Math 30-1: Workbook€¦ · Trigonometry One Trigonometry Two Permutations and ... A box with no lid can be made by cutting out squares from each corner of a rectangular piece of

Topics CoveredPolynomial, Radical, and Rational FunctionsTransformations and OperationsExponential and Logarithmic FunctionsTrigonometry OneTrigonometry TwoPermutations and Combinations

Mathematics 30-1

Copyright © 2014 | www.math30.ca

A workbook and animated series by Barry Mabillard

This page has been left blank for correct workbook printing.

Trigonometry IFormula Sheet

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The Unit Circle

Note: The unit circle is NOT included on the official formula sheet.

Trigonometry II

Exponential andLogarithmic Functions

Permutations &Combinations

Transformations& Operations

Polynomial, Radical& Rational Functions

Curriculum AlignmentMath 30-1: Alberta | Northwest Territories | NunavutPre-Calculus 12: British Columbia | YukonPre-Calculus 30: SaskatchewanPre-Calculus 40S: Manitoba

Mathematics 30-1Formula Sheet

Unit 1: Polynomial, Radical, and Rational FunctionsLesson 1: Polynomial FunctionsLesson 2: Polynomial DivisionLesson 3: Polynomial FactoringLesson 4: Radical Functions

Unit 2: Transformations and Operations

Lesson 5: Rational Functions ILesson 6: Rational Functions II

Lesson 1: Basic TransformationsLesson 2: Combined TransformationsLesson 3: InversesLesson 4: Function Operations

Unit 3: Exponential and Logarithmic FunctionsLesson 5: Function Composition

Lesson 1: Exponential FunctionsLesson 2: Laws of LogarithmsLesson 3: Logarithmic Functions

Unit 4: Trigonometry ILesson 1: Degrees and RadiansLesson 2: The Unit CircleLesson 3: Trigonometric Functions ILesson 4: Trigonometric Functions II

Unit 5: Trigonometry IILesson 5: Trigonometric EquationsLesson 6: Trigonometric Identities ILesson 7: Trigonometric Identities II

Unit 6: Permutations and CombinationsLesson 1: PermutationsLesson 2: CombinationsLesson 3: The Binomial Theorem

7:45 (16 days)1:38 (3 days)1:29 (3 days)1:13 (3 days)0:52 (2 days)

4:38 (11 days)

1:00 (2 days)1:33 (3 days)

0:57 (2 days)0:50 (2 days)0:42 (2 days)0:48 (2 days)

5:55 (12 days)1:21 (3 days)

1:52 (4 days)2:11 (4 days)1:52 (4 days)

9:59 (17 days)2:22 (4 days)2:15 (4 days)2:24 (5 days)1:58 (4 days)



40:19 (78 days)Total Course

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Table of Contents

Mathematics 30-1

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Polynomial, Radical, and Rational FunctionsLESSON ONE - Polynomial Functions

Lesson Notes321

i) x5 + 3 iv) 4x2 - 5x - 112ii) 5x + 3 v) 1x2 + x - 4

3vii) 5 - 1x viii) 7 x + 2 1

x + 3ix)iii) 3 vi) |x|

b) Determine which expressions are polynomials. Explain your reasoning.

For each polynomial function given below, state the leading coefficient, degree, and constant term.

the leading coefficient is ______, the degree of the polynomial is ______, and the constant term of the polynomial is ______.

Example 2: End Behaviour of Polynomial Functions.a) The graphs of several even-degree polynomials are shown. Study these graphs and generalize the end behaviour of even-degree polynomials.b) The equations and graphs of several odd-degree polynomials are shown. Study these graphs and generalize the end behaviour of odd-degree polynomials.

