4 games. In how many ways can you choose which games to play?

3) You are buying a new car. There are 7 different colors to choose from and 10 different types

of optional equipment you can buy. You can choose only 1 color for your car and can afford

only 2 of the options. How many combinations are there for your car?

4) There are 6 artists each presenting 5 works of art in an art contest. The 4 works judged best

will be displayed in a local gallery. In how many ways can these 4 works all be chosen from

the same artist’s collection?



5) An amusement park has 20 different rides. You want to ride at least 15 of them. How many

different combinations of rides can you go on?

6) A summer concert series has 12 different performing artists. You decide to attend at least 4

of the concerts. How many different combinations of concerts can you attend? Decide

whether the problem requires combinations or permutations to find the answer. Then solve

the problem.

7) You are buying a flower arrangement. The florist has 12 types of flowers and 6 types of

vases. If you can afford exactly 3 types of flowers and need only 1 vase, how many different

arrangements can you buy?

8) Eight members of a school marching band are auditioning for 3 drum major positions. In how

many ways can students be chosen to be drum majors?

9) Your school yearbook has an editor-in-chief and an assistant editor-in-chief. The staff of the

yearbook has 15 students. In how many ways can students be chosen for these 2 positions?

10) A relay race has 4 runners who run different parts of the race. There are 16 students on your

track team. In how many ways can your coach select students to compete in the race? 58.

11) You must take 6 elective classes to meet your graduation requirements for college. There are

12 classes that you are interested in. In how many ways can you select your elective classes?

12) A group of 20 high school students is volunteering to help elderly members of their

community. Each student will be assigned a job based on requests received for help. There

are 8 requests for raking leaves, 7 requests for running errands, and 5 requests for washing

windows. a. One way to count the number of possible job assignments is to find the number

of permutations of 8 L’s (for “leaves”), 7 E’s (for “errands”), and 5 W’s (for “windows”).

a) Use this method to write the number of possible job assignments first as an

expression involving factorials and then as a simple number.

b) Another way to count the number of possible job assignments is to first choose the 8

students who will rake leaves, then choose the 7 students who will run errands from

the students who remain, and then choose the 5 students who will wash windows

from the students who still remain. Use this method to write the number of possible

job assignments first as an expression involving factorials and then as a simple

number.

c) Writing How do the answers to parts (a) and (b) compare to each other? Explain why

this makes sense.

13) Expand a) 6)4( x b)

6)3( yx c) 72 )( yx d)

73)2( yx

14) Use the binomial theorem to write the binomial expansion. a) 5)3( yx b)

6)2( yx c)

53 )3( x d) 72 )2( yx d)

4)33( x e)72 )2( yx f)

423 )( yx

15) Find the coefficient of 6x in the expansion of; a) 8)2( x b)

12

2

1

xx c)

10

2

3 3

2

x

x. Also

find the constant term for part c and d.



Functions and Their Graphs

Relations and Functions In natural language relations are a kind of links existing between objects. Examples: ‘mother of’,

‘neighbor of’, “part of”, ‘is older than’, ‘is an ancestor of’, ‘is a subset of’, etc. These are binary

relations. Mathematically A relation is a mapping, or pairing, of input values with output

values. The set of input values is called the domain and the set of output values is called the

range.

Definition A function f from a set X to a set Y is a rule that assigns to each element of X a

unique element of Y . We could write f ∶ 𝑎 → 𝑏 to indicate that f sends a to b. But the most

common notation (called the function notation) is f(𝑎) = 𝑏: read as “the value of ƒ at x,” or

simply as “ƒ of x.” This notation means that when the input a is inserted into the function f, the

output is b.

The set X of inputs is called the domain of the function f. The set Y of all conceivable outputs is

the codomain of the function f. The set of all outputs is the range of f. The range is a subset of Y.

A relation is a function provided there is exactly one output for each input. It is not a function if

at least one input has more than one output.

Relations (and functions) between two quantities can be represented in many ways, including

mapping diagrams, tables, graphs, equations, and verbal descriptions

Example 1 Identify the domain and range. Then tell whether the relation is a function.

Solution

a. The domain consists of -3, 1, and 4, and

the range consists of -2, 1, 3, and 4.

The relation is not a function because the

input 1 is mapped onto both -2 and 1.

b. The domain consists of -3, 1, 3, and 4,

and the range consists of -2, 1, and 3. The relation is a function; each input in the domain is mapped onto exactly one output in the range.

A relation can be represented by a set of ordered pairs of the form (x, y). In an ordered pair the

first number is the x-coordinate and the second number is the y-coordinate. To graph a relation,

plot each of its ordered pairs in a coordinate plane such as the one shown. A coordinate plane is

divided into four quadrants by the x-axis and the y-axis. The axes intersect at a point called the

origin.



Example 2 Graph the relations given in Example 1.

Solution

a) Write the relation as a set o ordered

pairs: (-3, 3), (1, -2), (1, 1), (4, 4). Then

plot the points in a coordinate plane.

b) Write the relation as a set of ordered

pairs: (-3, 3), (1, 1), (3, 1), (4, -2). Then

plot the points in a coordinate plane.

In Example 2 notice that the graph of the relation that is not a function (the graph on the left) has

two points that lie on the same vertical line. You can use this property as a graphical test for

functions.

Vertical Line Test for Functions: A relation is a function if and only if no vertical line

intersects the graph of the relation at more than one point.

Variables other than x and y are often used when working with relations in real-life situations, as

shown in the next example.

Example 3 The graph shows the ages a

and diameters d of several pine trees at Lund

breck Falls in Canada. Are the diameters of

the trees a function of their ages? Explain.

Solution

The diameters of the trees are not a function

of their ages because there is at least one

vertical line that intersects the graph at more

than one point. For example, a vertical line

intersects the graph at the points (75, 1.22)

and (75, 1.58). So, at least two trees have the

same age but different diameters.

Graphing and Evaluating Functions Many functions can be represented by an equation in two variables, such as 72 xy . An

ordered pair (x, y) is a solution of such an equation if the equation is true when the values of x

and y are substituted into the equation. For instance, (2, 3) is a solution of 72 xy because

72 xy is a true statement.

In an equation, the input variable is called the independent variable. The output variable is

called the dependent variable and depends on the value of the input variable. For the equation y

= 2x-7, the independent variable is x and the dependent variable is y.

The graph of an equation in two variables is the collection of all points (x, y) whose coordinates

are solutions of the equation.



Graphing Equations in Two Variables

To graph an equation in two variables, follow these steps:

i) Construct a table of values. (ii) Graph enough solutions to recognize a pattern.

(iii) Connect the points with a line or a curve.

Example 3 Graph the function y = x + 1.

Solution

Begin by constructing a table of values

Choose x. -2 -1 0 1 2

Evaluate y -1 0 1 2 3

Plot the points. Notice the five points lie on

a line.

Draw a line through the points

The function in Example 4 is linear function a linear function because it is of the form

cmxy where m and c are constants. The graph of a linear function is a line. By naming a

function “ƒ” you can write the function using function notation f(𝑥) = 𝑚𝑥 + 𝑐

The symbol ƒ(x) is read as “the value of ƒ at x,” or simply as “ƒ of x.” Note that ƒ(x) is another

name for y. The domain of a function consists of the values of x for which the function is

defined. The range consists of the values of ƒ(x) where x is in the domain of ƒ. Functions do not

have to be represented by the letter ƒ. Other letters such as g or h can also be used.

Example 5 Decide whether the function is linear. Then evaluate the function when x =-2.

a) 53)f( 2 xxx b) 62)( xxg

Solution

a) f(x) is not a linear function because it has an 2x -term. Now 55)2(3)2()2f( 2

b) g(x) is a linear function because it has the form g(x) = mx + b. Now 26)2(2)2( g

Example 6 In March of 1999, Bertrand Piccard and Brian Jones attempted to become the first

people to fly around the world in a balloon. Based on an average speed of 97.8 kilometers per

hour, the distance d (in kilometers) that they traveled can be modeled by d = 97.8t where t is the

time (in hours). They traveled a total of about 478 hours. The rules governing the record state

that the minimum distance covered must be at least 26,700 kilometers

a) Identify the domain and range and

determine whether Piccard and Jones set

the record.

b) Graph the function. Then use the graph

to approximate how long it took them to

travel 20,000 kilometers.

Solution

a) Because their trip lasted 478 hours, the

domain is 0 ≤ t ≤ 478. The distance they

traveled was 700,46)478(8.97 d km,

so the range is 0 ≤ d ≤ 46,700. Since

46,700 > 26,700, they did set the record



b) The graph of the function is shown. Note that the graph ends at (478, 46,700). To find how

long it took them to travel 20,000 kilometers, start at 20,000 on the d-axis and move right

until you reach the graph. Then move down to the t-axis. It took them about 200 hours to

travel 20,000 kilometers

Functions defined by equations

Most of the functions that we will explore in mathematics and science are defined by some

equation. For example, consider the function f ℝ → ℝ defined by mapping a real number to the

number that is one more than its square, that is f(𝑥) = 𝑥2 + 1

This function maps real numbers (the set ℝ) to real numbers and so the domain of f is ℝ and the

codomain of f is ℝ. However, the range of f is the set of all real numbers greater than or equal to

1. In interval notation, the domain of f is ),( and the range is ),1[ .

We can often define a function implicitly in an equation involving two variables. Traditionally

we use the letter x for the input variable and the letter y for the output variable. The equation

𝑥2 − 𝑦 = −1 defines the function f(𝑥) = 𝑥2 + 1 discussed above.

As another example, consider the linear equation 0432 yx

This equation can be viewed as defining a function with inputs x and outputs y. From this

viewpoint, we can solve for y and get )24(3

1 xy

We might then explicitly define the function )24()(f3

1 xx

Note: Our choice of x as input and y as output was somewhat arbitrary. We could have decided

(Contrary to custom) that y is the input and x is the output. Then, solving for x, we have

)34(2

1 yx and so we can create the function x = g(y) (with input y and output x)

)34()(g2

1 yy

Example 2 Does the equation 𝑥2𝑦 = 4 define y as a function of x? (If it does, give the domain

of the implied function.)

Solution. We attempt to solve for y. We may divide both sides of the equation by 𝑥2 as long as x

is not zero. This gives us 2

4

xy :

Is there a problem with x = 0? No, x = 0 does not allow 𝑥2𝑦 = 4, so x will never be zero in this

equation. YES; 2

4

xy :

The domain of this function is all real numbers except zero. In interval notation this is

),0()0,(

Example 3 Does the equation 42 xy define y as a function of x?

Solution.

If we attempt to solve for y, we divide both sides of the equation by x (as long as 0x ) and so

we have xx

yy 442 ie y could be positive or negative { there will generally be two

choices here.

NO; for example, if x = 1 then we don't know if y = 2 or y = -2:

Example 4 Does the equation 02 yx define y as a function of x? (Why/why not?)



Solution.

Although it might be tempting to solve for y, first notice that if x is zero then y could be 0 or 1 or

2.71828 or anything! So the input x = 0 does not give a unique output. This is not a function.

NO; if x = 0 then y could be anything.

(Note how this is different than example 2. In example 2, x = 0 is not a possible input in the

equation. But here x = 0 is a possibility for a solution to the equation!)

Let us practice the function notation, f(x).A formula for f(x) tells us how the input x leads to the

output f(x):

Example 5 Suppose 9)(f 2 xx . Compute: f(0), f(1), f(-1), f(-5), f(-x) , f(x + h), f( x ),

f(2a + 1) and -f(x) + 2

Solution

If 9)(f 2 xx then 990)0(f 2 891)1(f 2 , 89)1()1(f 2

169)5()5(f 2 99)()(f 22 xxx 929)()(f 222 hxhxhxhx

99)()(f 2 xxx 8449)12()12(f 22 aaaa 22 112)9(2)(f- xxx

Question: Apply the mapping g: 𝑥 → 3𝑥 to the domain {0, 2, 4, 6, 8}. List the image set

Given that f: 𝑥 →1

𝑥 and g: 𝑥 → 1 − 𝑥 write down the values of f(1) f(0) and f(−𝑥)

The domain of a function

The domain of a function is generally viewed as the largest possible set of inputs into the

function. For example, the domain of the function xx )(f is all real numbers greater than or

equal to zero. In interval notation we write ),0[ . Note that we cannot evaluate f(x) at negative

numbers (if we assume we are always working with real numbers.)

We often need to find the domain of a function and if the inputs are real numbers, express the

domain in interval notation. When we do this, it is often easier to ask the question, \What is not

in the domain?"

For example, in the square root function, xx )(f we might ask the question, \Which numbers

do not have a square root?" Since the square of a real number cannot be negative, then our

answer is \We cannot take the square root of negative numbers." So the domain must be numbers

which are not negative, that is, zero and positive real numbers. So the domain of xx )(f is

),0[ : (We can indeed take the square root of 0 so we want to include 0 in the domain.)

Example 6 Find the domain of the function 512

32

2

1)(g

x

x

x

xx

Solution

What numbers cannot serve as input to g(x)? Since we cannot have denominators equal to zero

then 2x cannot be an input; neither can 2

1x : So the domain of this function g is all real

numbers except 2x and 2

1x .

There are several ways to write the domain of g. Using set notation, we could write the domain

as {𝑥𝜖ℝ: 𝑥 ≠ −2 , −1

2} This is a precise symbolic way to say, “All real numbers except -2 &

21 "

We could also write the domain in interval notation: ),(),2()2,(2

1

2

1



This notation says that the domain includes all the real numbers smaller than -2, along with all

the real numbers between -2 and 21 , along with (in addition) the real numbers larger than

21 :

Example 7 Find the domain of the function;

a) xx 2)(f b) 1)(f xx c) 3

1)(f

x

xx d)

86

1)(f

2

xx

xx

Solution

a) Since the square root function requires nonnegative inputs, we must have 02 x :

Add x to both sides of the inequality to get x2 : In interval notation this is ]2,(

Answer: The domain is ]2,(

b) Since the square root function requires nonnegative inputs, we must have 01x : If we add

1 to both sides of the inequality we have 1x :

Answer: The domain is ),1[

c) Again, as in the part b, we must have 1x but we must also prevent the denominator from

being zero, so x cannot be 3, either.

Answer: The domain is ),3()3,1[

d) We must have 1x and we must prevent the denominator from being zero. The denominator

factors as )4)(2(862 xxxx , so x cannot be 2 or 4. So our answer is all real numbers

at least as big as 1 and not equal to 2 or 4. In interval notation, our answer is:

),4()4,2()2,1[ .

The domain is ),4()4,2()2,1[

Exercise

1) Is a function always a relation? Is a relation always a function? Explain your reasoning.

2) Explain why a vertical line, rather than a horizontal line, is used to determine if a graph

represents a function.

3) Decide whether the function is linear. Then evaluate the function for the given value of x.

a) )4(f;11)(f xx b) )4(f;21)(f x c) )6(f;5)(f xx

d) )(f;43)(f2

1 xx e) )2(f;29)(f 23 xxx f) )6(f;5)(f 2

3

2 xxx

4) The volume of a cube with side length s is given by the function 3)(v ss . Find )(v2

3 ).

Explain what )(v2

3 represents.

5) The volume of a sphere with radius r is given by the function 3

3

4)(v rr . Find V )2(v ).

Explain what V(2) represents.

6) Identify the domain and range of the

relation shown. Then tell whether the

relation is a function.

7) Graph the function a xy 4 b) 1 xy

b) 2

1

2

3 xy c) xy 9 d) xy3

24

8) Evaluate the function when x = 3

a) xx 6)(f b) 72)(f xx c) 2)(f xx d) 210)(f xx e) xxx 7)(f 3



9) A car has a 16 gallon gas tank. On a long highway trip, gas is used at a rate of about 2 gallons

per hour. The gallons of gas g in the car’s tank can be modeled by the equation 𝑔 = 16 − 2𝑡

where t is the time (in hours).

a) Identify the domain and range of the function. Then graph the function.

b) At the end of the trip there are 2 gallons of gas left. How long was the trip?

10) Identify the domain and range.

11) Graph the relation. Then tell whether the

relation is a function

x 0 0 2 2 4 4

y 4 -4 -3 3 -1 1

x -5 -4 -3 0 3 4 5

y -6 -4 -2 -1 -2 -4 -6

x -2 -2 0 2 2

y 1.5 -3.5 0 1.5 -3.5

12) Use a mapping diagram to represent the relation. Then tell whether the relation is a function.

13) Use the vertical line test to determine whether the relation is a function.

14) Why does y = 3 represent a function, but

x = 3 does not?

15) The graph shows the ages and finishing

places of the top three competitors in

each of the four categories of the 100th

Boston Marathon. Is the finishing place

of a competitor a function of his or her

age? Explain your reasoning.

16) Evaluate the expression for the given values of x and y.

a) 9

6

x

y when x= 3 & y= -2 c)

3

)5(

x

y when x=2 & y=5 e)

)4(

)1(

x

y when x=6 & y=4

b) 9

11

x

y when x= 4 & y= 5 d)

)4(

)1(

x

y when x = 6 & y = 4 f)

x

y

14

10 when x=6 & y = 8



17) The table below shows the number of shots attempted and the number of shots made by 9

members of the Utah Jazz basketball team in Game 1 of the 1998 NBA Finals.

Player Shots

attempted, x

Shots

made, y

Bryon Russell 12 6

Karl Malone 25 9

Greg Foster 5 1

Jeff Hornacek 10 2

John Stockton 12 9

Howard Eisley 6 4

Chris Morris 6 3

Greg Ostertag 1 1

Shandon

Anderson

5 3

a) Identify the domain and range of the

relation. Then graph the relation.

b) Is the relation a function? Explain.

