Alg 3(10) 1 Ab Value and Piecewisefaculty.pingry.org/vmcgrath/documents/8GraphingPiecewise...Alg 3(10) 1 Ab Value and Piecewise Ch 2.2 Graphing SKETCHPAD I A Def: An equation whose

Alg 3(10) 1

Ab Value and Piecewise

Ch 2.2 Graphing SKETCHPAD

I A Def: An equation whose graph is a line is a linear equation.

Linear:

3 2 6

4

2

3 1

x y

x

y

y x

Not Linear

23 2

1 14

x y

x y

What is a graph? B What methods can I use to graph line? 1) 2) II EXAMPLES GRAPH THE FOLLOWING 1) y = -3x + 4 2) 3x - 2y = ½

III GRAPHING

A slope = m =

1) What if there is no change in y?

2) What if there is no change in x?

3) Can lines have the same slope?

4) Can lines have the same y-intercept?

Alg 3(10) 2


x-intercept

y-intercept

parallel lines

perpendicular lines

vertical lines horizontal lines

IV EXAMPLES

1. Find the x and y intercepts for

a) 3x + y = 6 b) 1

4x +

1

3y = 6

2) Find “t” if the line through ( -1,1) and (3,2) is parallel to the line through (0,6) and (-8, t). f) Show the figure with the following vertices is a parallelogram. A (1, 2) B (4, 6) C (8, 3) D (5, -1)

Alg 3(10) 3


Algebra 3 Assignment # 1

Graph using the x and y intercepts:

(1) 3 6 24 0x y (2) 4 2 12 0x y

(3) 5 x 2 15 = 0

Graph the points to satisfy the given domain:

(4) 3 5 1 4y x x

Find the slope:

(5) 6,7 , 106, 7 (6) 2, 6 , 13,8

(7) 1,1 , 1, 17

(8) Why is the line determined by (6, -5) and (8, -8) parallel to the line through (-3, 12) and (1, 6)

(9) The line through ( -4, 6 ) ( 3, 2 ) is parallel to the line through (0, 10 ) ( -8, y ). Find y.

(10) What kind of figure has these vertices?

A ( 7, 5) B (8, 3) C (0, -1) D (-1, 1)

Alg 3(10) 4


Answers:

1. 0,4 , 8,0

2. 0,6 , 3,0

3. 1,0 ,no y int ercept

4. See solved sheet

75. -

50

146.

11

7. no slope

1029.

7

Alg 3(10) 5


2.3 Line Equations A Slope – Intercept Form __________________ m = _________ b = ______ Convert 10x – 2y = 3 to slope/intercept form

B Standard Form ______________________ __________________ Convert y = 1.5 x – 6 to standard form. ________________ C Point – Slope Form _____________________ ___________________ D To write the equation of a line you need the slope and a point on the line. 1. slope = 3, y – intercept = 10 y = 2. slope = 3, x – intercept = 10 y = 3. slope = 3, through point (10,10) __________________ 4. 2 points (1,2) (3,4) __________________ 5. 2 intercepts x-int = -2 __________________ y-int = 3

Alg 3(10) 6


6. Parallel to 2x + 3y = 7 and passes through (-1, -1) _______________

7. Perpendicular to 4x – y = 7 and passes through (-1, -1) _______________

8. Perpendicular to y = 3 and passes through (12, 4) ________________

9. Parallel to x + 2y = 1 and passes through the point of intersection of the lines y = 3x – 2 and y = 2x + 1.

__________________ 10. vertical line and point (2,3) ____________ 11. horizontal line and point (2,3) ____________ 12. If f(x) = x+ 3, graph f(2), f (-1) and f(x) 13. Graph triangle ABC with coordinates A ( -6, 4 ) B ( 2 , 8 ) and C ( 6, 2 ). a) write the equation of all the medians b) find where they intersect

Alg 3(10) 7


Assignment # 2

Write the equation in slope intercept form:

(1) 4x-3y-7=0 (2) 1/4x-1/2y=1

Write equation of the line in standard form:

(3) (2,3) (3,2) (4) (3,0) (0,-3)

(5) (-1,-13) (-8,1) (6) (10,27) (12,27)

(7) Two lines, parallel to the coordinate axes intersect at the point (5,-7). What are their

equations?