Example 3: Zeros, Roots, and x-intercepts of a Polynomial Function.a) Define “zero of a polynomial function”. Determine if each value is a zero of P(x) = x2 - 4x - 5. i) -1 ii) 3 b) Find the zeros of P(x) = x2 - 4x - 5 by solving for the roots of the related equation, P(x) = 0.c) Use a graphing calculator to graph P(x) = x2 - 4x - 5. How are the zeros of the polynomial related to the x-intercepts of the graph?d) How do you know when to describe solutions as zeros, roots, or x-intercepts?

quadratic quadratic

quartic quartic

quadratic quadratic

quartic quartic

i ii

viv

iii

vii

iv

viii

Even-Degree Polynomials Example 2a

linear cubic

cubic quintic

linear cubic

cubic

i ii

viv

iii

vii

iv

viii

quintic

Odd-Degree Polynomials Example 2b

Example 4: Multiplicity of zeros in a polynomial function.a) Define “multiplicity of a zero”.

b) P(x) = -(x + 3)(x - 1) c) P(x) = (x - 3)2 d) P(x) = (x - 1)3 e) P(x) = (x + 1)2(x - 2)

a) P(x) = (x - 5)(x + 3) 12

b) P(x) = -x2(x + 1)

a) P(x) = (x - 1)2(x + 2)2

b) P(x) = x(x + 1)3(x - 2)2

a) P(x) = -(2x - 1)(2x + 1)

b) P(x) = x(4x - 3)(3x + 2)

Examples 5 - 7: Find the requested data for each polynomial function, then use this information to sketch the graph. i) Find the zeros and their multiplicities. ii) Find the y-intercept. iii) Describe the end behaviour. iv) What other points are required to draw the graph accurately?

Example 5: Example 6: Example 7:

For the graphs in parts (b - e), determine the zeros and state each zero’s multiplicity.

i) f(x) = 3x - 2 ii) y = x3 + 2x2 - x - 1 iii) P(x) = 5

Example 1: Introduction to Polynomial Functions.a) Given the general form of a polynomial function, P(x) = anx

n + an-1xn-1 + an-2x

n-2 + ... + a1x1 + a0,

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r

4h

Example 14: Three students share a birthday on the same day. Quinn and Ralph are the same age, but Audrey is two years older. The product of their ages is 11548 greater than the sum of their ages.a) Find polynomial functions that represent the age product and age sum.b) Write a polynomial equation that can be used to find the age of each person.c) Use a graphing calculator to solve the polynomial equation from part (b). Indicate your window settings. How old is each person?

Example 15: The volume of air flowing into the lungs during a breath can be represented by the polynomial function V(t) = -0.041t3 + 0.181t2 + 0.202t, where V is the volume in litres and t is the time in seconds.a) Use a graphing calculator to graph V(t). State your window settings.b) What is the maximum volume of air inhaled into the lung?At what time during the breath does this occur?c) How many seconds does it take for one complete breath?d) What percentage of the breath is spent inhaling?

Example 16: A cylinder with a radius of r and a height of h is inscribed within a sphere that has a radius of 4 units. Derive a polynomial function, V(h), that expresses the volume of the cylinder as a function of its height.

(0, 4)

(0, -1)

(2, -6)

152

4,

(0, -9)

(-6, 0)

(0, -6)

16 cm

20 cm

x

x

Example 12: Given the characteristics of a polynomial function, draw the graph and derive the function.

a) Characteristics of P(x):x-intercepts: (-1, 0) and (3, 0); sign of leading coefficient: (+);polynomial degree: 4; relative maximum at (1, 8)

b) Characteristics of P(x):x-intercepts: (-3, 0), (1, 0), and (4, 0); sign of leading coefficient: (-);polynomial degree: 3; y-intercept at: (0, -3/2)

Example 13: A box with no lid can be made by cutting out squares from each corner of a rectangular piece of cardboard and folding up the sides. A particular piece of cardboard has a length of 20 cm and a width of 16 cm. The side length of a corner square is x.a) Derive a polynomial function that represents the volume of the box.b) What is an appropriate domain for the volume function?c) Use a graphing calculator to draw the graph of the function. Indicate your window settings.d) What should be the side length of a corner square if the volume of the box is maximized?e) For what values of x is the volume of the box greater than 200 cm3 ?

Polynomial, Radical, and Rational FunctionsLESSON ONE - Polynomial FunctionsLesson Notes 321

Examples 8 - 10: Determine the polynomial function corresponding to each graph. You may leave your answer in factored form.