18) Water pressure can be measured in

atmospheres, where 1 atmosphere equals

14.7 pounds per square inch. At sea level

the water pressure is 1 atmosphere, and

it increases by 1 atmosphere for every 33

feet in depth. Therefore, the water

pressure p can be modeled as a function

of the depth d by this equation:

1300,133

1 ddp


function. Then graph the function.

b) What is the water pressure at a depth

of 100 feet?

19) The graph shows the number of Independent representatives for the 100th–105th Congresses.

Is the number of Independent representatives a function of the Congress number? Explain

your reasoning.

20) Your cap size is based on your head

circumference (in inches). For head

circumferences from 8

720 inches to 25

inches, cap size s can be modeled as a

function of head circumference c by this

equation: )1(33

1 cs


function. Then graph the function.

b) If you wear a size 7 cap, what is your head circumference?

21) For the numbers 2 through 9 on a

telephone keypad, draw two mapping

diagrams: one mapping numbers onto

letters, and the other mapping letters

onto numbers. Are both relations

functions? Explain.

Composite Functions

Composition of functions is when one function is inside another function. For example, if we

look at the function 2)12()(h xx . We can say that this function, h(x), was formed by the

composition of two other functions, the inside function 2𝑥 − 1 and the outside function z2. The

letter z was used just to represent a different variable, we could have used any letter that we

wanted. Notice that if we put the inside function, 2𝑥 − 1, into the outside function, z2, we would

get 𝑧2 = (2𝑥 − 1)2, which is our original function h(x).

The notation used for the composition of functions looks like this, )(g)(f x . So what does this

mean? )(g)(f x , the composition of the function f with g, is defined as )(gf)(g)(f xx ,

notice that in the case the function g is inside the function f.



Also )(fg)(f)(g xx the f function is inside of the g function

Therefore )(g)(f x and )(f)(g x are often different because in the composite )(g)(f x , f(x) is

the outside function and g(x) is the inside function. Whereas in the composite )(f)(g x , g(x) is

the outside function and f(x) is the inside function.

In composite functions it is very important that we pay close attention to the order in which the

composition of the functions is written. In many cases )(g)(f x is not the same as )(fg x

)(gf)(g)(f xx , the g fun

Example 1 If 4x-9)(f x and 7-2x)(g x , find )(g)(f x and )(f)(g x

Solution

xxxxx 8372889)72(49)(gf)(g)(f

xxxxx 81178187)49(2)(fg)(f)(g

Notice that )(g)(f x and )(f)(g x produces different answers )(f)(g)(g)(f xx

Example 2 If 5-3x)(h x and 7x-2x)(g 2x find )(g)(h x and )(h)(g x

Solution

52165723g(x)h)(g)(h 22 xxxxx

8581183521506018

3521)25309(2)53(7)53(2h(x)g)()(g

22

22

xxxxx

xxxxxxh

Exercise

1) For the given functions find the indicated composite function

a) If 24x-x)(f 2 x and 7-3x)(g x , find )(g)(f x and )(f)(g x

b) If 6x-5)(g x and 11-9x)(h x , find )(g)(h x and )(h)(g x

c) If 52)(f xx and 3-5x)(g 2x , find )(g)(f x and )(f)(g x

d) If xx 29)(f and 35xx4)(g 2 x , find )(g)(f x and )(f)(g x

e) If 3)(f xx and 93x-x4)(g 2 x , find )(g)(f x and )(f)(g x

f) If 3 4-x)(f x and 4x)(g 3 x , find )(g)(f x and )(f)(g x

2) Let 42 72)(g and6)(f xxxxx Perform the indicated operation and state the domain

a) )(g)(f xx b) )(g)(f xx c) )(g)(f xx d) )(g)(f xx

3) Let 8)(g and3)(f 1 xxxx Perform the indicated operation and state the domain.a)

)(gf x b) )(fg x c) )(ff x d) )(gg x

Inverse Functions In the previous section you learned that a relation is a mapping of input values onto output

values. An inverse relation maps the output values back to their original input values. This

means that the domain of the inverse relation is the range of the original relation and that the

range of the inverse relation is the domain of the original relation.



The graph of an inverse relation is the reflection of the graph of the original relation on y = x.

To find the inverse of a relation that is given by an equation in x and y, switch the roles of x and y

and solve for y (if possible).

Example 1 Find an equation for the inverse of the relation 42 xy .

Solution

The original relation is 42 xy . Switch x and y to get 42 yx then solve for y. 22

1 xy

The inverse relation is 22

1 xy

In Example 1 both the original relation and the inverse relation happen to be functions. In such

cases the two functions are called inverse functions.

Remarks: Functions ƒ and g are inverses of each other provided xx )(gf and xx )(fg

The function g is denoted by -1f , read as “ƒ inverse.”

Given any function, you can always find its inverse relation by switching x and y. For a linear

function f(x) = mx + b where m ≠ 0, the inverse is itself a linear function.

Example 2 Verify that 42)(f xx and 2)(f2

1-1 xx are inverses

Solution

We need to show that xx )(ff -1 and xx )(ff -1 Now

xxxxx 222)42()42(f)(ff2

1-1-1 and

xxxxx 444222f)(ff2

1

2

1-1 )(f x and )(f -1 x are inverses

Example 3 When calibrating a spring

scale, you need to know how far the spring

stretches based on given weights. Hooke’s

law states that the length a spring stretches is

proportional to the weight attached to the

spring. A model for one scale is

30.5wl where l is the total length (in

inches) of the spring and w is the weight (in

pounds) of the object.

a) Find the inverse model for the scale.



b) If you place a melon on the scale and the spring stretches to a total length of 5.5”, how much

does the melon weigh?

Solution

a) 62l0.53)-l(w30.5wl

b) To find the weight of the melon, substitute 5.5 for l to get 562(5.5)62lw

Thus the melon weighs 5 pounds

The graphs of the power functions 2)(f xx

and 3)(g xx are shown below along with

their reflections in the line y = x. Notice that

the inverse of 3)(g xx is a function, but

that the inverse of 2)(f xx is not a

function. But if the domain of 2)(f xx is

restricted, to only nonnegative real numbers,

then the inverse of f is a function.

Example 4 Find the inverse of the function 2)(f xx , x ≥ 0.

Solution 2)(f xxy Switch x and y to get xyyx 2

. Because the domain of ƒ is restricted to

nonnegative values, the inverse function is xx )(f -1

(You would choose xx )(f -1if the

domain had been restricted to x ≤ 0.)

Check To check your work, graph ƒ and -1f

as shown. Note that the graph of

xx )(f -1is the reflection of the graph of

2)(f xx , x ≥ 0 in the line y = x.

In the graphs at the top of the page, notice that the graph of 2)(f xx can be intersected twice

with a horizontal line and that its inverse is not a function. On the other hand, the graph of 3)(g xx cannot be intersected twice with a horizontal line and its inverse is a function. This

observation suggests the horizontal line test.

Horizontal Line Test: If no horizontal line intersects the graph of a function ƒ more than once,

then the inverse of ƒ is itself a function.

Example 5 Consider the function 2)(f 3

2

1 xx . Determine whether the inverse of ƒ is a

function. Then find the inverse.

Solution Begin by graphing the function and noticing that no horizontal line intersects the graph

more than once. This tells you that the inverse of ƒ is itself a function. To find an equation for -1f , complete the following steps.

Write original function as 2y 3

2

1 x ƒ(x) is Replaced with y. Now Switch x and y.

33

2

1 422x xyy Thus the inverse function is 3-1 42)(f xx



Example 6 Near the end of a star’s life the

star will eject gas, forming a planetary

nebula. The Ring Nebula is an example of a

planetary nebula. The volume V (in cubic

kilometers) of this nebula can be modeled

by 326)1001.9( tV where t is the age (in

years) of the nebula. Write the inverse

model that gives the age of the nebula as a

function of its volume. The graph of the function in Eg5

Then determine the approximate age of the Ring Nebula given that its volume is about 1 38105.1 cubic kilometers.

Solution

393

26

326 )1004.1()1001.9(

)1001.9( VV

ttV

. Substitute 138105.1 for V to get

5500105.1)1004.1()1004.1(3 38939 Vt . The Ring Nebula is about 5500 years old.

Exercise

1) Explain the steps in finding an equation for an inverse function.

2) Find the inverse relation.

X 1 2 3 4 5

Y -1 -2 -3 -4 -5

X -2 -1 0 2 4

y 2 1 0 1 2

X 1 4 1 0 1

y 3 -1 6 -3 9

X 1 -2 4 2 -2

y 0 3 -2 2 -1

3) Find an equation for the inverse relation.

a) x5y b) 12y x c) x3

26y d) 4

13y x e) 138y x f) x7

3

7

5y

4) Verify that ƒ and g are inverse functions a) 22)(g,1)(f2

1 xxxx b)

2

1

6

1)(g,36)(f xxxx c) 21

2

13 )(g,8)(f xxxx d) xxxxx 3)(g,0)(f 2

3

1 e)

31

)1()(g,13)(f2

13 xxxx f) 55

7

1 27)(g,1)2()(f xxxx g)

4

4

14 7)(g,01256)(f xxxxx

5) Find the inverse function; a) 2)(f 3 xx b) 9)(f 3

5

3 xx c) 0,3)(f 4 xxx

d) 5

3

1 2)(f xx e) 0,2)(f 2 xxx

f) 5

6

1

3

2)(f xx g) 0,)(f2

14 xxx

6) The graph of xx 1)(f is shown. Is

the inverse of ƒ a function? Explain

7) Find the inverse power function a) 7)(f xx b) 0,)(f2

16 xxx c) 0,3)(f 4 xxx

d) 5

32

1)(f xx e) 310)(f xx f) 0,)(f 2

4

9 xxx

8) Match the graph with the graph of its inverse



a) b) c)

9) Graph the function ƒ. Then use the graph to determine whether the inverse of ƒ is a function

a) xx 23)(f b) 3)(f xx c) 1)(f 2 xx d) 22)(f xx e) 3)(f 3 xx

f) 2)(f xx g) 32)(f xx h) )3)(1()(f xxx j) 196)(f 4 xxx

10) The Federal Reserve Bank of New York reports international exchange rates at 12:00 noon

each day. On January 20, 1999, the exchange rate for Canada was 1.5226. Therefore, the

formula that gives Canadian dollars in terms of United States dollars on that day is

USC DD 5226.1 where CD represents Canadian dollars and

USD represents United States

dollars. Find the inverse of the function to determine the value of a United States dollar in

terms of Canadian dollars on January 20, 1999.

11) The formula to convert temperatures from degrees Fahrenheit to degrees Celsius is:

)32(9

5 FC Write the inverse of the function, which converts temperatures from degrees

Celsius to degrees Fahrenheit. Then find the Fahrenheit temperatures that are equal t-29°C,

10°C, and 0°C.

12) In bowling a handicap is a change in score to adjust for differences in players’ abilities. You

belong to a bowling league in which each bowler’s handicap h is determined by his or her

average a using this formula )200(9.0 ah (If the bowler’s average is over 200, the

handicap is 0.) Find the inverse function. Then find your average if your handicap is 27.

13) You and a friend are playing a number-guessing game. You ask your friend to think of a

positive number, square the number, multiply the result by 2, and then add 3. If your friend’s

final answer is 53, what was the original number chosen? Use an inverse function in your

solution.

14) The weight w (in kilograms) of a hake, a type of fish, is related to its length l (in centimeters)

by this function: )1037.9( 6

3

1 w Find the inverse of the function. Then determine the

approximate length of a hake that weighs 0.679 kilogram.

15) The weight w (in pounds) that can be supported by a shelf made from half-inch Douglas fir

plywood can be modeled by

39.82

dw where d is the distance (in inches) between the

supports for the shelf. Find the inverse of the function. Then find the distance between the

supports of a shelf that can hold a set of encyclopedias weighing 66 pounds.



Graphing Quadratic Function

A quadratic function has the form 𝑦 = 𝑎𝑥2 + 𝑏𝑥 + 𝑐 where𝑎 ≠ 0. The graph of a quadratic

function is U-shaped and is called a parabola. Eg the graphs of 𝑦 = 𝑥2 and 𝑦 = −𝑥2 are shown.

The origin is the lowest point on the graph

of 2xy and the highest point on the graph

of 2xy . The lowest or highest point on

the graph of a quadratic function is called

the vertex. The graphs of 2xy and 𝑦 =

−𝑥2are symmetric about the y-axis, called

the axis of symmetry. In general, the axis of

symmetry for the graph of a quadratic

function is a vertical line through the vertex.

Activity: Graph each of these functions on the same axis: of 𝑦 = 0.5𝑥2 , 𝑦 = 𝑥2, 𝑦 =

2𝑥2 and 𝑦 = 3𝑥2 Repeat the exercise for these functions: 𝑦 = −0.5𝑥2 , 𝑦 = −𝑥2, 𝑦 =

−2𝑥2 and 𝑦 = −3𝑥2

i) What are the vertex and axis of symmetry of the graph of , 𝑦 = 𝑎𝑥2?

ii) Describe the effect of a on the graph of , 𝑦 = 𝑎𝑥2

In the above activity, you examined the graph of the simple quadratic function 𝑦 = 𝑎𝑥2. The

graph of the more general function 𝑦 = 𝑎𝑥2 + 𝑏𝑥 + 𝑐 is a parabola with these characteristics:

• The parabola opens up if 𝑎 > 0 and opens down if 𝑎 < 0. The parabola is wider than the

graph of 𝑦 = 𝑥2 if |𝑎| < 1 and narrower than the graph of 𝑦 = 𝑥2 if |𝑎| > 1.

• The x-coordinate of the vertex is a

b2

and the axis of symmetry is the vertical line a

bx

2 .

Example 1 Graph 𝑦 = 2𝑥2 − 8𝑥 + 6 . Solution

Note that the coefficients for this function are a = 2, b =-8, and c = 6. Since a > 0, the parabola

opens up. Find and plot the vertex. The x-coordinate is 22

ab .

The y-coordinate is 𝑦 = 2 × 22 − 8 × 2 +

6 = −2 So, the vertex is (2, -2).

Draw the axis of symmetry 𝑥 = 2.

Plot two points on one side of the axis of

symmetry, such as (1, 0) and (0, 6). Use

symmetry to plot two more points, such as

(3, 0) and (4, 6).

Draw a parabola through the plotted points The quadratic function 𝑦 = 𝑎𝑥2 + 𝑏𝑥 + 𝑐 is written in standard form Two other useful forms

for quadratic functions are given below.

Vertex form: 𝑦 = 𝑎(𝑥 − ℎ)2 + 𝑘 The vertex is (h, k). The axis of symmetry is 𝑥 = ℎ.

Intercept form:𝑦 = 𝑎(𝑥 − 𝑝)(𝑥 − 𝑞) The x-intercepts are p and q. The axis of symmetry is

halfway between (p, 0) and (q, 0).

Note For both forms, the graph opens up if 𝑎 > 0 and opens down if 𝑎 < 0.



Example 2 Graph 𝑦 = −1

2(𝑥 + 3)2 + 4

Solution

The function is in vertex form 𝑦 = 𝑎(𝑥 −

ℎ)2 + 𝑘 where 𝑎 = −1

2, ℎ = −3, and 𝑘 =

4. Since 𝑎 < 0, the parabola opens down.

To graph the function, first plot the vertex

(h, k) = (-3, 4). Draw the axis of symmetry x

= -3 and plot two points on one side of it,

such as (-1, 2) and (1, -4). Use symmetry to

complete the graph

Example 3 Graph 𝑦 = −(𝑥 + 2)(𝑥 − 4)

Solution

The function is in intercept form 𝑦 = 𝑎(𝑥 −𝑝)(𝑥 − 𝑞)where 𝑎 = −1, 𝑝 = −2, and 𝑞 =4. the x-intercepts occur at (-2, 0) and (4, 0).

The axis of symmetry lies halfway between

these points, at 𝑥 = 1. So, the x-coordinate

of the vertex is 𝑥 = 1 and the y-coordinate

of the vertex is: 𝑦 = −(1 + 2)(1 − 4) = 9

The graph of the function is shown.

Example 4 Researchers conducted an experiment to determine temperatures at which people

feel comfortable. The percent y of test subjects who felt comfortable at temperature x (in degrees

Fahrenheit) can be modeled by: 𝑦 = −3.678𝑥2 + 527.3𝑥 − 18,807

a) What temperature made the greatest percent of test subjects comfortable?

b) At that temperature, what percent felt comfortable?

Solution

a) Since a is negative, the graph opens down and the function has a maximum value.

The maximum value occurs at: 72)678.3(2

3.527

2

a

bx

b) The corresponding value of y is: : 𝑦 = −3.678(72)2 + 527.3(72) − 18,807 ≈ 92 The

temperature that made the greatest percent of test subjects comfortable was about 72°F. At

that temperature about 92% of the subjects felt comfortable.

Example 5 The Golden Gate Bridge in San Francisco has two towers that rise 500 feet above

the road and are connected by suspension cables as shown. Each cable forms a parabola with

equation 𝑦 =1

8960(𝑥 − 2100)2 + 8 where x and y are measured in feet. Source: Golden Gate

Bridge, Highway and Transportation District

a) What is the distance d between the two towers?

b) What is the height ¬ above the road of a cable at its lowest point? S

Solution

a) The vertex of the parabola is (2100, 8), so a cable’s lowest point is 2100 feet from the left

tower shown above. Since the heights of the two towers are the same, the symmetry of the

parabola implies that the vertex is also 2100 feet from the right tower. Therefore, the towers

are 𝑑 = 2(2100) = 4200 feet apart.



b) The height ¬ above the road of a cable at its lowest point is the y-coordinate of the vertex.

Since the vertex is (2100, 8), this height is 𝑙 = 8 feet.

Exercise

1) Graph the quadratic function. Label the vertex and axis of symmetry. 742 xxy

4)1(2 2 xy )1)(2( xxy 322

3

1 xxy 6)4( 2

5

3 xy )3(2

5 xxy

2) Is )8)(5(2 xxy in standard form, vertex form, or intercept form?