(8) Write the equation of the line parallel to 2x+3y=6 that passes through the point (1,-1).

Write the equation of the line that is perpendicular to the given line and passes through

the indicated point.

(9) y = 3x - 1 ; (4,7) (10) y - 2x = 5 ; (-5,1)

(11) Write the equation of the altitude to BC if the triangle coordinates are

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

(12) Graph triangle ABC with coordinates A ( -6, 4 ) B ( 2 , 8 ) and C ( 6, 2 ).

A) write the equation of all the altitudes

B) find where they intersect

Alg 3(10) 8


Assignment # 2 Answers:

4 7

1) 3 3

12) 2

2

3) 5

4) 3

5) 2 15

6) y=27

7) 5 and y=-7

2 18) y = -

3 3

1 19) y = - 8

3 3

1 110) y = - 1

2 2

11) y = 3x 2

y x

y x

y x

y x

x y

x

x

x

x

12 ) 1 1

4 , 92 2

Alg 3(10) 9


Perpendicular Bisectors, Distance

A) Given the triangle ABC with coordinates A ( -6, 4 ) B ( 2 , 8 ) and C ( 6, 2 ).

1) Find the equations of the perpendicular bisectors.

2) Find their intersection called the circumcenter.

B) Given the diagram, derive the distance formula

d =

1 1x ,y

2 2x , y

d

Alg 3(10) 10


C) Examples: 1). Triangle ABC has vertices A (-5,2), B (1,6) C (7,-4). Find a The slope of each side of ABC b. The slope of the altitude to side AB. c. The equation of the altitude to side AB. d. The equation of the median to AB 2) Find the distance between the point (5, -1) and the line whose equation is 3x + 4y = -4. 3) Find the distance between the parallel lines whose equations are y = 2x + 3 and y = 2x – 1.

Alg 3(10) 11


ANSWERS 1. 2. 3.

a. 2 13 2 34 6 5

b. 2 5 1

,... ,...3 3 2

c. 3 3

,... ,...22 5

d. (-2,4), (4,1) (1,-1) e. 7

ASSIGNMENT #3

(1) Find the distance between each of the following pairs of points.

(a) ( 6 , 3 ) and ( 2 , 3 ) (c) ( 9 , 17 ) and ( 3 , 7 )

(b) ( 3 , 6 ) and ( 2 , 6 ) (d) , 65

31

and , 43

21

(2) Determine whether each of the following sets of points is collinear.

(a) ( 2 , 2 ) , ( 6 , 4 ) , ( 4 , 11 ) (b) ( 3 , 9 ) , ( 1 , 1 ) , ( 2 , 7 )

(3) Triangle ABC has vertices A( 2 , 1 ) , B( 5 , 8 ) , C( 1 , 2 ). Determine whether ABC is a right triangle.

(4) Find all values of the constant k if the line connecting the points ( 4 , 1 ) and ( k , 5 ) is to be; (a) parallel

(b) perpendicular to the line connecting the points ( 3 , k ) and ( 3 , 6 ).

(5) Triangle ABC has vertices A( 5 , 2 ) , B( 1 , 6 ) , C( 7 , 4 ). Find:

(a) the perimeter of ABC.

(b) the slope of each side of ABC.

(c) the slope of the altitude to each side of ABC.

(d) the midpoint of each side of ABC.

(e) the length of the median to the longest side of ABC.

Alg 3(10) 12


(6) Using triangle ABC from problem #6, find a. the equation of AC b. the length of the median to AB c the equation of the median to AB d. the equation of the altitude to AC. (7) Using the triangle ABC from problem #66, find a the equations of the perpendicular bisectors of AB and AC b the point of intersection of the perpendicular bisectors in part “a” (8) Using triangle ABC again, and your previous work, find a the distance from the point of intersection of the perpendicular bisectors to A and to B b Write the equation of the circle circumscribed around triangle ABC.