Example 8: Example 9:

Example 10:

a) b)

a) b)

a) b)

Example 11: Use a graphing calculator to graph each polynomial function. Find window settings that clearly show the important features of each graph. (x-intercepts, y-intercept, and end behaviour)

a) P(x) = x2 - 2x - 168b) P(x) = x3 + 7x2 - 44xc) P(x) = x3 - 16x2 - 144x + 1152

a)

b) Label the division components (dividend, divisor, quotient, remainder) in your work for part (a).

c) Express the division using the division theorem, P(x) = Q(x)•D(x) + R. Verify the division theoremby checking that the left side and right side are equivalent.

x3 + 2x2 - 5x - 6x + 2

d) Another way to represent the division theorem is .P(x)D(x)

RD(x)

Express the division using this format.

e) Synthetic division is a quicker way of dividing than long division. Divide (x3 + 2x2 - 5x - 6) by (x + 2)

using synthetic division and express the result in the form .P(x)D(x)

= Q(x) +R

D(x)

Example 2: Divide using long division.P(x)D(x)

= Q(x) +R

D(x)Express answers in the form .

a) (3x3 - 4x2 + 2x - 1) ÷ (x + 1) b) x3 - 3x - 2x - 2

c) (x3 - 1) ÷ (x + 2)

Example 3: Divide using synthetic division.P(x)D(x)

= Q(x) +R

D(x)Express answers in the form .

a) (3x3 - x - 3) ÷ (x - 1) b) 3x4 + 5x3 + 3x - 2x + 2

c) (2x4 - 7x2 + 4) ÷ (x - 1)

Example 4: Polynomial division only requires long or synthetic division when factoring is not an option. Try to divide each of the following polynomials by factoring first, using long or synthetic division as a backup.

b) (6x - 4) ÷ (3x - 2) c) (x4 - 16) ÷ (x2 + 4) d)x3 + 2x2 - 3x

x - 3a) x2 - 5x + 6

x - 3

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Example 1: Divide using long division and answer the related questions.

3 -4 -5 23 -7 2

1

3 -7 2 0

Polynomial, Radical, and Rational FunctionsLESSON TWO - Polynomial Division

Lesson Notes

Example 5: When 3x3 - 4x2 + ax + 2 is divided by x + 1, the quotient is 3x2 - 7x + 2 and the remainder is zero. Solve for a using two different methods.

a) Solve for a using synthetic division.b) Solve for a using P(x) = Q(x)•D(x) + R.

Example 6: A rectangular prism has a volume of x3 + 6x2 - 7x - 60. If the height of the prism is x + 4, determine the dimensions of the base.

V = x3 + 6x2 - 7x - 60

x + 4

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Polynomial, Radical, and Rational FunctionsLESSON TWO - Polynomial DivisionLesson Notes

Example 8: If f(x) ÷ g(x) = , determine f(x) and g(x).4x2 + 4x - 3 - 6x - 1

Example 12: Use the remainder theorem to find the value of k in each polynomial.

a) (kx3 - x - 3) ÷ (x - 1) Remainder = -1 b)3x3 - 6x2 + 2x + k

x - 2Remainder = -3

c) (2x3 + 3x2 + kx - 3) ÷ (2x + 5) Remainder = 2 d) 2x3 + kx2 - x + 62x - 3

(2x - 3 is a factor)

Example 15: Given the graph of P(x) = x3 + kx2 + 5 and the point (2, -3), determine the value of a on the graph.

P(x) = x3 + kx2 + 5

(2, -3)

(4, a)

Example 15

3 -4 -5 23 -7 2

1-

3 -7 2 0

Example 7: The graphs of f(x) and g(x) are shown below.a) Determine the polynomial corresponding to f(x).b) Determine the equation of the line corresponding to g(x).c) Determine Q(x) = f(x) ÷ g(x) and draw the graph of Q(x).

(3, 8) f(x) g(x)

Example 9: The Remainder Theorema) Divide 2x3 - x2 - 3x - 2 by x - 1 using synthetic division and state the remainder.b) Draw the graph of P(x) = 2x3 - x2 - 3x - 2 using technology. What is the value of P(1)?c) How does the remainder in part (a) compare with the value of P(1) in part (b)?d) Using the graph from part (b), find the remainder when P(x) is divided by:i. x - 2 ii. x iii. x + 1e) Define the remainder theorem.