3) Write the quadratic function in standard form )1)(2( xxy )3)(4(2 xxy

5)1(4 2 xy 7)2( 2 xy )8)(6(2

1 xxy 4)9( 2

3

2 xy

4) The equation given in Example 5 is based on temperature preferences of both male and

female test subjects. Researchers also analyzed data for males and females separately and

obtained the equations below.

Males: 217736.6124290 2 xxy Females: 330929.9086224 2 xxy

What was the most comfortable temperature for the males? for the females?

5) Match the quadratic function with its graph a) )3)(2( xxy b) 2)3( 2 xy c)

1162 xxy

6) Graph the quadratic function. Label the vertex and axis of symmetry

.a) 122 xxy b) 19122 2 xxy c) 242 xxy d) 53 2 xy e)

542

2

1 xxy f) 32

6

1 xxy g) 5)4(3 2 xy h) 4)3(2 2 xy i)

3)1( 2

3

1 xy j) 2

4

5 )3( xy

7) Graph the quadratic function. Label the vertex, axis of symmetry, and x-intercepts.

a) )6)(2( xxy b) )1)(1(4 xxy c) )5)(3( xxy

d) )1)(4(3

1 xxy e) )3)(2(2

1 xxy f) )2(3 xxy

8) Write the quadratic function in standard form.

a) )5)(2( xxy b) )4)(3( xxy c) )7)(4(3 xxy d)

)14)(85( xxy e) 2)3( 2 xy f) 2)5(11 xy g) 9)2(6 2 xy

h) 20)7(8 2 xy i) 2)29(4 xxy j) )3)(6(3

7 xxy k) 2

32

2

1 )18( xy

9) In parts (a) and (b), use a graphing calculator or computer to examine how b and c affect the

graph of cbxaxy 2

a) Graph cxy 2 c for c = -2, -1, 0, 1, and 2. Use the same viewing window for all the

graphs. How do the graphs change as c increases?

b) Graph y = x2 + bx for b = -2, -1, 0, 1, and 2. Use the same viewing window for all the

graphs. How do the graphs change as b increases?

10) Although a football field appears to be flat, its surface is actually shaped like a parabola so

that rain runs off to either side.



The cross section of a field with

synthetic turf can be modeled by

5.1)80(000234.0 2 xy where x

and y are measured in feet. What is the

field’s width? What is the maximum

height of the field’s surface? 11) Scientists determined that the rate y (in calories per minute) at which you use energy while

walking can be modeled by 15050,3.51)2.90(00849.0 2 xxy where x is your

walking speed (in meters per minute).

a) Graph the function on the given domain.

b) Describe how energy use changes as walking speed increases.

c) What speed minimizes energy use?

12) The woodland jumping mouse can hop

surprisingly long distances given its

small size. A relatively long hop can be

modeled by )6(9

2 xxy where x and

y are measured in feet. How far can a

woodland jumping mouse hop? How

high can it hop?

13) The engine torque y (in foot-pounds) of one model of car is given by

8.382.2375.3 2 xxy where x is the speed of the engine (in thousands of revolutions

per minute). Find the engine speed that maximizes torque. What is the maximum torque?

14) A kernel of popcorn contains water that expands when the kernel is heated, causing it to pop.

The equations below give the “popping volume” y (in cubic centimeters per gram) of

popcorn with moisture content x (as a percent of the popcorn’s weight).

Hot-air popping: 8.944.21761.0 2 xxy Hot-oil popping: 0.767.17652.0 2 xxy

a) For hot-air popping, what moisture content maximizes popping volume? What is the

maximum volume?

b) For hot-oil popping, what moisture content maximizes popping volume? What is the

maximum volume?

c) The moisture content of popcorn typically ranges from 8% to 18%. Graph the equations

for hot-air and hot-oil popping on the interval 8 ≤ x ≤ 18.

d) Based on the graphs from part (c), what general statement can you make about the

volume of popcorn produced from hot-air popping versus hot-oil popping for any

moisture content in the interval 8 ≤ x ≤ 18?

15) Write khxay 2)( and ))(( qxpxay in standard form. Knowing that the vertex

of the graph of cbxaxy 2 occurs at a

bx2

, show that the vertex for khxay 2)(

occurs at x = h and that the vertex for ))(( qxpxay occurs at 2

qpx

.

Graphing Absolute Value Functions

The absolute value of x is defined by

0for

0for)(f

xx

xxxx

The graph of this piece wise function consists of two rays, is V-shaped, and opens up.



The corner point of the graph, called the vertex, occurs at the origin.

Notice that the graph of y = |x| is symmetric about the y-axis because for every point (x, y) on the

graph, the point (-x, y) is also on the graph

Activity

1) In the same coordinate plane, graph y = a|x| for a -2, , 21

2

1 and 2. What effect does a have

on the graph of y = a|x|? What is the vertex of the graph of y = a|x|?

2) In the same coordinate plane, graph y = |x- h| for h = -2, 0, and 2. What effect does h have on

the graph of y = |x-h|? What is the vertex of the graph of y = |x-h|?

3) In the same coordinate plane, graph y = |x| + k for k = -2, 0, and 2. What effect does k have on

the graph of y = |x| + k? What is the vertex of the graph of y = |x| + k?

Although in the above exercise you investigated the effects of a, h, and k on the graph of

𝑦 = 𝑎|𝑥 − ℎ| + 𝑘 separately, these effects

can be combined. Eg, the graph of y = |x| is

shown in blue along with the graph of.

342 xy in red. Notice that the vertex

of the red graph is (4, 3) and that the red

graph is narrower than the blue graph. The graph of 𝑦 = 𝑎|𝑥 − ℎ| + 𝑘 has the following characteristics.

i) The graph has vertex (h, k) and is symmetric in the line x = h.

ii) The graph is V-shaped. It opens up if a > 0 and down if a < 0.

iii) The graph is wider than the graph of y = |x| if |a| < 1.

iv) The graph is narrower than the graph of y = |x| if |a| > 1.

To graph an absolute value function you may find it helpful to plot the vertex and one other

point. Use symmetry to plot a third point and then complete the graph.

Example 1 Graph 𝑦 = −|𝑥 + 2| + 3.

Solution

To graph 𝑦 = −|𝑥 + 2| + 3, plot the vertex

at (-2, 3). Then plot another point on the

graph, such as (-3, 2). Use symmetry to plot

a third point, (-1, 2). Connect these three

points with a V-shaped graph. Note that 𝑎 =−1 < 0 and |a| = 1, so the graph opens

down and is the same width as the graph of y

=|x|.

Example 2 Write an equation of the graph shown.

Solution

The vertex of the graph is (0, -3), so the equation has the form: 𝑦 = 𝑎|𝑥 − 0| + (−3) or

3 xay



To find the value of a, substitute the

coordinates of the point (2, 1) into the

equation and solve.

2321 aa . So the equation of the

graph is 32 xy .

Note The graph opens up and is narrower

than the graph of y = |x|, so 2 is a reasonable

value for a.

Example 3 The front of a camping tent can be modeled by the function 5.35.24.1 xy

where x and y are measured in feet and the x-axis represents the ground. a. Graph the function.

b. Interpret the domain and range of the function in the given context.

Solution

a) The graph of the function is shown. The

vertex is (2.5, 3.5) and the graph opens

down. It is narrower than the graph of y

= |x|.

b) The domain is 0 ≤ x ≤ 5, so the tent is 5

feet wide. The range is 0 ≤ y ≤ 3.5, so

the tent is 3.5 feet tall.

Example 4 While playing pool, you try to shoot the eight ball into the corner pocket as shown.

Imagine that a coordinate plane is placed over the pool table.

The eight ball is at ),5(4

5 and the pocket you

are aiming for is at (10, 5). You are going to

bank the ball off the side at (6, 0).

a) Write an equation for the path of the

ball.

b) Do you make your shot?

Solution

a) The vertex of the path of the ball is (6, 0), so the equation has the form 𝑦 = 𝑎|𝑥 − 6|.

Substitute the coordinates of the point ),5(4

5 into the equation and solve for a.

b) 4

5

4

565 aa . So the equation for the path of the ball is 64

5 xy .

a. You will make your shot if the point (10, 5) lies on the path of the ball. 61054

5

Substitute 5 for y and 10 for x. 55 _ Simplify. The point (10, 5) satisfies the equation, so

you do make your shot

Exercise

1) Graph the function. Then identify the vertex, tell whether the graph opens up or down, and

whether the graph is wider, narrower, or the same width as the graph of y = |x|. a) xy2

1

b) 2

5 xy c) 2

3

3

2 xy d) 1062 xy e) 22

1 xy f) 433

1 xy

2) Write an equation for the function whose graph is shown.



3) Which statement is true about the graph

of the function y =-|x + 2| + 3?

i) Its vertex is at (2, 3).

ii) Its vertex is at (-2, -3).

iii) It opens down.

iv) It is wider than the graph of y = |x|.

4) Suppose that the tent in Example 3 is 7 feet wide and 5 feet tall. Write a function that models

the front of the tent. Let the x-axis represent the ground. Then graph the function and identify

the domain and range of the function.

5) Match the function with its graph. a) xy 3 b) xy 3 c) xy31

6) Use the Trace feature of the graph of absolute functions to find the corresponding x-value(s)

for the given y-value; a) 10;4 yxy b) 9;14 yxy c) 2

3

8

15 ; yxy d)

0;57

4 yxy e)2

1;52 yxy f) 2;732 yxy g)

4

25

2

3 ;63 yxy h) 5;55.175.3 yxy

7) Match the function with its graph. a) 2 xy b) 2 xy c) 2 xy



8) A musical group’s new single is released. Weekly sales s (in thousands) increase steadily for

a while and then decrease as given by the function 40202 ts where t is the time (in

weeks) Graph the function hence find the maximum number of singles sold in one week

9) A rainstorm begins as a drizzle, builds up to a heavy rain, and then drops back to a drizzle.

The rate r (in inches per hour) at which it rains is given by the function 2

1

2

1 1 ts

where t is the time (in hours).

a) Graph the function.

b) For how long does it rain and when does it rain the hardest?

10) Suppose a musical piece calls for an orchestra to start at fortissimo (about 90 decibels),

decrease in loudness to pianissimo (about 50 decibels) in four measures, and then increase

back to ortissimo in another four measures. The sound level s (in decibels) of the musical

piece can be modeled by the function 5410 ms where m is the number of measures.

a) Graph the function for 0 ≤ m ≤ 8.

b) After how many measures should the orchestra be at the loudness of mezzo forte

(about 70 decibels)?

11) You are sitting in a boat on a lake. You can get a sunburn from sunlight that hits you directly

and from sunlight that reflects off the water. Sunlight reflects off the water at the point (2, 0)

and hits you at the point (3.5, 3). Write and graph the function that shows the path of the

sunlight.

12) The Transamerica Pyramid, shown at the right, is an office building in San Francisco. It

stands 853 feet tall and is 145 feet wide at its base. Imagine that a coordinate plane is placed

over a side of the building. In the coordinate plane, each unit represents one foot

, and the origin is at the center of the

building’s base. Write an absolute

function whose graph is a V-shaped

outline of the sides of the building,

ignoring the “shoulders” of the

building.

13) You are trying to make a hole in-one on the miniature golf green shown. Imagine that a

coordinate plane is placed over the golf green.

The golf ball is at (2.5, 2) and the

hole is at (9.5, 2). You are going to

bank the ball off the side wall of the

green at (6, 8). Write an equation for

the path of the ball and determine if

you make your shot.

Graphing Simple rational function

A rational function is a function of the form )(

)()(f

xq

xpx where 𝑝(𝑥) and 𝑞(𝑥) are polynomials

and 𝑞(𝑥) ≠ 0. In this lesson you will learn to graph rational functions for which p(x) and q(x) are

linear. For instance, consider the rational function: x

y 1 . The graph of this function is called a

hyperbola and is shown below. Notice the following properties.

• The x-axis is a horizontal asymptote. • The y-axis is a vertical asymptote.

• The domain and range are all nonzero real numbers.



The graph has two symmetrical parts called branches. For each point (x, y) on one branch, there

is a corresponding point (-x , -y) on the other branch

Activity Graph each function. xxxx

yyyy 2132 ,,,

a) Use the graphs to describe how the sign of a affects the graph of x

ay .

b) Use the graphs to describe how |a| affects the graph of x

ay

All rational functions of the form kyhx

a

have graphs that are hyperbolas with asymptotes at

x = h and y = k. To draw the graph, plot a couple of points on each side of the vertical asymptote.

Then draw the two branches of the hyperbola that approach the asymptotes and pass through the

plotted points..

Example 1 Graph 13

2

xy . State the

domain and range

Solution

Draw the asymptotes x =-3 and y = -1.

Plot two points to the left of the vertical

asymptote, Eg (-4, 1) & (-5, 0), and two

points to the right, such as (-1, -2) and (0,

35 ) Use the asymptotes and plotted points

to draw the branches of the hyperbola. The

domain is all real numbers except -3, and the

range is all real numbers except -1. Note All

rational functions of the form dcx

baxy

also

have graphs that are hyperbolas.

The vertical asymptote occurs at the x-value that makes the denominator zero. The horizontal asymptote is the line cay

Example 2 Graph 42

1

x

xy State the domain and range.

Solution

Draw the asymptotes. Solve 2𝑥 − 4 = 0 for

x to find the vertical asymptote x = 2. The

horizontal asymptote is the line 21

cay _.

Plot two points to the left of the vertical

asymptote, such as (0, 41 ) and (1, -1), and

2 points to the right, such as (3, 2) & (4, 45 )

Use the asymptotes and plotted points to

draw the branches of the hyperbola.

X y

x y

-4 41 4 4

1

-3 31 3 3

1

-2 21 2 2

1

-1 -1 1 1

21 -2 2

1 2



The domain is all real numbers except 2, and the range is all real numbers except 21

Example 3 For a fundraising project, your math club is publishing a fractal art calendar. The

cost of the digital images and the permission to use them is $850. In addition to these “one-time”

charges, the unit cost of printing each calendar is $3.25.

a) Write a model that gives the average cost per calendar as a function of the number of

calendars printed.

b) Graph the model and use the graph to estimate the number of calendars you need to print

before the average cost drops to $5 per calendar.

c) Describe what happens to the average cost as the number of calendars printed increases.

Solution

a) The average cost (A) is the total cost of making the calendars divided by the number of

calendars printed

Let x be number of calendars printed then, One-time charges =$850 and the

Unit cost=$3.25 per calendar x

xA

3.25 850

b) The graph of the model is shown at the right. The A-axis is the vertical asymptote and the

line A = 3.25 is the horizontal asymptote. The domain is 𝑥 > 0 and the range is 𝐴 > 3.25When A=5, x is about 500. So, you need

to print about 500 calendars before the

average cost drops to $5 per calendar.

c) As the number of calendars printed

increases, the average cost per calendar

gets closer and closer to $3.25. For

instance, when x = 5000 the average cost

is $3.42, and when x = 10,000 the

average cost is $3.34.

Exercise 1) If the graph of a rational function is a hyperbola with the x-axis and the y-axis as asymptotes,

what is the domain of the function? What is the range?

2) Explain why the graph shown below is not the graph of 73

6 x

y

3) Identify the horizontal and vertical

asymptotes of the graph of the function

a) 43

2

xy b)

4

32

x

xy

c) 3

3

x

xy d)

42

5

x

xy

e) 108

3

xy f) 5

6

4

xy

4) Look back at Example 3 on page 542. Suppose you decide to generate your own fractals on a

computer to save money. The cost for the software (a “one-time” cost) is $125. Write a

model that gives the average cost per calendar as a function of the number of calendars



printed. Graph the model and use the graph to estimate the number of calendars you need to

print before the average cost drops to $5 per calendar.

5) Identify the horizontal and vertical asymptotes of the graph of the function. Then state the

domain and range a) 23

xy b) 2

3

4

xy c) 2

3

2

xy d)

3

2

x

xy e)

13

22

x

xy f)

54

23

x

xy g) 17

43

22

xy h)

416

234

x

xy i) 19

6

4

xy

6) Match the function with its graph. a) 32

3

xy b) 3

2

3

xy c)

3

2

x

xy

7) Graph the function. State the domain and range a) x

y4

b) 13

3

xy c) 8

5

4

xy d)

37

1

xy e) 6

2

6

xy f)

4

5

xy g) 2

124

1

xy h)

xy

2

3 i) 5

63

4

xy

8) Graph the function. State the domain and range. a) 3

2

x

xy b)

34

x

xy c)

83

7

x

xy

d) 23

19

x

xy e)

124

310

x

xy f)

x

xy

4

25 g)

42

3

x

xy h)

15

7

x

xy i)

12

414

x

xy

9) Write a rational function that has the vertical asymptote 4x and the horizontal asymptote 3y .

10) You’ve paid $120 for a membership to a racquetball club. Court time is $5 per hour.

a) Write a model that represents your average cost per hour of court time as a function of

the number of hours played. Graph the model. What is an equation of the horizontal

asymptote and what does the asymptote represent?

b) Suppose that you can play racquetball at the YMCA for $9 per hour without being a

member. How many hours would you have to play at the racquetball club before your

average cost per hour of court time is less than $9?

11) Air temperature affects how long it takes sound to travel a given distance. The time it takes

for sound to travel one kilometer can be modeled by 3316.0

1000

Tt where t is the time (in

seconds) and T is the temperature (in degrees Celsius). You are 1 kilometer from a lightning

strike and it takes you exactly 3 seconds to hear the sound of thunder. Use a graph to find the

approximate air temperature. (Hint: Use tick marks that are 0.1 unit apart on the t-axis.)