Alg 3(10) 13


ANSWERS

(1) (a) 8 (c) 512

(b) 13 (d) 12

5

(2) (a) no (b) yes

(3) yes

(4) (a) k = 6 or k = 8 (b) k = 0

(5) (a) 342 56 132

(b) m AB = 32

, m AC = 21

, m BC = 35

(c) to AB = 23

, to AC = 2 , to BC = 53

(d) midpoint AB ( 2 , 4 ) , midpoint BC ( 4 , 1 ) , midpoint AC ( 1 , 1 )

(e) 7

(6) (7) (8)

a. 1

12

y x

b. 3 5

c. (4,-1), (2,5) = 2 10

d. y = 2x - 4

Alg 3(10) 14


ABSOLUTE VALUE / PIECEWISE FUNCTIONS

I Absolute Value V or

II Pieces rays, segments, partial graphs

III Step Functions

I Absolute Value y x

y x 3

y x – 4

y x 2

^

x y

x y

Alg 3(10) 15


y 4 x 1

1y x + 1

4

| x + 2 | + | y | = 3

Summary y = |x - h| + k SHIFTS

x y

x y

Alg 3(10) 16



Sketch a graph of each of the following please.

(1) 3 x y

(2) 1 x 2 y

(3) 2 x y

(4) 3 2 x y

(5) 13

y = x 1 + 2

(6) y = 2 3x 2 2

(7) x + y = 2

(8) x + y = 2

Alg 3(10) 17


II Piecewise Graphing y = f(x) y = 2x + 1 is the same as f(x) = 2x + 1 f(x) = 2x + 1 when x < 0 f(x) = x – 4 when 0 < x < 3 f(x) = 3x when x 3

(5) 0 x if 1 x

0 x if 1 x xf

(6)

x 1 if x 1

f x x if 1 x 1

x 2 if x 1

Alg 3(10) 18


2.4 Greatest Integer Function

y = [x] y is the greatest integer less than or equal to x

x y

0 0

1/4 0

1/2 0

3/4 0

1 1

1.25 1

1.5 1

1.75 1

2 2

2.25 2

2.75 2

3 3

3.999 3

4 4

“Round down”

The Price is Right – can’t go over the amount shown

y = [x] – 2

x y

0 -2

1/4 -2

1/2 -2

3/4 -2

1 -1

1.25 -1

1.5 -1

1.75 -1

2 0

2.25 0

2.75 0

3 1

3.999 1

4 2

Alg 3(10) 19


y = [x + 2 ]

y = 2[x]

x y

0+2=2 2

¼+2 2

½+2 2

¾+2 2

1+2 3

1.25 3

1.5 3

1.75 3

2 4

2.25 4

2.75 4

3 5

3.999 5

4 6

x y

0 0(2)=0

1/4 0

1/2 0

3/4 0

1 1(2)=2

1.25 2

1.5 2

1.75 2

2 2(2)=4

2.25 4

2.75 4

3 3(2)=6

3.999 6

4 4(2)=8

Alg 3(10) 20


y = [2x]

METHOD

1. Set inside [ ] between 0 and 1, 1 and 2, 2 and 3 etc

2. Solve for x

3. Solve for y

y = [3x + 1] – 1

1) 2) 1 3x 1 2 3) 2 3x 1 3

2) 1

x 03

3) y = -1 y = 0 y = 1

x y

0(2)=0

0

1/4(2)=1/2 0

1/2(2)=1 1

3/4(2)=1.5 1

1(2)= 2

1.25(2)= 2

1.5(2)= 3

1.75(2)= 3

2(2)= 4

2.25(2)= 4

2.75(2)= 5

3(2)= 6

3.999(2)= 7

4(2)= 8

0 3x 1 1

sub into original

use smaller # (-1/3)

Alg 3(10) 21


Algebra 3 Assignment # 6 Piecewise Functions

Sketch a graph of each of the following please.

(1) x 1 if x 0

y x 1 if x 0

(2)

2x 4 if x -2

y 1 x 3 if x -2

2

(3)

x 1 if x 1

y x if 1 x 1

x 2 if x 1

(4) y = x + 2

(5) y x + 2

(6) 12

y x

(7) 12

y x

Alg 3(10) 22


Graphing Quadratic Functions

Graph 2y = x + 2x 3

Standard Form for Parabola

2

y = 1 x + 1 4

vertex = (h,k) vertex 1, 4

axis of symmetry x = h axis x = - 1

Graph 2 points on either side of vertex

(-1, -4)

x = -1

x y

-3 0

-2 -3

-1 -4

0 -3

1 0

2f(x) (x )ha k

Alg 3(10) 23


Observations:

Sketch

21. f(x) (x 1) 4 = +

2y x 3 2y x 1 2y (x 2)

2y (x 3) 1

Alg 3(10) 24


2. 2y = 2x + 2x 2

3.