Example 10: The Factor Theorema) Divide x3 - 3x2 + 4x - 2 by x - 1 using synthetic division and state the remainder.b) Draw the graph of P(x) = x3 - 3x2 + 4x - 2 using technology. What is the remainder when P(x) is divided by x - 1?c) How does the remainder in part (a) compare with the value of P(1) in part (b)?d) Define the factor theorem.e) Draw a diagram that illustrates the relationship between the remainder theorem and the factor theorem.

Example 11: For each division, use the remainder theorem to find the remainder. Use the factor theorem to determine if the divisor is a factor of the polynomial.

b)x4 - 2x2 + 3x - 4

x + 2a) (x3 - 1) ÷ (x + 1) c) (3x3 + 8x2 - 1) ÷ (3x - 1) d) 2x4 + 3x3 - 4x - 9

2x + 3

Example 13: When 3x3 + mx2 + nx + 2 is divided by x + 2, the remainder is 8. When the same polynomial is divided by x - 1, the remainder is 2. Determine the values of m and n.

Example 14: When 2x3 + mx2 + nx - 6 is divided by x - 2, the remainder is 20.The same polynomial has a factor of x + 2. Determine the values of m and n.

Examples 2 - 8: Factor and graph.a) Factor algebraically using the integral zero theorem.b) Use technology to graph the polynomial. Can the polynomial be factored using just the graph?c) Can P(x) be factored any other way?

Example 2: P(x) = x3 + 3x2 - x - 3.

Example 4: P(x) = x3 - 3x + 2

Example 5: P(x) = x3 - 8

Example 6: P(x) = x3 - 2x2 - x - 6

Example 3: P(x) = 2x3 - 6x2 + x - 3

Example 7: P(x) = x4 - 16

Example 8: P(x) = x5 - 3x4 - 5x3 + 27x2 - 32x + 12

a) P(x) has zeros of -4, 0, 0, and 1.The graph passes through the point (-1, -3).

Example 9: Given the zeros of a polynomial and a point on its graph, find the polynomial function. You may leave the polynomial in factored form. Sketch each graph.

b) P(x) has zeros of -1, -1, and 2.The graph passes through the point (1, -8).

Example 10: A rectangular prism has a volume of 1050 cm3. If the height of the prism is 3 cm less than the width of the base, and the length of the base is 5 cm greater than the width of the base, find the dimensions of the rectangular prism.Solve algebraically.

Example 11: Find three consecutive integers with a product of -336. Solve algebraically.

Example 12: If k, 3k, and -3k/2 are zeros of P(x) = x3 - 5x2 - 6kx + 36, and k > 0, find k and write the factored form of the polynomial.

Example 13: Given the graph of P(x) = x4 + 2x3 - 5x2 - 6x and various points on the graph, determine the values of a and b. Solve algebraically.

(2, 0)

(a, 0)(b, 0)

(0, 0)

Example 13

Example 14: Solve each equation algebraically and check with a graphing calculator.a) x3 - 3x2 - 10x + 24 = 0b) 3x3 + 8x2 + 4x - 1 = 0

Quadratic FormulaFrom Math 20-1:The roots of a quadratic equation with the form ax2 + bx + c = 0 can be found with the quadratic formula:

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Polynomial, Radical, and Rational FunctionsLESSON THREE - Polynomial Factoring

Lesson Notes

Example 1: The Integral Zero Theorema) Define the integral zero theorem. How is this theorem useful in factoring a polynomial?b) Using the integral zero theorem, find potential zeros of the polynomial P(x) = x3 + x2 - 5x + 3.c) Which potential zeros from part (b) are actually zeros of the polynomial?d) Use technology to draw the graph of P(x) = x3 + x2 - 5x + 3. How do the x-intercepts of the graph compare to the zeros of the polynomial function?e) Use the graph from part (d) to factor P(x) = x3 + x2 - 5x + 3.

x3 - 5x2 + 2x + 8

(x + 1)(x - 2)(x - 4)

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Polynomial, Radical, and Rational FunctionsLESSON THREE - Polynomial FactoringLesson Notes

This page has been left blank for correct workbook printing.

x3 - 5x2 + 2x + 8

(x + 1)(x - 2)(x - 4)

x f(x)

-1

0

1

4

9

a) Fill in the table of values for the function f(x) = x

b) Draw and state the domain and range.f(x) = x

Example 1: Introduction to Radical Functions

b) f(x) = -xa) f(x) = x-

Example 2: Graph each function.