12) When the source of a sound is moving relative to a stationary listener, the frequency ƒl (in

hertz) heard by the listener is different from the frequency ƒs (in hertz) of the sound at its

source. An equation for the frequency heard by the listener is r

740

f740f s

lwhere r is the

speed (in miles per hour) of the sound source relative to the listener.



a) The sound of an ambulance siren has a frequency of about 2000 hertz. You are standing

on the sidewalk as an ambulance approaches with its siren on. Write the frequency that

you hear as a function of the ambulance’s speed.

b) Graph the function from Exercise 47 for 0 ≤ r ≤ 60. What happens to the frequency you

hear as the value of r increases?

13) Economist Arthur Laffer argues that beyond a certain percent pm, increased taxes will

produce less government revenue. His theory is illustrated in the graph below.

a) Using Laffer’s theory, an economist

models the revenue generated by one

kind of tax by 110

800080

p

pR

where R is the government revenue

(in tens of millions of dollars) and p

is the percent tax rate (55 ≤ p ≤ 100).

Graph the model.

b) Use your graph from part a to find

the tax rate that yields $600 million

of revenue.

14) In what line(s) is the graph of xy 1 asymmetric? What does this symmetry tell you about the

inverse of the function xx 1)(f ?

15) Show algebraically that the function 105

3)(f

xx and the function

5

4710)(g

x

xx are

equivalent.

Graphing General Rational Functions In this section you will learn how to graph rational functions for which p(x) and q(x) may be

higher-degree Let p(x) and q(x) be polynomials with no common factors other than 1.

The graph of the rational function 01

2

2

1

1

01

2

2

1

1

...

...

)(

)()(f

bxbxbxbxb

axaxaxaxa

xq

xpx

n

n

n

n

n

n

m

nm

has the

following characteristics.

i) The x-intercepts of the graph of ƒ are the real zeros of p(x).

ii) The graph of ƒ has a vertical asymptote at each real zero of q(x).

iii) The graph of ƒ has at most one horizontal asymptote if;

• m < n, the line y = 0 is a horizontal asymptote.

• m = n, the line n

m

b

ay is a horizontal asymptote.

• m > n, the graph has no horizontal asymptote The graph’s end behavior is the same as

the graph of nm

b

axy

n

m

Example 1 Graph 1

42

x

y . State the domain and range.



Solution

The numerator has no zeros, so there is no x-

intercept. The denominator has no real

zeros, so there is no vertical asymptote. The

degree of the numerator is < the degree of

the denominator, so the line y = 0 is a

horizontal asymptote. The bell-shaped graph

passes through the points (-3, 0.4), (-1, 2),

(0, 4), (1, 2), and (3, 0.4). The domain is all

real numbers, and the range is 0 < y ≤ 4

Example 2 Graph 4

32

2

x

xy ..

Solution

The numerator has 0 as its only zero, so the graph has one x-intercept at (0, 0). The denominator

can be factored as (x + 2)(x-2), so the denominator has zeros -2 and 2. This implies that the lines

x = -2 and x = 2 are vertical asymptotes of the graph.

The degree of the numerator (2) is equal to the degree of the denominator (2), so the horizontal

asymptote is 3n

m

b

ay . To draw the graph, plot points between and beyond the vertical

asymptotes.

Example 3 Graph 4

322

x

xxy ..

Solution

The numerator can be factored as (x-3)(x + 1), so the x-intercepts of the graph are 3 and -1. The

only zero of the denominator is -4, so the only vertical asymptote is x = -4. The degree of the

numerator (2) is greater than the degree of the denominator (1), so there is no horizontal

asymptote and the end behavior of the graph of ƒ is the same as the end behavior of the graph of

xxy 12. To draw the graph, plot points to the left and right of the vertical asymptote.

See graph next page.

Manufacturers often want to package their products in a way that uses the least amount of

packaging material. Finding the most efficient packaging sometimes involves finding a local

minimum of a rational function.



Example 4 A standard beverage can has a volume of 355 cubic centimeters.

a) Find the dimensions of the can that has this volume and uses the least amount of material

possible.

b) Compare your result with the dimensions of an actual beverage can, which has a radius of 3.1

centimeters and a height of 11.8 centimeters.

Solution

a) The volume must be 355 cubic centimeters, so you can write the height h of each possible

can in terms of its radius r ie 2

2 355355

rhhrv

Using the least amount of material is equivalent to having a minimum surface area S. You

can find the minimum surface area by writing its formula in terms of a single variable and

graphing the result.

rr

rrrhrrS

7102

35522)(2 2

2

2

Graph the function for the surface area S.

Using a graphing calculator, the

Minimum feature gives the minimum

value of S about 278, which occurs when

r ≈ 3.84 centimeters and

66.7)84.3(

3552

h cm

b) An actual beverage can is taller and narrower than the can with minimal surface area—

probably to make it easier to hold the can in one hand.



Exercise

1) Let )(

)()(f

xq

xpx where p(x) and q(x) are polynomials with no common factors other than 1.

Describe how to find the x-intercepts and the vertical asymptotes of the graph of ƒ.

2) Let )(

)()(f

xq

xpx where p(x) and q(x) are both cubic polynomials with no common factors

other than 1. The leading coefficient of p(x) is 8 and the leading coefficient of q(x) is 2.

Describe the end behavior of the graph of ƒ.

3) Graph the function.

a) 3

62

x

y b)1

42

x

xy c)

2

72

2

x

xy d)

72

3

x

xy e)

1

22

2

x

xy f)

162

x

xy

4) The can for a popular brand of soup has a volume of about 342 cubic centimeters. Find the

dimensions of the can with this volume that uses the least metal possible. Compare these

dimensions with the dimensions of the actual can, which has a radius of 3.3 centimeters and a

height of 10 centimeters

5) Identify the x-intercepts and vertical asymptotes of the graph of the function.

a) 92

x

xy b)

1

32 2

x

xy c)

16

5922

2

x

xxy d)

3

32

x

xy

e)

6

542

x

xxy

f) 8

101332

2

x

xxy g)

92

822

x

xy h)

1

22

2

x

xxy i)

x

xy

2

273

6) Match the function with its graph a) 2

72

2

x

xy b)

4

82

xy c)

42

3

x

xy

7) Give an example of a rational function whose graph has two vertical asymptotes: x=2 and x=7.

8) Graph the function. a) 2

32 2

x

xy b)

8

242

xy c)

1

142

x

xy d)

45

1322

2

xx

xxy

e) 63

2 2

x

xy f)

324

133

3

x

xy g)

27

12113

2

x

xxy h)

145

42

xx

xy

i) 45

1322

2

xx

xxy j)

16

42

2

x

xy k)

x

xxy

2

2092 l)

xx

xxy

4

152

23

9) Match the function with its graph. a) 27

33

x

y b) 92

3

x

xy c)

12

42

x

xxy



10) Suppose you want to make a rectangular garden with an area of 200 square feet. You want to

use the side of your house for one side of the garden and use fencing for the other three sides.

Find the dimensions of the garden that minimize the length of fencing needed.

11) The total energy expenditure E (in joules per gram mass per kilometer) of a typical

budgerigar parakeet can be modeled by v

vvE

75.4717.2131.0 2 where v is the speed of

the bird (in kilometers per hour). Graph the model. What speed minimizes a budgerigar’s

energy expenditure?

12) The mean temperature T (in degrees Celsius) of the Atlantic Ocean between latitudes 40°N

and 40°S can be modeled by 10007403

20000178002

dd

dT where d is the depth (in meters). Graph

the model. Use your graph to estimate the depth at which the mean temperature is 4°C.

13) For 1985 to 1995, the average daily cost per patient C (in dollars) at community hospitals in

the United States can be modeled by 10001225

462048204072

xx

xC where x is the number of years

since 1985. Graph the model. Would you use this model to predict patient costs in 2005?

14) For 1980 to 1995, the total revenue R (in billions of dollars) from parking and automotive

service and repair in the United States can be modeled by 1000268257..0

30432641642723

2

xxx

xxR

where x is the number of years since 1980. Graph the model. In what year was the total

revenue approximately $75 billion?

15) The acceleration due to gravity g§ (in meters per second squared) of a falling object at the

moment it is dropped is given by the function 1372

14

1007.4)1028.1(

1099.3'

hhg where h is

the object’s altitude (in meters) above sea level.

a) Graph the function.

b) What is the acceleration due to gravity for an object dropped at an altitude of

c) 5000 kilometers?

d) Describe what happens to g’ as h increases.

16) For 1980 to 1995, the total revenue R (in billions of dollars) from hotels and motels in the

United States can be modeled by x

xR

01.01

88.2676.2

where x is the number of years since

1980. Graph the model. Use your graph to find the year in which the total revenue from

hotels and motels was approximately $68 billion.

17) Consider the following two functions )5)(3(

)3)(2()(g and

)5)(3(

)2)(1()(f

xx

xxx

xx

xxx Notice

that the numerator and denominator of g have a common factor of x- 3.



a) Make a table of values for each function from x = 2.95 to x = 3.05 in increments of 0.01.

b) Use your table of values to graph each function for 2.95 ≤ x ≤ 3.05.

c) As x approaches 3, what happens to the graph of ƒ(x)? to the graph of g(x)?

Graphing Exponential and Power Functions

An Exponential Functions involves the expression 𝑏𝑥 where b is a positive number other than 1

To see the basic shape of the graph of an exponential function such as ƒ(𝑥) = 2𝑥 , you can make

a table of values and plot points, as shown

below

x -3 -2 -1 0 1 2 3

𝑦 = 2𝑥 0.125 0.25 0.5 1 2 4 8

Notice the end behavior of the graph. As

x )(f x , which means that the

graph moves up to the right. As

0)(f, xx which means that the

graph has the line y = 0 as an asymptote. An

asymptote is a line that a graph approaches

as you move away from the origin.

Activity Graph y = 0.2(2)𝑥 and y = 5(2)𝑥. Compare the graphs with the graph of y = 2𝑥.

Also Graph y = −0.2(2)𝑥and y = −5(2)𝑥 and compare the graphs with the graph of y = 2𝑥.

Describe the effect of a on the graph of y = 𝑎(2)𝑥 when a is positive and when a is negative

In the activity you may have observed the following about the graph of y = 2𝑥: The graph

passes through the point (0, a). That is, the y-intercept is a and the x-axis is an asymptote of the

graph.. The domain is all real numbers and the range is y > 0 if a > 0 and y < 0 if a < 0. These

characteristics of the graph of y = 2𝑥 applies for any graph of the form 𝑦 = 𝑎𝑏𝑥 . If 𝑎 >

0 and 𝑏 > 1, the function 𝑦 = 𝑎𝑏𝑥 is an exponential growth function

Example Graph the function; a) y = 0.5(3)𝑥 b) y = −1.5𝑥

Solution

Plot (0,0.5) and (1,1.5) . Then, from left to

right, draw a curve that begins just above the

x-axis, passes through the two points, and

moves up to the right. .

Plot (0, −1) and (1, −1.5) then, from left to

right, draw a curve that begins just below

the x-axis, passes through the two points,

and moves down to the right.



To graph a general exponential function, kaby hx , begin by sketching the graph of = 𝑎𝑏𝑥 .

Then translate the graph horizontally by h units and vertically by k units.

Example Graph 423 1 xy State the

domain and range.

Solution

Begin by lightly sketching the graph xy 23 , which passes through (0, 3) and

(1, 6). Then translate the graph 1 unit to the

right and 4 units down. Notice that the graph

passes through (1, -1) and (2, 2). The

graph’s asymptote is the line 4y . The

domain is all real numbers, and the range is

4y .

Modelling with an Exponential Function

Just as two points determine a line, two points also determine an exponential curve.

Example Write an exponential function 𝑦 = 𝑎𝑏𝑥 whose graph passes through (1, 6) & (3, 24).

Solution

Substitute the two given points into 𝑦 = 𝑎𝑏𝑥 to obtain two equations in 𝑎 and 𝑏.

Thus 6 = 𝑎𝑏1 and 24 = 𝑎𝑏3. To solve the system, divide equation iI by equation I to get 𝑏2 =

4 , 0r 𝑏 = ±. Using 𝑏 = 2, you then have 𝑎 = 3. So, = 3 (2)𝑥 . .

When you are given more than two points, you can decide whether an exponential model fits the

points by plotting the natural logarithms of the y-values against the x-values. If the new points

(x, ln y) fit a linear pattern, then the original points (x, y) fit an exponential pattern.

Example 2 The table gives the number y (in millions) of cell-phone subscribers from 1988 to

1997 where t is the number of years since 1987.

t 1 2 3 4 5 6 7 8 9 10

y 1.6 2.7 4.4 6.4 8.9 13.1 19.3 28.2 38.2 48.7

a) Draw a scatter plot of ln y versus x. Is an exponential model a good fit for the original data?

b) Find an exponential model for the original data.

Solution

Use a calculator to create a new table of values.

t 1 2 3 4 5 6 7 8 9 10

𝑙𝑛 𝑦 0.47 0.99 1.48 1.86 2.19 2.57 2.96 3.34 3.64 3.89



Then plot the new points as shown. The points lie close to a line, so an exponential model should

be a good fit for the original data.

To find an exponential model 𝑦 = 𝑎𝑏𝑥 ,

choose two points on the line, such as (2,

0.99) and (9, 3.64). Use these points to find

an equation of the line. Then solve for y.

teeey

y

)46.1(30.1

0.2339t37.0ln

9t37.00.2330.2339t37.0

A graphing calculator that performs

exponential regression does essentially what

is done in Example 2, but uses all of the

original data.

Power function

A power function has the form 𝑦 = 𝑎𝑥𝑏 . Because there are only two constants (a and b), only

two points are needed to determine a power curve through the points.

Example 3 Write a power function 𝑦 = 𝑎𝑥𝑏 whose graph passes through (2, 5) and (6, 9).

Solution

Substitute the coordinates of the 2 given points into 𝑦 = 𝑎𝑥𝑏 to obtain two equations in a and b

5 = 𝑎2𝑏 and 9 = 𝑎6𝑏 To solve the system, solve for a in the first equation to get 𝑎 = 5(2)−𝑏,

then substitute into the second equation 0.5353log

8.1log8.1396)2(5 bbbb

Using b = 0.535, you then have𝑎 = 5(2)−0.535 ≈ 3.45., so 0.5353.45xy

When you are given more than two points, you can decide whether a power model fits the points

by plotting the natural logarithms of the y-values against the natural logarithms of the x-values.

If the new points (ln x, ln y) fit a linear pattern, then the original points (x, y) fit a power pattern.



Example 4 The table gives the mean distance x from the sun (in astronomical units) and the

period y (in Earth years) of the six planets closest to the sun.

Planet Mercury Venus Earth Mass Jupitor Sartun

X 0.387 0.723 1.000 1.524 5.203 9.539

Y 0.214 0.615 1.000 1.881 11.852 29.458

a) Draw a scatter plot of ln y versus ln x. Is a power model a good fit for the original data?

b) Find a power model for the original data.

c) Use the model to estimate the period of Neptune, which has a mean distance from the sun

of 30.043 astronomical units.

Solution

a) Use a calculator to create a new table of values.

𝑙𝑛 𝑥 -0.949 -0.324 0.000 0.421 1.629 2.255

𝑛 𝑦 -1.423 -0.486 0.000 0.632 2.473 3.383

Then plot the new points, as shown at the

right. The points lie close to a line, so a

power model should be a good fit for the

original data.

b) To find a power model y = axb , choose

two points on the line, such as (0, 0) and

(2.255 , 3.383). Use these points to find

an equation of the line. Then solve for y.

5.1ln5.1ln xyxy

c) Substituting 30.043 for x in the model

gives 𝑦 = (30.043)1.5 ≈ 165 years for

the period of Neptune

Exercise

1) Write an exponential function of the form xaby whose graph passes through the given

points. a) (1, 3), (2, 36) b) (2, 18), (3, 108) c) (1, 4), (3, 16) d) (2, 3.5), (1, 5.2)

e) (-3 , 3), (4, 6561) f) ),1(,),4(2

21

81

112 g)(3 , 13.5) , (5 , 30.375) h) ),4(,),2(4

625

4

25

2) Write a power function of the form baxy whose graph passes through; a) (3, 27), (9, 243)

b). (2, 6), (4, 48) c) ),2(,),4(2

1

5

3 d). (4.5, 9.2), (1, 6.4) e) (2.2 , 10.4), (8.8, 20.3) f)

(2.9 , 9.4) , (7.3 , 12.8) g)(2.71 , 6.42) , (13.55 , 29.79)

3) Using xaby and

baxy , take the natural logarithm of both sides of each equation. What is

the slope and y-intercept of the line relating x and ln y for xaby ? of the line relating ln x

and ln y for baxy ?

4) Use the table of values to draw a scatter plot of ln y versus x. Then find an exponential model

for the data.

X 1 2 3 4 5 6 7 8

Y 14 28 56 112 224 448 896 1792



X 1 2 3 4 5 6 7 8

Y 10.2 30.5 43.4 64.2 89.7 120.6 210.4 302.5

X 2 4 6 8 10 12 14 16

Y 12.8 20.48 32.77 52.43 83.89 134.22 214.75 343.6

5) Use the table of values to draw a scatter plot of ln y versus ln x. Then find a power model for

the data..

X 1 2 3 4 5 6 7

Y 0.78 7.37 27.41 69.63 143.47 259.00 426.79

X 1 2 3 4 5 6 7

Y 1.2 5.4 9.8 14.3 25.6 41.2 65.8

X 2 4 6 8 10 12 14

Y 1.89 1.44 1.22 1.09 1.00 0.93 0.87

6) Write y as a function of x 5.424.0log xy b) 8.02.0log xy c) xy 88.012.0log

d) 4ln xy e) 548.0log48.0log xy f) 7.4ln3.2ln xy g)

98.038.2ln xy h) xy log751.348.1log i) xy 5.15.2ln j)

xy log4.3log2.1 k) xy loglog6

5

2

1 l) 8

3

4

1

8

1 ln4ln2 xy

7) Find equations of the line, the exponential curve, and the power curve that each pass through

the points (1, 3) and (2, 12). Graph the equations in the same coordinate plane and then

describe what happens when the equations are used as models to predict y-values for x-values

greater than 2.