Uhoh – what do we do??????

23. x = y + 2

Alg 3(10) 25



Graph each the following quadratic functions please. For each, state the: (a) coordinates of the vertex, (b)

equation of the axis of symmetry, and (c) range.

(1) 2

f x = x 3 2

(2) 2

f x = x + 3 + 2

(3) 21

4f x = x 1 + 4

(4) 2

f x = 3x 18x + 31

(5) 2

f x = x 8x + 13

(6) 2

f x = 3x + 12x 5

(7) 2

f x = x 2x + 1

Alg 3(10) 26


Algebra 3 Assignment # 7 Answers

(1) Vertex: (3 , –2) Axis: x = 3 Range: y ≥ –2

(2) Vertex: (–3 , 2) Axis: x = –3 Range: y ≤ 2

(3) Vertex: (–1 , 3) Axis: x = 1 Range: y ≥ 4

(4) Vertex: (3 , 4) Axis: x = 3 Range: y ≥ 4

(5) Vertex: (4 , –3) Axis: x = 4 Range: y ≥ –3

(6) Vertex: (2 , 7) Axis: x = 2 Range: y ≤ 7

(7) Vertex: (1 , 0) Axis: x = 1 Range: y ≥ 0

Alg 3(10) 27


Applications

1) I have 80 feet of fence to make a garden which will have one wall of my house as a border. Find the dimensions so that the area is a maximum.

2 ) Find two consecutive positive integers such that the sum of their squares is 113. (notice! No “maximum/minimum”)

Essentially you will see three things.

• There will be enough information for 2 equations with 2 unknowns.

• The equation setup will involve multiplication or squaring; it won’t be linear systems.

• You will be asked either to solve (i.e. set = 0) or find x so as to maximize or minimize the situation.

What are we looking for?

Alg 3(10) 28


3) The sum of two numbers is 40. Find the two numbers if their product is a maximum. 4 ) The sum of a number and its square is 72. Find the number. 5) The sum of 2 numbers is 12. Find the numbers if the product of one and twice the other is a maximum.

6) The length of a rectangle is 3 cm greater than the width. The area is 70 square centimeters. Find the dimensions.

Alg 3(10) 29


Algebra 3 Assignment # 8 Quadratic Word Problems

(1) One positive integer is 3 greater than another. The sum of the squares of the integers is 89. Find the

integers.

(2) A border of uniform with is to be added around a rectangle which measures 8 feet by 12 feet. The area of

the new rectangle is to be twice that of the original rectangle. How wide is the border?

(3) A rectangular pool measures 10 yards by 18 yards. The pool is surrounded by a walk of uniform width. The

area of the walk is 52 square yards. How wide is the walk?

(4) The length of a rectangle is three times its width. If the width is decreased by 1 foot, and the length is

increased by 3 feet, the area of the new rectangle will be 72 square feet. Find the dimensions of the new

rectangle.

(5) A strip of masking tape is placed around the edges of a rectangular window prior to painting its frame. If

the window measures 3 feet by 4 feet, and the masking tape covers half the area of the window, how wide

is the tape?

(6) A company’s daily profit P, in dollars is given by 2P = 2x + 120x 180 , where x is the number of

articles produced per day. Find the value of x so that the daily profit will be a maximum.

(7) While pulling a box trailer back to college, Karen approaches a covered bridge in the shape of a parabola

whose equation is 22

5y = x + 23(the roadway is represented by the x-axis).

The rectangular trailer is 10 feet wide and 14 feet high. Will the truck be able to pass safely through the

opening?

Alg 3(10) 30



(1) 5 & 8

(2) 2 feet

(3) 7 + 62 yards

(4) 4 feet by 18 feet

(5) 12

foot

(6) 30

(7) No

Alg 3(10) 31


Max and Min Problems

(1) The sum of two numbers is 12. Find the numbers if the sum of the square of one number and four times the

other number is to be as small as possible.