.


a) f(x) = 2 x b) f(x) = x12 c) f(x) = 2x d) f(x) = x1

2Example 4: Graph each function.

c) f(x) = x - 1 d) f(x) = x + 7a) f(x) = x - 5 b) f(x) = x + 2


a) f(x) = x - 3 + 2 c) f(x) = - x - 3 d) f(x) = -2x - 4b) f(x) = x + 42

Examples 6 - 8: Graph y = and state the domain and range.

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Polynomial, Radical, and Rational FunctionsLESSON FOUR - Radical Functions

Lesson Notesy = x

f(x)

Example 6: a) y = x + 4 b) y = -(x + 2)2 + 9

y = (x - 5)2 - 4Example 7: a) y = x2b)

Example 8: a) y = -(x + 5)2 y = x2 + 0.25b)

x + 2 = 3

Examples 9 - 12: Solve each radical equation in three different ways.a) Solve algebraically and check for extraneous roots.b) Solve by finding the point of intersection of a system of equations.c) Solve by finding the x-intercept(s) of a single function.

x + 2x =

x + 3 = x + 32

16 - x2 = 5

Example 9:

Example 10:

Example 11:

Example 12:

Math 30-1 students are expected to know that domain and range can beexpressed using interval notation.

() - Round Brackets: Exclude pointfrom interval.[] - Square Brackets: Include pointin interval.Infinity ∞ always gets a round bracket.

Examples: x ≥ -5 becomes [-5, ∞); 1 < x ≤ 4 becomes (1, 4];x ε R becomes (-∞ , ∞);

Interval Notation

-8 ≤ x < 2 or 5 ≤ x < 11 becomes [-8, 2) U [5, 11),where U means “or”, or union of sets;x ε R, x ≠ 2 becomes (-∞ , 2) U (2, ∞);-1 ≤ x ≤ 3, x ≠ 0 becomes [-1 , 0) U (0, 3].

A set is simply a collection of numbers, such as {1, 4, 5}. We use set-builder notation to outline the rules governing members of a set.

Set-Builder Notation

10-1

{x | x ε R, x ≥ -1}State thevariable.

In words: “The variable is x, such that x can beany real number with the condition that x ≥ -1”.

List conditionson the variable.

As a shortcut, set-builder notation can be reducedto just the most important condition.

10-1x ≥ -1

While this resource uses the shortcut for brevity, as set-builder notation is covered in previous courses,Math 30-1 students are expected to know how to read and write full set-builder notation.

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Polynomial, Radical, and Rational FunctionsLESSON FOUR - Radical FunctionsLesson Notes

y = x

(2, 2) (2, 3)

(8, -1)(1, 3)

Example 13: Write an equation that can be used to find the point of intersection for each pair of graphs.

a) b) c) d)

Example 14: A ladder that is 3 m long is leaning against a wall. The base of the ladder is d metres from the wall, and the top of the ladder is h metres above the ground.

a) Write a function, h(d), to represent the heightof the ladder as a function of its base distance d.

c) How far is the base of the ladder from the wall when the top of the ladder is metres above the ground? 5

a) If a ball is dropped from twice its original height, how will that change the time it takes to fall?b) If a ball is dropped from one-quarter of its original height, how will that change the time it takes to fall?c) The original height of the ball is 4 m. Complete the table of values and draw the graph. Do your results match the predictions made in parts (a & b)?

Example 15: If a ball at a height of h metres is dropped, the length of time it takes to hit the ground is:

t = h4.9

where t is the time in seconds.

b) Graph the function and state the domain and range. Describe the ladder’s orientation when d = 0 and d = 3.

a) Derive a function, V(r), that expresses the volume of the paper cup as a function of r.b) Graph the function from part (a) and explain the shape of the graph.

Example 16: A disposable paper cup has the shape of a cone. The volume of the cone is V (cm3), the radius is r (cm), the height is h (cm), and the slant height is 5 cm.