8) You have just created your own Web site. You are keeping track of the number of hits (the

number of visits to the site). The table shows the number y of hits in each of the first 10

months where x is the month number

X 1 2 3 4 5 6 7 8 9 10

Y 22 39 70 126 227 408 735 1322 2380 4285

a) Find an exponential model for the data.

b) According to your model, how many hits do you expect in the twelfth month?

c) According to your model, how many hits would there be in the thirty-fourth month?

What is wrong with this number?

9) The table shows the number C of cranes in Izumi, Japan, from 1950 to 1990 where t

represents the number of years since 1950. Source: Yamashina Institute of Ornithology

T 0 5 10 15 20 25 30 35 40

C 293 299 438 1573 2336 3649 5602 7610 9959

a) Draw a scatter plot of ln C vs t. Is an exponential model a good fit for the original data?

b) Find an exponential model for the original data. Estimate the number of cranes in Izumi,

Japan, in the year 2000.

10) The table shows the cumulative number s of different stamps in the United States from 1889

to 1989 where t represents the number of years since 1889.

T 0 10 20 30 40 50 60 70 80 90 100

S 218 293 374 541 681 858 986 1138 1318 1794 2438

a) Draw a scatter plot of ln s vs t. Is an exponential model a good fit for the original data?



b) Find an exponential model for the original data. Estimate the cumulative number of

stamps in the United States in the year 2000.

11) The table shows the population y (in

millions) and the population rank x for

nine cities in Argentina in 1991.

a) Draw a scatter plot of ln y versus ln

x. Is a power model a good fit for the

original data?

b) Find a power model for the original

data. Estimate the population of the

city Vicente Lopez, which has a

population rank of 20.

City Rank

x Population

(millions), y

Cordoba 2 1.21

Rosario 3 1.12

La Matanza 4 1.11

Mendoza 5 0.77

La Plata 6 0.64

Moron 7 0.64

San Miguel de

Tucuman

8 0.62

Lomas de Zamoras 9 0.57

Mar de Plata 10 0.51

12) The femur is a large bone found in the leg or hind limb of an animal. Scientists use the

circumference of an animal’s femur to estimate the animal’s weight

The table at the right shows the femur

circumference C (in millimeters) and the

weight W (in kilograms) of several

animals.

a) Draw two scatter plots, one of ln W

versus C and another of ln W versus

ln C.

b) Looking at your scatter plots, tell

which type of model you think is a

better fit for the original data.

Explain your reasoning.

Animal C (mm) W (kg)

Meadow mouse 5.5 0.047

Guinea pig 15 0.385

Otter 28 9.68

Cheetah 68.7 38

Warthog 72 90.5

Nyala 97 134.5

Grizzly bear 106.5 256

Kudu 135 301

Giraffe 173 710

c) Using your answer from part (b), find a model for the original data.

d) The table at the right shows the

femur circumference C (in

millimeters) of four animals. Use the

model you found in part (c) to

estimate the weight of each animal.

Animal C (mm)

Raccoon 28

Cougar 60.25

Bison 167.5

Hippopotamus 208

13) The table shows the atomic number x and the melting point y (in degrees Celsius) for the

alkali metals.

Alkali metal Lithium Sodium Potassium Rubidium Cesium

Atomic number, x 3 11 19 37 55

Melting point, y 180.5 97.8 63.7 38.9 28.5

a) Draw a scatter plot of ln y versus ln x. Is a power model a good fit for the original data?

b) Find a power model for the original data.

c) One of the alkali metals, francium, is not shown in the table. It has an atomic number of

87. Using your model, predict the melting point of francium.



Graphing and Evaluating Polynomial Functions

A polynomial function is a function of the form 01

2

2

1

1 .......)(f axaxaxaxax n

n

n

n

where 0na , the exponents are all whole numbers, and the coefficients are all real numbers. For

this polynomial function, na is the leading coefficient, 0a is the constant term, and n is the

degree. You are already familiar with some types of polynomial functions. Eg, the linear

function f(𝑥) = 3𝑥 + 2 is a polynomial function of degree 1. The quadratic function f(𝑥) =

𝑥2 + 3𝑥 + 2 is a polynomial function of degree 2.

Evaluating Polynomial Functions

One way to evaluate a polynomial function is to use direct substitution. For instance,

7532.)(f 24 xxxx can be evaluated when x = 3 as 987)3(5)3(3)3(2.)3(f 24

Another way of evaluating a polynomial function is to use synthetic substitution. This method

involves fewer operations than direct substitution.

Example 1 The time t seconds it takes a camera battery to recharge after flashing n times can

be modeled by 3.525.00034.0000015.0. 23 nnnt Find the recharge time after 100 flashes.

Solution

Substitute n=100 to get 3.113.5)100(25.0)100(0034.0)100(000015.0. 23 t

The recharge time is about 11 seconds.

Example 2 Use synthetic substitution to evaluate 8452.)(f 24 xxxx when 3x .

Solution

Step 1 Write the coefficients of f(x) in order of decreasing exponent. write the value at which

f(x) is being evaluated to the left as shown below

Step 2 Bring down the leading coefficient.

Multiply the leading coefficient by the x

value and write the product under the 2nd

coefficient. Then add.

Step 3 Multiply the previous sum by the x

value and write the product under the 3rd

coefficient. Then add. Repeat this procedure

for all the remaining coefficients. The final

sum is the value of f(x) at the given x value.

Synthetic substitution gives f(3)=23 which

agrees with the direct substitution method.

End Behavior of Polynomial Functions

The end behavior of a polynomial’s graph is the behavior of the graph as x approaches positive

infinity or negative infinity. The expression x is read as “x approaches positive infinity.”



The end behavior of a polynomial function is determined by the function’s degree and the sign of

the leading coefficient. The end behavior of the graph of 01

1

1 .......)(f axaxaxax n

n

n

n

:

• For 0na and n even, )(f x as x and )(f x as x

• For 0na and n odd, )(f x as x and )(f x as x

• For 0na and n even, )(f x as x and )(f x as x

• For 0na and n odd, )(f x as x and )(f x as x ‡

Degree; odd leading coefficient positive Degree; odd leading coefficient negative

Degree; Even leading coefficient positive Degree; Even leading coefficient negative

Example State whether the degree of the polynomial is even or odd. Also state whether the

leading coefficient is positive or negative.

Solution



From the graph )(f x as x so

the degree is even and leading coefficient is

negative.

Graphing Polynomial Functions

To graph a polynomial function, first plot points to determine the shape of the graph’s middle

portion. Then use the knowledge of the end behavior to sketch the ends of the graph.

Example 1 Graph (a) 14.)(f 23 xxxx b) xxxxx 422.)(f 234

Solution

a) To graph the function, make a table of values and plot the corresponding points.

X -3 -2 -1 0 1 2 3

f(𝑥) -7 3 3 -1 -3 3 23

Connect the points with a smooth curve and check the end behavior. The degree is odd and

the leading coefficient is positive, so )(f x as x and )(f x as x . s

b) To graph the function, make a table of values and plot the corresponding points. Connect the

points with a smooth curve and check the end behavior.

X -3 -2 -1 0 1 2 3

f(𝑥) -21 0 -1 0 3 -16 -105

The degree is even and the leading coefficient is negative, so )(f x as x and

)(f x as x

(a) (b)

Example 2 A rainbow trout can grow up to 40 inches in length. The weight y (in pounds) of a

rainbow trout is related to its length x (in inches) according to the model = 0.0005𝑥3 . Graph the

model. Use your graph to estimate the length of a 10 pound rainbow trout.

Solution Make a table of values. The model makes sense only for positive values of x.

X 0 5 10 15 20 25 30 35 40

f(𝑥) 0 0.0625 0.5 1.69 4 7.81 13.5 21.4 32

Plot the points and connect them with a

smooth curve, as shown at the right. Notice

that the leading coefficient of the model is

positive and the degree is odd, so the graph

rises to the right. Read the graph backwards

to see that x ≈ 27 when y = 10. A 10 pound

trout is approximately 27 inches long.



Example 3 The energy E in foot-pounds) in each square foot of a wave is modelled b y 𝐸 =0.0029𝑠4 where s is the wind speed (in knots). Graph the model hence estimate the wind speed

need to generate a wave with 1000 foot-pounds of energy per square foot

Solution

Make a table of values. The model only deals with positive values of s

S 0 10 20 30 40

E 0 29 464 2349 7424

Plot the points and join them with a smooth

curve. Because the leading coefficient is

positive and the degree is even the graph

raises upward. From the graph, when

E=1000 the wind speed 24s knots.

Exercise

1) Identify the degree, type, leading coefficient and the constant term for the polynomial

function f(𝑥) = 6 + 2𝑥2 − 5𝑥4

2) state whether the function is a polynomial if so write it in standard form then state the degree,

type, leading coefficient and the constant term term a) 28)(f xx b) 386)(f 4 xxx

c) 6)(g 4 xx d) 1510)(h 23 xxx e) 103)(h 3

2

5 xxx f) xxxx 223 48)(g

3) Use direct substitution to evaluate the polynomial function for the given value of x term

a) 1,151025)(f 23 xxxxx b) 2;358)(f 324 xxxxxx

c) 3;24)(g 53 xxxx d) 5;20256)(h 3 xxxx

e) 4;10)(h 3

4

34

2

1 xxxxx f) 2;51064)(g 235 xxxxxx

4) Use synthetic substitution to evaluate the polynomial function for the given value of x

a) 3;16825)(f 23 xxxxx b) 2;9563128)(f 2234 xxxxxxx

c) 6;3578)(g 23 xxxxx d) 4;35148)(h 3 xxxx

e) 2;23813)(h 43 xxxxx f) 3;27106)(g 35 xxxx

g) 3;7114)(h 32 xxxx h) 4;203)(h 4 xxxx

5) Describe the degree and the leading coefficient for the polynomial function graphed below

6) From 1992 to 2003, the number of people in the United States who participated in

skateboarding can be modelled by 5552.062.014.00076.0)(S 234 ttttt where S is

is in millions and t is the number of years since 1992. Graph the model hence estimate the

first year that the number of skateboarding participants was greater than 8 million.



7) The weight of an ideal round-cut

diamond can be modelled by 𝑤 =0.0071𝑑3 − 0.090𝑑2 + 0.48𝑑where w

is the diamond’s weight (in carats) and d

is the diameter (in mm). according to the

model what is the weight of a diamond

with a diameter of 15mm

8) A cubic polynomial function f has leading coefficient 2 and constant term -5. If f(1)=0 and

f(2)=3, find f(-5)

9) From 1987 to 2003, the number of indoor movie screens M in the United States can be

modelled by 32 11267592600,21)(M tttt where t is the number of years since 1987.

a) State the degree and type of function

b) Make a table of values for the function hence graph it

10) From 1992 to 2003, the number of people in the United States who participated in

snowboarding can be modelled by 2.1037.084.0021.00013.0)(S 234 ttttt where S is

the number of participants (in millions) and t is the number of years since 1992. Graph the

model hence estimate the first year that the number of snowboarding participants was greater

than 2 million.

11) From 1980 to 2002, the number of quarterly periodicals P published in the United States can

be modelled by 4502398.8624.0138.0)(P 234 ttttt where t is the number of years

since 1980.

a) Describe the end behavior of the graph of the model.

b) Graph the model on the domain 220 t

c) Use your graph to estimate the number of quarterly periodicals in the year 2010. Is it

appropriate to make these predictions?

12) The weights of Sarus crane chicks S and hooded crane chicks H (both in grams) during the

10 days for following hatching can be modelled by the functions

1366.1449.3122.0 23 tttS and 1246.20471.3115.0 23 tttH where t is the

number of days after hatching.

a) According to the models what is the difference in weight between 5-day old Sarus crane

chicks and hooded crane chicks?

b) Graph the two models

c) A biologist finds that the weight of crane chick 3 days after hatching is 130 games. What

species of crane is the chick more likely to be? Explain how you arrived at that answer.

13) The weight y (in pounds) of a rainbow trout can be modelled by 3000304.0 xy where x is

the length of the trout (in inches)

a) Write a function that relates the weight y and length x of a rainbow trout if y is measured

in kilograms and x is measured in centimeters. Use the fact that 1kg 20.2 pounds and

1cm 394.0 inches

b) Graph the original function and the function from part (a) on the same coordinate plane.

What type of transformation can you apply to the graph 3000304.0 xy to produce the

graph from part (a)?



Finding Rational Zeros of polynomials

The polynomial function 10534120.64)(f 23 xxxx has 8

7

4

5

2

3 and as its zeros.

Notice that the numerators of these zeros (-3, -5, and 7) are factors of the constant

term, -105. Also notice that the denominators (2, 4 and 8) are factors of the leading coefficient,

64. These observations are generalized by the rational zero theorem.

The Rational Zero Theorem

If 0

2

2

1

1 .....)(f axaxaxax n

n

n

n

has integer coefficients, then every rational zero of ƒ

has the following form:na

a

q

p

t coefficien leading offactor

ermconstant t offactor 0

Example 1 list all the rational zeros of;

a) 20118.)(f 23 xxxx b) 1093..4)(f 234 xxxxx

Solution

a) Factors of the constant term; 20,10,5,4,2,1 Factors of the leading coefficient; 1

Possible rational zeros; 1

20,

1

10,

1

5,

1

4,

1

2,

1

1x

b) Factors of the constant term; 10,5,2,1 Factors of the leading coefficient; 4,2,1

Possible rational zeros; 4

5,

4

1,

2

5,

2

1,

1

10,

1

5,

1

2,

1

1x

Example 2 Find the rational zeros o 12112.)(f 23 xxxx

Solution

List the possible rational zeros. The leading coefficient is 1 and the constant term is º12. So, the

possible rational zeros are: 1

12

1

6

1

43

1

2

1

1x

Test x = 1: 20121121)1(f Test x =-1: 0121121)1(f

We can also Test these zeros using synthetic division as shown below

Since -1 is a zero of f, you can write )12)(1(121142.)(f 223 xxxxxxx

Factor the trinomial and use the factor theorem. )4)(3)(1.()12)(1.()(f 2 xxxxxxx

The zeros of f are -1, 3, and -4

In Example 1, the leading coefficient is 1. When the leading coefficient is not 1, the list of

possible rational zeros can increase dramatically. In such cases the search can be shortened by

sketching the function’s graph—either by hand or by using a graphing calculator

Example 3 Find all real zeros of 12529310)(f 234 xxxxx



Solution

The possible rational zeros of f are;

10

3

10

1

5

12

5

6

5

4

5

3

5

2

5

1

2

3

2

1

1

12

1

6

1

4

1

3

1

2

1

1x

Choose values to check. With so many

possibilities, it is worth your time to sketch

the graph of the function. From the graph, it

appears that some reasonable choices are

2

3 and

5

4

5

3

2

3 xxxx

Check these values using direct substitution

Test 2

3x : 012520310f2

32

2

33

2

34

2

3

2

3 Factor out a binomial 23x

)495)(32()821810)((12520310.)(f 2323

2

3234 xxxxxxxxxxxxx

Repeat the steps above for 495)(g 23 xxxx any zero of g will also be a zero of ƒ. The

possible rational zeros of g are 5

4

5

2

5

1

1

4

1

2

1

1x .

The graph of f shows that 5

4 may be a zero. Now 0495g

5

42

5

43

5

4

5

4 Factor out a

binomial 5

4x So )1)(45)(32()555)()(32()(f 22

5

4 xxxxxxxxx

Find the remaining zeros of f by using the quadratic formula to solve 012 xx

The real zeros of f are; 2

51 and

2

51

5

4,

2

3

xxxx

Example 4 You are designing a candle-making kit. Each kit will contain 25 cubic inches of

candle wax and a mold for making a model of the pyramid-shaped building at the Louvre

Museum in Paris, France. You want the height of the candle to be 2 inches less than the length of

each side of the candle’s square base. What should the dimensions of your candle mold be?

Solution

The volume is BhV3

1 where B is the area of the base and h is the height. Let the side of

square base be x inches then height is x-2 and since V=25 then

0752275)2(25 23232

3

1 xxxxxx

The possible rational solutions are 752515531 x . Use the possible solutions.

Note that in this case, it makes sense to test only positive x-values

So x = 5 is a solution. The other two solutions, which satisfies 01532 xx are

2

513

x and can be discarded because they are imaginary.

The base of the candle mold should be 5 inches by 5 inches. The height of the mold should be

5 − 2 = 3 inches.



Example 5 Some ice sculptures are

modelled by filling a mold with water and

then freezing it. You are making such an ice

sculpture for a school dance. It is toi be

shaped like a pyramid with height that is 1

foot greater than the length of each side of

the square base. The volume of the ice

sculpture is 4 cubic feet. What are the

dimensions of the mold?v

Solution

Write an equation for the volume; )1(4 height area) (base= volume 2

3

1

3

1 xx

Thus 012- 12 2323 xxxx . List the possible rational zeros

1

12

1

6

1

4

1

3

1

2

1

1x Test possible solutions only positive values make sense

2 x is a root of the function. Thus 0)63x)(2(12- 223 xxxx . The other 2 solutions

that satisfies 063x2 x are imaginary therefore they can be discarded. The only reasonable solution

is 2x the base of the mold is 2 feet by 2 feet and the height is 3 feet.

Modeling with Polynomial Functions

You know that two points determine a line and that three points determine a parabola. In

Example 1 you will see that four points determine the graph of a cubic function.

Example 1 Write the cubic function whose graph is shown at the right.