(2) A club is selling water bottles. They are currently charging $1.00 per bottle and are selling 70 bottles per

day. They estimate that for every 10¢ increase in the price, they will sell 5 fewer bottles per day. What

price should they charge to maximize daily receipts?

Alg 3(10) 32


Algebra 3 Assignment # 9 Max/Min Word Problems

(1) A manufacturer finds that the daily cost in dollars, C, of producing n statues is given by the formula

2C = n 120n + 4200. How many statues should be produced per day so that the cost will be a

minimum? What is the minimum daily cost?

(2) The sum of two numbers is 8. Find the numbers if the sum of the square of one of the numbers and 4 times

the other number is to be as small as possible. Find the smallest sum.

(3) When a department store sold a certain style of shirt for $20, the average number of shirts sold per week

was 100. The store observed that for every $1 decrease in price, 10 more shirts were sold weekly. What

price should the store charge to maximize the store’s weekly revenue?

(4) It is estimated that 14,000 people will attend a game when the admission price is $7.00. For each 25¢

increase in the price, the attendance will decrease by 280. What admission price should be charged to

maximize the gate receipts?

(5)

(6)

x x x x

y

A farmer has 120 yards of fencing to enclose a

rectangular region as shown in the figure to the right.

One side of the region will lie against a building, so no

fencing will be used on that side. Find the dimensions

which should be used to maximize the area of the entire

region. Find the maximum area which may be enclosed.

P (x , y)

(0 , 4)

(2 , 0)

Each point P on the line segment connecting the points (0 , 4) and (2 , 0)

determines a rectangle as shown in the figure to the right. Find the

coordinates of the point P if the area of the rectangle is to be as large as

possible. Find the area of the largest rectangle.

Alg 3(10) 33



(1) 60, $600

(2) 2 & 6, sum is 28

(3) $15

(4) $9.75

(5) 15 yards by 60 yards , 900 square yards

(6) P(1 , 2) , 2

Alg 3(10) 34


Algebra 3 Assignment # 10 ─ Review Worksheet

(1) Sketch a graph of each of the following please.

(a) y x 2 3 (b) y = 2 x + 3 4

(c) y = 2x + 1 (d) y = 2x + 1

(e) y = 3 x 2 (f) y 3 x 2

(g) 2x 3 if x 0

y x 2 if x 0

(h) 2

x + 4 if x 2y

x 2 if x 2

(2) Sketch a graph of each of the following. Label the vertex, the axis of symmetry, and at least two other

points.

(a) 8 6x x y 2 (c) 1 6x x3 y 2

(b) 1 2x y 2 (d)

2y = x + 4x 5

(3) A bus service is currently transporting 100 passengers at a cost of $2.00 per person. The owner estimates

that for every 20¢ increase in price there will be 5 fewer passengers. What price should be charged to

maximize revenue?

(4) Two integers have a sum of 13. The sum of their squares is 125. What are the numbers?

(5) The difference of 2 numbers is 14. Find the numbers if their product is to be a minimum and also find this

product.

(6) A rectangle has a length that is 5 less than 3 times the width. Find the dimensions of the rectangle if the

area is 28 cm².

Alg 3(10) 35


Algebra 3 Assignment # 10 ─ Review Worksheet Answers

(2) (a) Vertex: (3 , –1) , Axis of symmetry is x = 3

(b) Vertex: (0 , 1) , Axis of symmetry is x = 0

(c) Vertex: (–1 , 4) , Axis of symmetry is x = –1

(d) Vertex: (2 , –1) , Axis of symmetry is x = 2

(3) $3.00 (5 increases of 20 cents)

(4) 11 and 2

(5) –7 and 7, Product –49

(6) 4 by 7

Alg 3(10) 36


Extra Review Graph the following completely

1. y 2x 1 4 2. y 2x 1 4

3. y 2x 5 4. y 4 x 4 1

5. y 2 2x 4 1 6. 2x 3 if x 0

y2x 3 if x < 0

7.

6 if 2 x 4

y 3 if 0 < x 2

1 if -3 x 0

Documents

Alg 3(10) 1 Ab Value and Piecewisefaculty.pingry.org/vmcgrath/documents/8GraphingPiecewise...Alg 3(10) 1 Ab Value and Piecewise Ch 2.2 Graphing SKETCHPAD I A Def: An equation whose