Cone Volume

r

h5 cm

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Example 2: Reciprocal of a Linear Function. Given the graph of y = f(x), draw the graph of y = .1

f(x)a) y = x - 5

b) y = - x + 212

i. Domain & Range of y = f(x) ii. Domain & Range of y =f(x)1 iii. Asymptote Equation(s)

i. Domain & Range of y = f(x) ii. Domain & Range of y =f(x)1 iii. Asymptote Equation(s)

1x

a) Fill in the table of values for the function y = .

x y

-2

-1

-0.5

-0.25

0

0.25

0.5

1

2

c) Draw the graph of y = x in the same grid used for part (b).1x

Compare the graph of y = x to the graph of y = .

1xb) Draw the graph of the function y = .

d) Outline a series of steps that can be used to draw the1xgraph of y = , starting from y = x.

State the domain and range.

Example 1: Reciprocal of a Linear Function.

y = x1 Polynomial, Radical, and Rational Functions

LESSON FIVE - Rational Functions ILesson Notes

1x2 - 4

a) Fill in the table of values for the function y = .

x y

-3

-2

-1

0

1

2

3

x y

-2.05

-1.95

x y

1.95

2.05

b) Draw the graph of the function y = .1

x2 - 4State the domain and range.

c) Draw the graph of y = x2 - 4 in the same grid used for part (b).

Compare the graph of y = x2 - 4 to the graph of y = .1

x2 - 4

d) Outline a series of steps that can be used to draw the1

x2 - 4graph of y = , starting from y = x2 - 4.

Example 3: Reciprocal of a Quadratic Function.

a) b) c) d) e) f)

Example 4: Reciprocal of a Quadratic Function. Given the graph of y = f(x), draw the graph of y =1

f(x).

i. Domain & Range of y = f(x) ii. Domain & Range of y =f(x)1

iii. Asymptote Equation(s)

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Polynomial, Radical, and Rational FunctionsLESSON FIVE - Rational Functions ILesson Notes

y =x1

a) b) c) d)

Example 5: Given the graph of y = 1

f(x)draw the graph of y = f(x).,

Example 6: For each function, determine the equations of all asymptotes. Check with a graphing calculator.

a) f(x) =2x - 3

1b) f(x) =

x2 - 2x - 24

1c) f(x) =

6x3 - 5x2 - 4x

1d) f(x) =

4x2 + 9

1

Example 7: Compare each of the following functions to y = 1/x by identifying any stretches or translations, then draw the graph without using technology.

a) y =4x d) y =

2x - 3

+ 2b) y = - 31x c) y =

3x + 4

Example 8: Convert each of the following functions to the form

Identify the stretches and translations, then draw the graph without using technology.

.

a) y =1 - 2x

x b) y =x - 2x - 1

d) y =33 - 6xx - 5

c) y =6 - 2xx - 1

Example 9: The ideal gas law relates the pressure, volume, temperature, and molar amount of a gas with the formula PV = nRT. An ideal gas law experiment uses 0.011 mol of a gas at a temperature of 273.15 K.

a) If the temperature and molar amount of the gas are held constant, the ideal gas law follows a reciprocal relationship and can be written as a rational function, P(V). Write this function.b) If the original volume of the gas is doubled, how will the pressure change?c) If the original volume of the gas is halved, how will the pressure change?d) If P(5.0 L) = 5.0 kPa, determine the experimental value of the universal gas constant R.e) Complete the table of values and draw the graph for this experiment.f) Do the results from the table match the predictions in parts b & c?

))) ) ) )

) ) )) ) )

)))

)))

)))

)))

)))

V

P

1 2 3 4 5 6 7 8 9 10

5

10

15

20

25

30

35

40

45

50PV

0.5

1.0

2.0

5.0

10.0

(kPa)(L)

d

I

1 2 3 4 5 6

10

20

30

40

50

60

70

80

90

100

110

120

130

7 8 9 10 11 12

Id

2

4

8

12

(W/m2)(m)

1

a) If the original distance of the screen from the bulb is doubled, how does the illuminance change?b) If the original distance of the screen from the bulb is tripled, how does the illuminance change?c) If the original distance of the screen from the bulb is halved, how does the illuminance change?d) If the original distance of the screen from the bulb is quartered, how does the illuminance change?e) A typical household fluorescent bulb emits 1600 lumens. If the original distance from the bulb to the screen was 4 m, complete the table of values and draw the graph.f) Do the results from the table match the predictions made in parts a-d?