Solution

Use the three given x-intercepts to write the

following: ƒ(𝑥) = 𝑎(𝑥 + 3)(𝑥 − 2)(𝑥 − 5)

To find a, substitute the coordinates of the

fourth point. −15 = 𝑎(0 + 3)(0 − 2)(0 −

5), 𝑠𝑜 𝑎 = −0.5

Check the graph’s end behavior. The degree

of ƒ is odd and 0na , so )(f x as

x and )(f x as x .

To decide whether y-values for equally-

spaced x-values can be modeled by a

polynomial function, you can use finite

differences.

Example 2 The first three triangular numbers are shown at the right.



A formula for the nth triangular number is

ƒ(𝑛) = 1 2 (𝑛2 + 𝑛). Show that this

function has constant 2nd order differences. Solution

Write the first several triangular numbers. Find the first-order differences by subtracting

consecutive triangular numbers. Then find the second-order differences by subtracting

consecutive first-order differences.

Each 2nd order difference is 1, so the 2nd order difference is a constant.

Example 3 The profit P (in $”000,000) for a shoe manufacturer can be modeled by

xxP 4621 3 where x is the number of shoe produced (in millions). The company now

produces 1 million shoes and makes a profit if $25,000,000, but would like to cut back

production. What lesser number of shoe could the company produce and still make the same

profit?

Solution

Substitute 25 for P t0 get 0254621462125 33 xxxx . But 1x is one solution of

the equation 1 x is a factor of 0254621 3 xx . Use synthetic division to get the other

factors. So )252121)(1(254621 23 xxxxx .

Use quadratic formula to find that 7.0x is

the other positive solution.

The company could still make the same

profit producing about 700,000 shoe

Exercise

1) List the possible rational zeros of ƒ using the rational zero theorem.

a) f(𝑥) = 𝑥3 + 14𝑥2 + 41𝑥 − 56

b) f(𝑥) = 𝑥3 − 17𝑥2 + 54𝑥 + 72

c) f(𝑥) = 2𝑥3 + 7𝑥2 − 𝑥 + 30

d) f(𝑥) = 5𝑥4 + 12𝑥3 − 16𝑥2 + 10

e) f(𝑥) = 𝑥4 + 2𝑥2 − 24

f) f(𝑥) = 2𝑥3 + 5𝑥2 − 6𝑥 − 1

g) f(𝑥) = 2𝑥5 + 𝑥2 + 16

h) f(𝑥) = 2𝑥3 + 9𝑥2 − 53𝑥 − 60

i) f(𝑥) = 6𝑥4 − 3𝑥3 + 𝑥 + 10

j) f(𝑥) = 4𝑥3 + 5𝑥2 − 3

k) f(𝑥) = 8𝑥2 − 12𝑥 − 3

l) f(𝑥) = 3𝑥4 + 2𝑥3 − 𝑥 + 15

2) For each polynomial function, decide whether you can use the rational zero theorem to find

its zeros. Explain why or why not. (a) f(𝑥) = 6𝑥2 − 8𝑥 + 4 b) f(𝑥) = 0.3𝑥2 + 2𝑥 + 4.5

c) f(𝑥) =1

4𝑥2 − 𝑥 +

7

8

3) Find all the real zeros of the function

a. f(𝑥) = 𝑥3 − 3𝑥2 − 6𝑥 + 8

b. f(𝑥) = 𝑥3 + 4𝑥2 − 𝑥 − 4

c. f(𝑥) = 2𝑥3 − 5𝑥2 − 2𝑥 + 5

d. f(𝑥) = 2𝑥3 − 𝑥2 − 15𝑥 + 18

e. f(𝑥) = 𝑥3 + 4𝑥2 + 𝑥 − 6

f. f(𝑥) = 𝑥3 + 5𝑥2 − 𝑥 − 5



4) Suppose you have 18 cubic inches of wax and you want to make a candle in the shape of a

pyramid with a square base. If you want the height of the candle to be 3 inches greater than

the length of each side of the base, what should the dimensions of the candle be?

5) Use synthetic division to decide which of the values 1, -1, 2, and -2 are zeros of the function

a) f(𝑥) = 𝑥3 + 7𝑥2 − 4𝑥 − 28

b) f(𝑥) = 𝑥3 + 5𝑥2 + 2𝑥 − 8

c) f(𝑥) = 𝑥4 + 3𝑥3 − 7𝑥2 − 27𝑥 − 18

d) f(𝑥) = 2𝑥4 − 9𝑥3 + 8𝑥2 + 9𝑥 − 10

e) f(𝑥) = 𝑥4 + 𝑥3 + 3𝑥2 − 3𝑥 − 4

f) f(𝑥) = 3𝑥4 + 3𝑥2 + 2𝑥2 + 5𝑥 − 10

g) f(𝑥) = 𝑥3 − 3𝑥2 + 4𝑥 − 12

h) f(𝑥) = 𝑥3 + 𝑥2 − 11𝑥 + 10

i) f(𝑥) = 𝑥6 − 2𝑥4 − 11𝑥 + 12

j) f(𝑥) = 𝑥5 − 𝑥4 − 2𝑥3 − 𝑥2 + 𝑥 + 2

6) Find all the real zeros of the function.

a) f(𝑥) = 𝑥3 − 8𝑥2 − 23𝑥 + 30

b) f(𝑥) = 𝑥3 + 2𝑥2 − 11𝑥 − 12

c) f(𝑥) = 𝑥3 − 7𝑥2 + 2𝑥 + 40

d) f(𝑥) = 𝑥3 + 𝑥2 − 𝑥 − 2

e) f(𝑥) = 𝑥3 + 72 − 5𝑥2 − 18𝑥

f) f(𝑥) = 𝑥3 + 9𝑥2 − 4𝑥 − 36

g) f(𝑥) = 𝑥4 − 5𝑥3 + 7𝑥2 + 3𝑥 − 10

h) f(𝑥) = 𝑥4 + 𝑥3 + 𝑥2 − 9𝑥 − 10

i) f(𝑥) = 𝑥4 + 𝑥3 − 11𝑥2 − 9𝑥 + 18

j) f(𝑥) = 𝑥4 − 3𝑥3 + 6𝑥2 − 2𝑥 − 12

k) f(𝑥) = 𝑥5 + 𝑥4 − 9𝑥3 − 5𝑥2 − 36

l) f(𝑥) = 𝑥5 − 𝑥4 − 7𝑥3 + 11𝑥2 −

8𝑥 + 12

7) Find the real zeros of the function. Then match each function with its graph.

(a) f(𝑥) = 𝑥3 + 2𝑥2 − 𝑥 − 2 (b) f(𝑥) = 𝑥3 − 3𝑥 − 2 (c) f(𝑥) = 𝑥3 − 𝑥2 + 2

8) Use the graph to shorten the list of possible rational zeros then find all the real zeros of;

a) f(𝑥) = 4𝑥3 − 12𝑥2 − 𝑥 + 15

b) f(𝑥) = −3𝑥3 + 20𝑥2 − 36𝑥 + 16

9) Find all the real zeros of the function.

a) f(𝑥) = 2𝑥3 + 4𝑥2 − 2𝑥 − 4

b) f(𝑥) = 2𝑥3 − 5𝑥2 − 14𝑥 + 8

c) f(𝑥) = 2𝑥3 − 5𝑥2 − 𝑥 + 6

d) f(𝑥) = 2𝑥3 + 𝑥2 − 50𝑥 − 25

e) f(𝑥) = 2𝑥3 − 𝑥2 − 32𝑥 + 16

f) f(𝑥) = 3𝑥3 + 12𝑥2 + 3𝑥 − 18

g) f(𝑥) = 2𝑥4 + 3𝑥3 − 3𝑥2 + 3𝑥 − 5

h) f(𝑥) = 3𝑥4 − 8𝑥3 − 5𝑥2 + 16𝑥 − 5

i) f(𝑥) = 2𝑥4 + 𝑥3 − 𝑥2 − 𝑥 − 1

j) f(𝑥) = 3𝑥4 + 11𝑥3 + 11𝑥2 + 𝑥 − 2

k) f(𝑥) = 2𝑥5 + 𝑥4 − 32𝑥 − 16

l) f(𝑥) = 3𝑥5 + 𝑥4 − 243𝑥 − 18



10) From 1990 to 1994, the mail order sales of health products in the United States can be

modeled by 𝑆 = 10𝑡3 + 115𝑡2 + 25𝑡 + 2505 where S is the sales (in millions of dollars)

and t is the number of years since 1990. In what year were about $3885 million of health

products sold? (Hint: First substitute 3885 for S, then divide both sides by 5.)

11) The profit P (in millions of dollars) for a manufacturer of MP3 players can be modeled by 32 41216 xxxP where x is the number of MP3 players produced (in millions). Currently

the company produces 3 million MP3 players and makes a profit if $48,000,000. What lesser

number of MP3 players could the company produce and still make the same profit?

12) You have a picture that you want to frame, but first you have to put a mat around it. The

picture is 12 inches by 16 inches. The area of the mat is 204 square inches. If the mat extends

beyond the picture the same amount in each direction, what will the final dimensions of the

picture and mat be?

13) You are designing a monument and a

base as shown at the right. You will use

90 cubic feet of concrete for both pieces.

Find the value of x.

14) At a factory, molten glass is poured into

molds to make paperweights. Each mold

is a rectangular prism whose height is 3

inches greater than the length of each

side of the square base. A machine pours

20 cubic inches of liquid glass into each

mold. Find the dimensions of the mold.

15) You are building a solid concrete wheelchair ramp. The width of the ramp is three times the

height, and the length is 5 feet more than 10 times the height. If 150 cubic feet of concrete is

used, what are the dimensions of the ramp?

16) You are designing a kit for making sand

castles. You want one of the molds to be

a cone that will hold 48π cubic inches of

sand. What should the dimensions of the

cone be if you want the height to be 5

inches more than the radius of the base?

17) You are designing an in-ground lap

swimming pool with a volume of 2000

cubic feet. The width of the pool should

be 5 feet more than the depth, and the

length should be 35 feet more than the

depth. What should the dimensions of

the pool be?

18) You are a landscape artist designing a patio. The square patio floor is to be made from 128

cubic feet of concrete. The thickness of the floor is 15.5 feet less than each side length of the

patio. What are the dimensions of the patio floor?



19) At a factory molten glass is poured into a mold to make paperweights. Each model is a

rectangular prism with a height 4 inches greater than the length of each side of the square

base. Each mold holds 63 cubic inches of molten glass. Find the dimensions of the mold.

20) Write a polynomial equation to model the situation then list the possible rational zeros of the

fequation

a) A rectangular prism has edges of length x, x-1 and x-2 and a volume of 24.

b) A pyramid has a square base with sides of length x, a height of 2x-5 and a volume of 3.

21) From 1994 t0 2003, the amount of athletic equipment E (in millions of dollars) sold

domestically can be modeled by 32 1014020150,18)(E tttt where t is the number of

years since 1994. Use the following steps to find the year when about $20,300 million of

athletic equipmen was sold.

a) Write a polynomial equation that can be used to find the answer.

b) List the possible whole number solutions of the equation in part a) that are <10

c) Use synthetic division to determine which of the possible solutions in part b) is an actual

solution, then calculate the year which corresponds to the solution.

22) From 1990 t0 2000, the amount of U.S travelers to foreigh countries F (in 1000’s) can be

modeled by 916,433924202826412)(F 234 ttttt where t is the number of years

since 1990. Use the following steps to find the year when there were about 56,300,000

travelers.

a) Write a polynomial equation that can be used to find the answer.

b) List the possible whole number solutions of the equation in part a) that are 10

c) Use synthetic division to determine which of the possible solutions in part b) is an actual

solution.

d) Graph the function F(t) and explain why there are no other reasonable solutions then

calculate the year which corresponds to the solution.

23) The profit P (in millions of dollars) for a T-shirt manufacturer can be modeled by 324 xxxP where x is the number of shoe produced (in millions). Currently the

company produces 4 million T-shirts and makes a profit if $4,000,000. What lesser number

of T-shirt could the company produce and still make the same profit?

24) From 1985 to 2003, the total attendace A (in thausands) at NCAA women’s basketball games and the

number of NCAA women’s basketball teams can be modeled by the function 32 95.11.701882150 xxxA and

32 95.11.708.14725 xxxA where x is the

number of years since 1985. Write a function for the average attendance per team from 1985

to 2003.

25) The price P (in dollars) that a radio manufacturer is able to charge for a radio is given by 2440 xP where x is the number (in millions) of radios produced.it costs the company

$15 to make a radio.

a) Write an expression for the company’s total revenue in terms of x

b) Write a function for the company’s profit P by subtracting the total cost of making x

radios from the expression in part a)

c) Currently, the company produces 1.5 million radios and makes a profit if $24,000,000.

Write and solve an equation to find a lesser number of radios that he company could

tproduce and still make the same profit.

d) Do all the solutions in part a) make sense in this situation? explain



Conic Sections Parabolas, circles, ellipses, and hyperbolas are all curves that are formed by the intersection of a

plane and a double-napped cone. Therefore, these shapes are called conic sections or simply

conic. The equation of any conic can be written in the form 𝐴𝑥2 + 𝐵𝑥𝑦 + 𝐶𝑦2 + 𝐷𝑥 + 𝐸𝑦 +

𝐹 = 0 which is called a general second-degree equation in x and y. The expression 𝐵2 − 4𝐴𝐶

is called the discriminant of the equation and can be used to determine which type of conic the

equation represents.

History of Conics

In 200 B.C conic sections were studied thoroughly for the first time by a Greek mathematician

named Apollonius. Six hundred years later, the Egyptian mathematician Hypatia simplified the

works of Apollonius, making it accessible to many more people. For centuries, conics were

studied and appreciated only for their mathematical beauty rather than for their occurrence in

nature or practical use. TODAY astronomers know that the paths of celestial objects, such as

planets and comets, are conic sections. For example, a comet’s path can be parabolic, hyperbolic,

or elliptical.

Circle

A Circle is the set of all points (x, y) that are equidistant from a fixed point, called the center of

the circle. The distance r between the center and any point (x, y) on the circle is the radius. The

distance formula can be used to obtain an equation of the circle whose center is the origin and

whose radius is r. Because the distance

between any point (x, y) on the circle and

the center (0, 0) is r, you can write the

following.

22222 )0()0( ryxyxr

The standard form of the equation of a circle

with center at (0, 0) and radius r is as 222 ryx Eg A circle with center at (0, 0)

and radius 3 has equation 922 yx

Example 1 Draw the circle given by 22 25 xy .

Solution

Write the equation in standard form.

2525 2222 yxxy

In this form you can see that the graph is a

circle whose center is the origin and whose

radius is 5

Plot several points that are 5 units from the

origin. The points (0, 5), (5, 0), (0, º5), and

(º5, 0) are most convenient. Draw a circle

that passes through the four points.



Example 2 The point (1, 4) is on a circle whose center is the origin. Write the standard form of

the equation of the circle.

Solution

Because the point (1, 4) is on the circle, the radius of the circle must be the distance between the

center and the point (1, 4) 17)04()01( 222r the equation of the circle is 1722 yx .

A theorem in geometry states that a line

tangent to a circle is perpendicular to the

circle’s radius at the point of tangency. In

the diagram, AB is tangent to the circle

with center C at the point of tangency B, so

AB is perpendicular to BC .

Example 3 Write an equation of the line that is tangent to the circle 1322 yx at (2, 3).

Solution

The slope of the radius through the point (2,

3) is 2

3

02

03

m

Because the tangent line at (2, 3) is

perpendicular to this radius, its slope must

be the negative reciprocal of 2

3 , or

3

2 .

So, an equation of the tangent line

1332)2(3

23 yxxy .

Translated Circle

The standard equation of a translated circle center (h , k) is of the form (𝑥– ℎ)2 + (𝑦– 𝑘)2 = 𝑟2

where r is the radius of a circle. To write the standard equation of a translated circle

i) Group x terms together, y-terms together, and move constants to the other side

ii) Complete the square for the x-terms and for the y-terms

Remember that whatever you do to one side, you must also do to the other

Example 1 Find the equation of a circle of radius10 centered at (16, 10)

Solution The equation is(𝑥– 16)2 + (𝑦– 10)2 = 100

Example 2 Obtain the standard equation of the circle 𝑥2 + 𝑦2 − 10𝑥 + 8𝑦 − 8 = 0 hence find

the center and radius

Solution

49))4()5(16258)168()2510(8)8()10( 222222 yxyyxxyyxx

Therefore the center is (5 , -4) and the radius 𝑟 = 7

Question Obtain the standard equation of the circle 𝑥2 + 𝑦2 − 10𝑥 + 8𝑦 − 8 = 0 hence find

the center and radius 𝑥2 + 𝑦2 + 6𝑥 − 12𝑦 + 20 = 0

Example 3 Find the center, the radius, and write the standard equation of the circle if the

endpoints of a diameter are (-8 , 2) and (2 , 0).



Solution

Center: (−8+2

2 ,

2+0

2) = (−3 , 1) Using midpoint formular

Radius: using distance formula with radius and an endpoint √((2 − (−3))2 + (0 − 1))2 = √26

Thus the equation is (𝑥 + 3)2 + (𝑦 − 1)2 = 26

Question Obtain the standard equation of the circle 4𝑥2 + 4𝑦2 − 16𝑦 + 8𝑥 − 50 = 0 hence

find the center and radius

Example 4 A cellular phone transmission tower located 10 miles west and 5 miles north of

your house has a range of 20 miles. A second tower, 5 miles east and 10 miles south of your

house, has a range of 15 miles.

a) Write an inequality that describes each tower’s range.

b) Do the two regions covered by the towers overlap?

Solution

a) Let the origin represent your house. The first tower is at (-10, 5) and the boundary of its

range is a circle with radius 20. Substitute -10 for h, 5 for k, and 20 for r into the standard

form of the equation of a circle. (𝑥 + 10)2 + (𝑦 − 5)2 = 400 . The second tower is at (5, -

10). The boundary of its range is a circle with radius 15. (𝑥 − 5)2 + (𝑦 − 10)2 = 225

b) One way to tell if the regions overlap is

to graph the inequalities. You can see

that the regions do overlap. You can also

check whether the distance between the

two towers is less than the sum of the

ranges.

35215

1520))10(5()310( 22

Parabolas

A parabola is the set of all points in the

plane equidistant from a fixed point P

(called the focus) and a fixed line l (called

the directrix). The standard equation of a

Parabola with a vertical axis and a vertex at

the origin is given by 𝑥2 = 4𝑝𝑦. The

standard equation of a Parabola with a

horizontal axis and a vertex at the origin is

given by 𝑦 2 = 4𝑝𝑥

A parabolas can have a vertical axis of symmetry implying it can either open up or down or a

horizontal axis of symmetry implying it can open left or right. See the four cases shown below,



Example 1 Identify the focus and directrix of the parabola given by 𝑦2 = −6𝑥 . Draw the

parabola.

Solution

Because the variable y is squared, the axis of symmetry is horizontal.

To find the focus and directrix, rewrite the

equation as 𝑦2 = 4(−1.5)𝑥. The focus is

(-1.5, 0) and the directrix is 𝑥 = 1.5 To

draw the parabola, make a table of values

and plot points. Because 𝑝 < 0, the parabola

opens to the left. Therefore, only negative x-

values should be chosen.

X -1 -2 -3 -4 -5

Y

±2.4

5

±3.4

6

±4.2

4

±4.9

0

±5.4

8

Example 2 Write an equation of the parabola shown at the right.

Solution

The graph shows that the vertex is (0, 0) and

the directrix is 𝑦 = −𝑝 = −2. Substitute 2

for p in the standard equation for a parabola

with a vertical axis of symmetry.

𝑥2 = 4𝑝𝑦 Standard form, Substitute 𝑝 = 2

and Simplify ie 𝑥2 = 4(2)𝑦 = 8𝑦



Example 3 Sunfire is a glass parabola used to collect solar energy. The sun’s rays are reflected

from the mirrors toward two boilers located at the focus of the parabola. When heated, the

boilers produce steam that powers an alternator to produce electricity.

a) Write an equation for Sunfire’s cross section.

b) How deep is the dish?

Solution

a) The boilers are 10 feet above the vertex of the dish. Because the boilers are at the focus and

the focus is p units from the vertex, you can conclude that p = 10. Assuming the vertex is at

the origin, an equation for the parabolic cross section is 𝑥2 = 4𝑝𝑦 substituting p=10 we get

is 𝑥2 = 40𝑦

b) The dish extends is 37

2= 18.5 feet on either side of the origin. To find the depth of the dish,

substitute 18.5 for x in the equation from part (a). (18.5)2 = 40𝑦this gives 𝑦 = 8.6 The dish

is about 8.6 feet deep.

Translated Parabola

The general equation of a translated Parabola with a vertical axis is given by

(𝑥 − ℎ)2 = 4𝑝(𝑦 − 𝑘)

The vertex is at (h, k), and the focus at (h, k + p). The graph is a parabola which opens up or

down, depending on the sign of p.

The general equation of a translated Parabola with a horizontal axis is given by

(𝑦 − 𝑘)2 = 4𝑝(𝑥 − ℎ)

The vertex is at (h, k), and the focus at (h + p, k). The graph is a parabola which opens right or

left, depending on the sign of p

Example 1 Write an equation of the parabola whose vertex is (-2, 1) and whose focus is (-3, 1).

Solution

Choose form: Begin by sketching the

parabola, as shown. Because the parabola

opens to the left, it has the form

(𝑦 − 𝑘)2 = 4𝑝(𝑥 − ℎ) where p < 0. Find

h and k: The vertex is at (-2, 1), so

ℎ = −2 and 𝑘 = 1. Find p: The distance

between the vertex (-2, 1) and the focus

(-3, 1) is 𝑝 = −1. The standard form of the

equation is (𝑦 − 1)2 = 4𝑝(𝑥 + 2)

Example 2 find the equation of the parabola whose focus is (3 , 4) and directrix is 𝑥 = 1

Solution

The vertex is halfway between the focus and directrix, at (2 , 4)

The parabola opens to the right, so the general form is (𝑦 − 4)2 = 4(𝑥 − 2)

In the special case where the focus is on the directrix, then the resulting parabola i



Example 3 Write the equation of a parabola with a vertex of ( –2, 4) and focus point (0,

4). Also sketch the parabola.

Solution

Example 4 Given the equation of the parabola 4𝑥2 − 24𝑥 − 40𝑦 − 4 = 0 From the Standard

Form, determine the co-ordinates of the vertex and the equation of the directrix.

Solution

The equation can be written as 440)6(40440244 22 yxxyxx

)1(4)1(10)3(

4040)3(4

2

52

2

yyx

yx

Vertex: (ℎ, 𝑘) = (3, −1) , Since ‘x’ is

squared, this parabola opens vertically.

Since ‘p’ is positive, it opens upward. 𝑝 =

2.5 → Focus: (3, 1.5) The Directrix is a

horizontal line below the parabola, ‘p’ units

from the vertex. Directrix: 𝑦 = −3.5 The

Parabola opens upward from the Vertex,

away from the Directrix, around the Focus.

Example 3 Determine the co-ordinates of the vertex and the equation of the directrix for the

parabola 𝑦2 + 2𝑥 = 0

Solution

𝑦2 = 4(−0.5)𝑥 Vertex: (h, k) = (0, 0) Since

‘y’ is squared, this parabola opens

horizontally. Since ‘p’ is negative, it opens

to the left. 𝑝 − 0.5 Focus: (−0.5, 0) The

Directrix is a vertical line to the right of the

parabola, ‘p’ units from the vertex.

Directrix: 𝑥 = 0.5 The Parabola opens to the

left from the Vertex, away from the

Directrix, around the Focus.



Exercise

1) Graph the equation. Identify the focus and directrix of the parabola a) yx 42 b) 26xy

c) 0122 xy d) yx 43 2 e) 0282 yx f) 049 2 xy g) 02

8

1 yx h)

02

20

1 yx i) 160208 2 xy

2) Graph the parabola 4(𝑦 + 2) − (𝑥 − 1)2 = −4

3) Tell whether the parabola opens up, down, left, or right. (a) 23xy (b) yx 29 2 (c)

xy 62 2 (d) 27yx (e) yx 162 (f) xy 83 2 (g)

25 yx (h) yx3

42

4) Write the standard form of the equation of the parabola with the given focus or directrix and

vertex at (0, 0).

Focus; i) (-2 , 0) ii)(0 , -0.5) iii) (0 , 8

3) b) Directrix; i) 4x ii) 2

5y iii) 02

1 x

5) Match the equation with its graph;

a) xy 42 b) yx 42 c) yx 42 xy 42 d) e) xy4

12 f) yx4

12

6) Find the equation of the parabola whose vertex is at (−7, 2) and whose focus is at (−4, 2)

7) Graph the parabola and identify the focus, axis of symmetry, the directrix, and the focal

diameter: a) 8𝑥2 − 24𝑦 = 0 b) 6𝑦 2 − 24𝑥 = 0 c) 𝑦 = −2𝑥2 − 12𝑥– 19

8) The cross section of a television antenna

dish is a parabola. For the dish at the

right, the receiver is located at the focus,

4 feet above the vertex. Find an equation

for the cross section of the dish.

(Assume the vertex is at the origin.) If

the dish is 8 feet wide, how deep is it?

9) A searchlight has a parabolic reflector (has a cross section that forms a “bowl”). The

parabolic “bowl” is 16 inches wide from rim to rim and 12 inches deep. The filament of the

light bulb is located at the focus.

a) What is the equation of the parabola used for the reflector?



b) How far from the vertex is the filament of the light bulb?

10) The filament of a lightbulb is a thin wire that glows when electricity passes through it.

11) The filament of a car headlight is at the

focus of a parabolic reflector, which

sends light out in a straight beam. Given

that the filament is 1.5 inches from the

vertex, find an equation for the cross

section of the reflector. If the reflector is

7 inches wide, how deep is it?

12) In the drawing shown at the left, the rays

of the sun are lighting a candle. If the

candle flame is 12 inches from the back

of the parabolic reflector and the

reflector is 6 inches deep, then what is

the diameter of the reflector?

13) The cables of the middle part of a

suspension bridge are in the form of a

parabola, and the towers supporting the

cable are 600 feet apart and 100 feet

high. What is the height of the cable at a

point 150 feet from the center of the

bridge?

14) For an equation of the form2axy , discuss what effect increasing |a| has on the focus and

directrix.

15) You can make a solar hot dog cooker

using foil-lined cardboard shaped as a

parabolic trough. The drawing at the

right shows how to suspend a hot dog

with a wire through the focus of each

end piece. If the trough is 12 inches wide

and 4 inches deep, how far from the

bottom should the wire be placed?

16) A flashlight has a parabolic reflector. An equation for the cross section of the

reflector is xy7

322 . The depth of the reflector is 2

3 inches.

Explain why the value of p must be less than the depth of the reflector of a flashlight.

How wide is the beam of light

projected by the flashlight?

Write an equation for the cross

section of a reflector having the same

depth but a wider beam than the

flashlight shown. How wide is the

beam of the new reflector?

Write an equation for the cross section of a reflector having the same depth but a

narrower beam than the flashlight shown. How wide is the beam of the new reflector?



17) The latus rectum of a parabola is the line

segment that is parallel to the directrix,

passes through the focus, and has

endpoints that lie on the parabola. Find

the length of the latus rectum of a

parabola given by pyx 42 .

Ellipses An ellipse is the locus of all points pthe sum of whose distances from 2 ixed points called foci is

constant. 𝑑1 + 𝑑2 = 2𝑎 The line through the foci intersects the ellipse at two points, the vertices.

The line segment joining the vertices is the

major axis, and its midpoint is the center of

the ellipse. The line perpendicular to the

major axis at the center (called minor axis

which has length 2b) intersects the ellipse at

two points called the co-vertices..

The two types of ellipses we will discuss are those with a horizontal major axis and those with a

vertical major axis

The standard form of the equation of an ellipse with center at (0, 0) and major and minor axes of

lengths 2a and 2b, where 𝑎 > 𝑏 > 0, is as follows.

Equation Major axis Vertex Co-vertex

12

2

2

2

b

y

a

x

Horizontal (±𝑎 , 0) (0 , ±𝑏 )

12

2

2

2

a

y

b

x

Vertical (0 , ±𝑎 ) (±𝑏 , 0)

The foci of the ellipse lie on the major axis, c units from the center where 𝑐2 = 𝑎2 − 𝑏2

Example 1 Draw the ellipse given by 9𝑥2 + 16𝑦2 = 144. Identify the foci.



Solution

First rewrite the equation in standard form 9𝑥2 + 16𝑦2 = 144 1916

22

yx

Because the denominator of the 𝑥2 -term is

greater than that of the 𝑦2 -term, the major

axis is horizontal. So, 𝑎 = 4 and 𝑏 = 3. Plot the vertices and co-vertices. Then draw

the ellipse that passes through these four

points. The foci are at (c, 0) and (-c, 0). To

find the value of c, use the equation

𝑐2 = 𝑎2 − 𝑏2 = 16 − 9 = 7 .

The foci are at (±√(7 ,0)

Example 2 Write an equation of the ellipse with the given characteristics and center at (0, 0).

a)Vertex: (0, 7) Co-vertex: (-6, 0) b) Vertex: (-4, 0) Focus: (2, 0)

Solution

In each case, you may wish to draw the

ellipse so that you have something to check

your final equation against. a. Because the

vertex is on the y-axis and the co-vertex is

on the x-axis, the major axis is vertical with

a = 7 and b = 6.

Thus the equation is 14936

22

yx

Because the vertex and focus are points on a

horizontal line, the major axis is horizontal

with 𝑎 = 4 and 𝑐 = 2. To find b, use the

equation 𝑐2 = 𝑎2 − 𝑏2

1241162 b thus the equation is

11216

22

yx

Example 3 A portion of the White House lawn is called The Ellipse. It is 1060 feet long and

890 feet wide. a) Write an equation of The Ellipse. b) The area of an ellipse is 𝐴 = 𝜋𝑎𝑏. What

is the area of The Ellipse at the White House?

Solution

a) The major axis is horizontal with 𝑎 = 10 60 ÷ 2 = 530 and 𝑏 = 890 ÷ 2 = 445. Thus the

equation is 1445530 2

2

2

2

yx

. b) The area is A = π (530) (445) ≈ 741,000 square feet. Modeling

Remark: If 𝑎 = 𝑏 we get the equation of a circle



Example 4 In its elliptical orbit, Mercury ranges from 46.04 million kilometers to 69.86

million kilometers from the center of the sun. The center of the sun is a focus of the orbit. Write

an equation of the orbit.

Solution

Using the diagram shown, you can write a

system of linear equations involving a and c.

𝑎 − 𝑐 = 46.04 𝑎 + 𝑐 = 69.86 Adding

the two equations gives 2𝑎 = 115.9, so

𝑎 = 57.95. Substituting this a-value into

the second equation gives 57.95 + c = 69.86,

so c = 11.91. From the relationship 𝑐2 = 𝑎2 − 𝑏2 , you can conclude the following:

An equation of the elliptical orbit is (57 x .9

2 5) 2 + (56 y .7 2 1) 2 = 1 where x and y are

in millions of kilometers.

Translated Ellipse

The standard equation of a translated ellipse centered at point (h , k) and major and minor axes of

lengths 2a and 2b, where 𝑎 > 𝑏 > 0 is as follows

Equation Major axis Vertex Co-vertex

1)()(

2

2

2

2

b

ky

a

hx

Horizontal

(ℎ± 𝑎 , 𝑘) (ℎ , 𝑘 ± 𝑏 )

1)()(

2

2

2

2

a

ky

b

hx

Vertical

(ℎ , 𝑘± 𝑎 ) (ℎ ± 𝑏 , 𝑘)

Example 1

Write an equation of the ellipse with foci at (3, 5) and (3, -1) and vertices at (3, 6) and (3, -2).

Solution

Plot the given points and make a rough sketch. The ellipse has a vertical major axis, so its

equation is of this form:

Find the center: The center is halfway between the vertices. (ℎ, 𝑘) = (3+3

2,

6−2

2) = (3, 2)



Find a: The value of a is the distance

between the vertex and the center. 𝑎 =

√(3 − 3)2 + (6 − 2)2 = √02 + 42 = 4.

Find c: The value of c is the distance

between the focus and the center. 𝑐 =

√(3 − 3)2 + (5 − 2)2 = √02 + 32 = 3

Find b: Substitute the values of a and c into

the equation 𝑏2 = 𝑎2 − 𝑐2 = 16 − 9 = 7

The standard form of the equation is

Example 2 Identify the vertices, co-vertices, foci, and sketch the ellipses;

136

)2(

4

)3( 22

yx

Solution

Clearly 6a and 2b

32436 c . The center is (-3 ,

2)and the major axis is vertical the

major axis is 3x . The foci are at

(−3 , ±√32 ), the vertices are at (−3 , 8 )

and (−3 , −4 ) and the co-vertices are at (−5 , 2) and (−1 , 2 )

The graph of the ellipse look like this

Exercise

1) Write an equation of an ellipse with the given characteristics and center (0 , 0)

a) a)Vertices:: (0, 5) co-vertices: (-4 , 0) d) Foci: )0,102( Vertices: (-7, 0)

b) Vertices:: (9, 0) co-vertices: (0 , 2) e) Co-vertices: )0,91( , Foci: (0 , 3)

c) Foci: (0 , -5) Vertices: (0, 13) f) Co-vertices: )33,0( , Foci: (4 , 0)

2) Graph the equation. Identify the vertices, co-vertices and the foci of the ellipse

(a) 13625

22

yx

b) 1494

22

yx

c) 1925

4 22

xx

d) 14

9

9

4 22

yx

e) 164

22

xy

f) 494

22

yx

g) 14

9

64

22

yx

h) 4595 22 yx i) 259 22 yx

3) Write the equation in standard form. Then identify the vertices, co-vertices, and foci of the

ellipse. a) 6416 22 yx b) 36412 22 yx c) 2502510 22 yx d) e) f)

4) Suppose a satellite’s orbit is an ellipse with Earth’s center at one focus. If the satellite’s least

distance from Earth’s surface is 150 miles and its greatest distance from Earth’s surface is

600 miles, write an equation for the ellipse. (Use 4000 miles as Earth’s radius.)



5) An elliptical garden is 10 feet long and 6 feet wide. Write an equation for the garden. Then

graph the equation. Label the vertices, co-vertices, and foci. Assume that the major axis of

the garden is horizontal

6) Statuary Hall is an elliptical room in the United States Capitol in Washington, D.C.

The room is also called the Whispering

Gallery because a person standing at one

focus of the room can hear even a

whisper spoken by a person standing at

the other focus. This occurs because any

sound that is emitted from one focus of

an ellipse will reflect off the side of the

ellipse to the other focus. Statuary Hall

is 46 feet wide and 97 feet long.

a) Find an equation that models the

shape of the room.

b) How far apart are the two foci?

c) What is the area of the room’s floor?

7) The first artificial satellite to orbit Earth was Sputnik I, launched by the Soviet Union in

1957. The orbit was an ellipse with Earth’s center as one focus. The orbit’s highest point

above Earth’s surface was 583 miles, and its lowest point was 132 miles. Find an equation of

the orbit. (Use 4000 miles as the radius of Earth.) Graph your equation.

8) Australian football is played on an elliptical field. The official rules state that the field must

be between 135 and 185 meters long and between 110 and 155 meters wide. _Source: The

Australian News Network

a) Write an equation for the largest and for the smallest allowable playing field.

b) Write an inequality that describes the possible areas of an Australian football field.

9) A tour boat trvaels between two islands that are 12 miles apart. For a trip between the

islands, there is enough fuel for a 20-mile tour.

a) The region in which the boat can travel is bounded by an ellipse. Explain why this is so

b) Let (0, 0) be the center of the ellipse.

Find the coordinates of each island.

c) Suppose the boat travels from one

island, straight past the other island

to the vertex of the ellipse, and back

to the second island. How many

miles does the boat travel? What are

the coordinates of the vertex?

d) Use your answers to parts (b) and (c) to write an equation for the ellipse that bounds the

region the boat can travel in.

10) Show that 222 bac for any ellipse given by the equation 12

2

2

2

b

y

a

xwith foci at (c,0) and (-c, 0).

Hyperbolas

A hyperbola is the set of all points in the plane, the difference of whose distances from two fixed

points is a constant. The line through the foci intersects the hyperbola at two points, the vertices.



The line segment joining the vertices is the

transverse axis and its midpoint is the center

of the hyperbola. A hyperbola has two

branches and two asymptotes.

The asymptotes contain the diagonals of a rectangle centered at the hyperbola’s center, as shown

The standard form of the equation of a hyperbola with center at (0, 0) is as follows

Equation Transverse axis Vertex Foci Asymptotes

12

2

2

2

b

y

a

x Horizontal )0,( a )0,( c xy

ab

12

2

2

2

b

x

a

y Vertical ),0( a ),0( c xy

b

a

The foci of the hyperbola lie on the transverse axis, c units from the center where 𝑎2 + 𝑏2 = 𝑐2

Example 1 Draw the hyperbola given by 4𝑥2 − 9𝑦2 = 36

Solution

First rewrite the equation in standard form 149

22

yx

. From the equation 𝑎2 = 9 and 𝑏2 = 4,

so 𝑎 = 3 and 𝑏 = 2 . Because the 𝑥2-term is positive, the transverse axis is horizontal and the

vertices are at (-3, 0) and (3, 0). To draw the hyperbola, first draw a rectangle that is centered at

the origin, 2a= 6 units wide and 2b= 4 units high. Then show the asymptotes by drawing the

lines that pass through opposite corners of the rectangle. Finally, draw the hyperbola.



Example 2 Write an equation of the hyperbola with foci at (0, -3) and (0, 3) and vertices at (0, -

2) and (0, 2).

Solution

The transverse axis is vertical because the

foci and vertices lie on the y-axis. The

center is the origin because the foci and the

vertices are equidistant from the origin.

Since the foci are each 3 units from the

center, c= 3. Similarly, because the vertices

are each 2 units from the center, 𝑎 = 2 You

can use these values of a and c to find b.

Now 𝑐2 = 𝑎2 + 𝑏2 thus 𝑏2 = 9 − 4 = 5

Because the transverse axis is vertical, the

standard form of the equation is

154

22

xy

Example 3 A hyperbolic mirror can be used to take panoramic photographs. A camera is

pointed toward the vertex of the mirror and is positioned so that the lens is at one focus of the

mirror. An equation for the cross section of the mirror is 1916

22

xy

where x and y are

measured in inches. How far from the mirror is the lens?

Solution

Notice from the equation that 162 a and 92 b , so using these values and the equation

222 bac to find the value of c now 5259162 cc

Since 𝑎 = 4 and 𝑐 = 5, the vertices are at (0, -4) and (0, 4) and the foci are at (0, -5) and (0, 5) .

The camera is below the mirror, so the lens is at (0, º5) and the vertex of the mirror is at (0, 4).

The distance between these points is 4 − (−5) = 9 The lens is 9 inches from the mirror.

Example 4 The diagram at the right shows the hyperbolic cross section of a sculpture located at

the Fermi National Accelerator Laboratory in Batavia, Illinois.

a) Write an equation that models the curved

sides of the sculptur

b) At a height of 5 feet, how wide is the

sculpture? (Each unit in the coordinate

plane represents 1 foot.)

Solution

a) From the diagram you can see that the

transverse axis is horizontal and 1a .

So the equation has the form 12

22

b

yx

Because the hyperbola passes through

the point (2, 13), we can substitute 2x

and 13y into the equation and solve

for b to get 𝑏 = 7.5.



An equation of the hyperbola is 15.7 2

22

yx

b) At a height of 5 feet above the ground, 𝑦 = −8. To find the width of the sculpture, substitute

this value into the equation and solve for x. You get 𝑥 ≈ 1.46. At a height of 5 feet, the width

is 2𝑥 ≈ 2.92 feet.

Translated Hyperbola The standard equation of a translated hyperbola centered at point (h , k) and major and minor

axes of lengths 2a and 2b, where 𝑎 > 𝑏 > 0 is as follows

Equation Transverse axis Vertex Foci Asymptotes

1)()(

2

2

2

2

b

ky

a

hx Horizontal ),( kah

),( kch

. )(( hxky

a

b

1)()(

2

2

2

2

b

hx

a

ky Vertical ),( akh

),( ckh

)(( hxky

b

a

Example 1 Graph 14

)1()1(

22

xy

Solution The 2yy -term is positive, so the transverse axis is vertical. Since 4 and1 22 ba ,

Plot the center at (h, k) = (-1, -1). Plot the

vertices 1 unit above and below the center at

(-1, 0) and (-1, -2). Draw a rectangle that is

centered at (-1,-1) and is 2a = 2 units high

and 2b = 4 units wide. Draw the asymptotes

through the corners of the rectangle. Draw

the hyperbola so that it passes through the

vertices and approaches the asymptotes. If the origin is placed halfway between the foci, the equation is found using the distance formula.

Exercise

1) Graph the equation. Identify the vertices, the foci and asymptotes a) 99 22 xy

b) 13625

22

xy

c) 164

22

yx

d) 175100

22

xy

e) 3002512 22 xy

f) 18149

22

yx

g) 144436 22 yx h) 36418 22 yx i) 160208 22 xy

2) Write an equation of the hyperbola with the given foci and vertices )53,0(

c) Foci: (0, -5), (0, 5) Vertices: (0, -3), (0, 3)

d) Foci: (-8, 0), (8, 0) Vertices: (-7, 0), (7, 0)

e) Foci: )0,34( , )0,34( ) Vertices: (-5, 0), (5, 0)

f) Foci: (0, -9), (0, 9)Vertices: )53,0( , )53,0(

g) Foci: ( 8 , 0) , Vertices: )0,34( f) Foci: )65,0( , Vertices: (0, 4),)



3) Suppose the mirror in example 3 above has a cross-section modeled by the equation

1925

22

yx

where x and y are measured in inches. If you place a camera with its lens at the

focus, how far is the lens from the vertex of the mirror?

4) Match the equation with its graph

a) 1416

22

yx

b) 124

22

xy

c) 1416

22

xy

d) 124

22

yx

5) Write the equation of the hyperbola in standard form a) 324936 22 yx b) 8181 22 xy

c) 9436 22 xy d) 093616 22 xy e) 436

22

xy f) 9

9

4

9

22

yx

6) Identify the vertices and foci of the hyperbola a) 1649

22

yx

b) 149

22

xy

c) 14121

22

yx

d) 324814 22 xy e) 100425 22 xy f) 3601036 22 yx

7) Graph the equation. Identify the foci and asymptotes a) 112125

22

yx

b) 136

22

yx

c) 14925

22

xy

d) 11009

22

xy

e) 164

22

xy

f) 18125

16 22

yx

g) 14

9

64

22

yx

h) 810081100 22 yx i) 259 22 yx

8) The sundial at the left was designed by

Professor John Shepherd. The shadow of

the gnomon traces a hyperbola

throughout the day. Aluminum rods

form the hyperbolas traced on the

summer solstice, June 21, and the winter

solstice, December 21.

a) One focus of the summer solstice

hyperbola is 207 inches above the

ground. The vertex of the aluminum

branch is 266 inches above the

ground. If the x-axis is 355 inches

above the ground and the center of

the hyperbola is at the origin, write an equation for the summer solstice hyperbola



b) One focus of the winter solstice hyperbola is 419” above the ground. The vertex of the

aluminum branch is 387 inches above the ground and the center of the hyperbola is at the

origin. If the x-axis is 355 inches above the ground, write an equation for the winter

solstice hyperbola.

c) Use your equations from Exercises 64 and 65 to draw the lower branch of the summer

solstice hyperbola and the upper branch of the winter solstice hyperbola

9) When an airplane travels faster than the speed of sound, the sound waves form a cone behind

the airplane. If the airplane is flying parallel to the ground, the sound waves intersect the

ground in a hyperbola with the airplane directly above its center. A sonic boom is heard

along the hyperbola. If you hear a sonic boom that is audible along a hyperbola with the

equation 14100

22

yx

where x and y are measured in miles, what is the shortest horizontal

distance you could be to the airplane?

10) Suppose you are making a ring out of clay for a necklace. If you have a fixed volume of clay

and you want the ring to have a certain thickness, the area of the ring becomes fixed.

However, you can still vary the inner radius x and the outer radius y.

a) Suppose you want to make a ring with an area of 2 square inches. Write an equation

relating x and y.

b) Find three coordinate pairs (x, y) that satisfy the relationship from part (a). Then find the

width of the ring, y-x, for each coordinate pair.

c) How does the width of the ring, y-x, change as x and y both increase? Explain why this

makes sense.

11) Use the diagram at the right to show that

|d2 - d1| = 2a.

12) Two microphones, 1 mile apart, record

an explosion. Microphone A receives the

sound 2 seconds after Microphone B. Is

this enough information to decide where

the sound came from? Use the fact that

sound travels at 1100 feet per second

Classifying a Conic Sections from its Equation

The equation of any conic can be written in the form 𝐴𝑥2 + 𝐵𝑥𝑦 + 𝐶𝑦2 + 𝐷𝑥 + 𝐸𝑦 + 𝐹 = 0

The discriminant 𝐵2 − 4𝐴𝐶 of the equation can be used to determine which type of conic the

equation represents.

If 𝐵2 − 4𝐴𝐶 < 0 B = 0, and A = C the equation is a circle

If 𝐵2 − 4𝐴𝐶 < 0 and either B ≠ 0 or 𝐴 ≠ 𝐶 the equation is a ellipse

If 𝐵2 − 4𝐴𝐶 = 0 the equation is a parabola

If 𝐵2 − 4𝐴𝐶 > 0 the equation is a hyperbola

Note If B = 0, each axis of the conic is horizontal or vertical. If B ≠ 0, the axes are neither

horizontal nor vertical.

Example 1 Classify and graph the conic given by 2𝑥2 + 𝑦2 − 4𝑥 − 4 = 0.

Solution

Since𝐴 = 2, 𝐵 = 0 and 𝐶 = 1, the discriminant is 𝐵2 − 4𝐴𝐶 = 02 − 4 × 2 × 1 = −8



Because 𝐵2 − 4𝐴𝐶 < 0 and A ≠ C, the graph is an ellipse To graph the ellipse,

First complete the square

163

)1(6)1(2

24)12(24)42(

2222

2222

yxyx

yxxyxx

Clearly center is (1 , 0) and 36 ba

3 c . A sketch of the ellipse is shown

Example 2 Classify and graph the conic

given by 05481443294 22 yxyx

Solution

Since A=4 B=0 and C=-9, the discriminant is 𝐵2 − 4𝐴𝐶 = 02 − 4 × 4 × −9 = 144 > 0

Thus the graph is a hyperbola. To graph the hyperbola, first complete the square

14

)8(

9

)4(36)8(9)4(4

57664548)6416(9)168(4548)1449()324(

2222

2222

yxyx

yxxyxx

Comparing this with 1)()(

2

2

2

2

b

ky

a

hx,

ℎ = −4, 𝑘 = −8, 𝑎 = 3, and 𝑏 = 2 To

draw the hyperbola, plot the center at

(ℎ, 𝑘) = (−4, −8) and the vertices at (-7, -

8) and (-1, -8). Draw a rectangle 2𝑎 = 6

units wide and 2𝑏 = 4 units high and

centered at (-4, -8). Draw the asymptotes

through the corners of the rectangle. Then

draw the hyperbola so that it passes through

the vertices and approaches the asymptotes.

Exercise

1) Write an equation for the conic section.

a) Circle with center at; i) (4, -1) and radius 7 ii) (9, 3) and radius 4 iii) (4, 2) and radius 3

b) Parabola with vertex at; i) (1, -2) and focus at (1, 1) ii) (-3, 1) and directrix 𝑥 = −8

c) Ellipse with vertices at (2, -3) and (2, 6) and foci at (2, 0) and (2, 3)

d) Ellipse with vertices at (-2, 2) and (4, 2) and co-vertices at (1, 1) and (1, 3)

e) Hyperbola with vertices at (5, -4) and (5, 4) and foci at (5, -6) and (5, 6)

f) Hyperbola with vertices at (-4, 2) and (1, 2) and foci at (-7, 2) and (4, 2)

g) Ellipse with foci at (2, -4) and (5, -4) and vertices at (-1, -4) and (8, -4)

h) Parabola with vertex at (3, -2) and focus at (3, -4)

i) Hyperbola with foci at (5, 2) and (5, -6) and vertices at (5, 0) and (5, -4)

2) Match the equation with its graph a) 036243649 22 yxyx b)

09422 xyy c) 036243649 22 yxyx d) 046422 yxxy e)

046422 yxyx f) 061541694 22 yxyx



3) Classify the conic section.

a) 04422 yxx

b) 02653 22 yxyx

c) 084722 yxyx

d) 01325 22 yxyx

e) 01021022 yxyx

f) 0302634 22 yxyx

g) 018394 22 yxyx

h) 036243649 22 yxyx

i) 0222251636 22 yxyx

j) 06041644 22 yxyx

k) 0625429 22 yxxy

l) 0820182516 22 yxyx

m) 09822 yxx

010482 2 xyx

n) 03740122012 22 yxyx

o) 05510549 22 yxyx

p) 042422 yxyx

q) 0233624169 22 yxxy

r) 06364216 22 yxxy

s) 0171642 yxx

4) Look back at Example *. Suppose there is a tower 25 miles east and 30 miles north of your

house with a range of 25 miles. Does the region covered by this tower overlap the regions

covered by the two towers in Example 5? Illustrate your answer with a graph.

5) Graph the equation. Identify the important characteristics of the graph, such as the center,

vertices, and foci.

a) 1)1()7( 22 yx

b) )2(3)4( 2 xy

c) 4)2()6( 22 yx

d) )3(12)7( 2 yx

e) 14

)3(

16

)8( 22

xy

f) 149

)6(

2

)3( 22

yx

g) 1916

)1( 22

yx

h) 1)4(16

22

yx

6) Classify the conic section and write its equation in standard form. Then graph the equation.



a) 044122 xyy

b) 0248622 yxyx

c) 01198729 22 yxyx

d) 0484484 22 yxyx

e) 01824 22 yxyx

f) 036241222 yxyx

g) 01281616 22 yyx

h) 07489 22 yxyx

i) 036121222 yxyx

j) 0942022 yxy

k) 012842 yxx

l) 0164163649 22 yxyx

7) A Gregorian telescope contains two mirrors whose cross sections can be modeled by the equations

0254,295729405 22 yx and 01440120 2 xy What types of mirrors are they?

8) The whisper dish shown at the left can be

seen at the Thronateeska Discovery Center

in Albany, Georgia. Two dishes are

positioned so that their vertices are 50 feet

apart. The focus of each dish is 3 feet from

its vertex. Write equations for the cross

sections of the dishes so that the vertex of

one dish is at the origin and the vertex of

the other dish is on the positive x-axis.

9) To practice making a figure eight, a figure

skater will skate along two circles etched

in the ice. Write equations for two

externally tangent circles that are each 6

feet in diameter so that the center of one

circle is at the origin and the center of the

other circle is on the positive y-axis.

10) When a pencil is sharpened the tip becomes a cone. On a pencil with flat sides, the

intersection of the cone with each flat side is a conic section. What type of conic is it?

11) A new crayon has a cone-shaped tip.

When it is used for the first time, a flat

spot is worn on the tip. The edge of the

flat spot is a conic section, as shown.

What type(s) of conic could it be?

12) Which of the following is an equation of the hyperbola with vertices at (3, 5) and (3, -1) and

foci at (3, 7) and (3, o3)? (a) 19

)2(

25

)3( 22

yx

(b) 125

)3(

9

)2( 22

xy

(c) 17

)3(

9

)2( 22

xy

(d) 116

)2(

9

)2( 22

yx

(e) 116

)3(

9

)2( 22

xy

13) What conic does 076210025 22 yxyx represent? (a) Parabola (b) Circle c) Ellipse

d) Hyperbola e) Not enough information



14) A degenerate conic occurs when the intersection of a plane with a double-napped cone is

something other than a parabola, circle, ellipse, or hyperbola.

a) Imagine a plane perpendicular to the axis of a double napped cone. As the plane passes

through the cone, the intersection is a circle whose radius decreases and then increases.

At what point is the intersection something other than a circle? What is the intersection?

b) Imagine a plane parallel to the axis of a double-napped cone. As the plane passes through

the cone, the intersection is a hyperbola whose vertices get closer together and then

farther apart. At what point is the intersection something other than a hyperbola? What is

the intersection?

c) Imagine a plane parallel to the nappe passing through a double-napped cone. As the plane

passes through the cone, the intersection is a parabola that gets narrower and then flips

and gets wider. At what point is the intersection something other than a parabola? What is

the intersection?

15) Tell what type of path each comet follows. Which comet(s) will pass by the sun more than

once?

a) 0000,137100200,14350 2 yxx

b) 0900,12400,18200,1346002200 22 yxyx

c) 0000,695000,52000,2026500000,5 2 yxyx

Documents

Lecture Notes STA 2101 ALGEBRA FOR STATISTICS AND FINANCE · 2017-09-28 · ii) Recognize equations and graphs involving the conic sections. iii) Recognize graphs and properties of