Example 10: The illuminance of light can be described with the reciprocal-square relation ,

where I is the illuminance (SI unit = lux), S is the amount of light emitted by a source (SI unit = lumens), and d is the distance from the light source in metres. In an experiment to investigate the reciprocal-squarenature of light illuminance, a screen can be moved from a baseline position to various distances from the bulb.

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Polynomial, Radical, and Rational FunctionsLESSON SIX - Rational Functions II

Lesson Notesy =

x + 2

x2 + x - 2

Example 1: Numerator Degree < Denominator Degree. Predict if any asymptotes or holes are present in the graph of each rational function. Use a graphing calculator to draw the graph and verify your prediction.

a) y = xx2 - 9

y =x2 + 1x + 2b) y =

x2 - 16x + 4c) y =

x3 - x2 - 2xx2 - x - 2

d)

Example 2: Numerator Degree = Denominator Degree. Predict if any asymptotes or holes are present in the graph of each rational function. Use a graphing calculator to draw the graph and verify your prediction.

a) y = 4xx - 2

b) y =x2 - 1

x2

c) y =x2 + 93x2

d) y =x2 - x - 6

3x2 - 3x - 18

Example 3: Numerator Degree > Denominator Degree. Predict if any asymptotes or holes are present in the graph of each rational function. Use a graphing calculator to draw the graph and verify your prediction.

a) y =x + 4

x2 + 5x + 4 b) y =x - 3

x2 - 4x + 3 d) y = x2 - x - 6x + 1

c) y = x2 + 5x - 1

Example 4: Graph without using the graphing feature of your calculator.y = xx2 - 16

Example 5: Graph without using the graphing feature of your calculator.y = 2x - 6x + 2

Example 6: Graph without using the graphing feature of your calculator.y = x2 + 2x - 8x - 1

Example 7: Graph without using the graphing feature of your calculator.y = x2 - 5x + 6x - 2

i. Horizontal Asymptote ii. Vertical Asymptote(s) v. Domain and Rangeiii. y - intercept iv. x - intercept(s)

i. Horizontal Asymptote ii. Vertical Asymptote(s) v. Domain and Rangeiii. y - intercept iv. x - intercept(s)

i. Horizontal Asymptote ii. Vertical Asymptote(s) v. Domain and Rangeiii. y - intercept iv. x - intercept(s) vi. Oblique Asymptote

iii. y - intercept iv. x - intercept(s)ii. Holesi. Can this rational function be simplified? v. Domain and Range

b) vertical asymptote(s)

horizontal asymptote

x-intercept(s)

hole(s)

x = 0

y = 0

none

(-1, -1)

Example 8: Find the rational function with each set of characteristics and draw the graph.

a) vertical asymptote(s)

horizontal asymptote

x-intercept(s)

hole(s)

x = -2, x = 4

y = 1

(-3, 0) and (5, 0)

none

Example 9: Find the rational function shown in each graph.

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Polynomial, Radical, and Rational FunctionsLESSON SIX - Rational Functions IILesson Notes

y =x + 2

x2 + x - 2

Example 10: Solve the rational equation

a) Solve algebraically and check for extraneous roots.

b) Solve the equation by finding the point of intersection of a system of functions.

c) Solve the equation by finding the x-intercept(s) of a single function.

=3xx - 1

4 in three different ways.

Example 11: Solve the rational equation 9x - 1

- =6x

-6 in three different ways.

Example 12: Solve the equation4

x + 1- =x

x - 26

x2 - x - 2in three different ways.

Example 13: Cynthia jogs 3 km/h faster than Alan. In a race, Cynthia was able to jog 15 km in the same time it took Alan to jog 10 km. How fast were Cynthia and Alan jogging?

a) Fill in the table and derive an equationthat can be used to solve this problem.b) Solve algebraically.c) Check your answer by either(i) finding the point of intersection of two functions OR (ii) finding the x-intercept(s) of a single function.

d s t

Cynthia

Alan

Example 14: George can canoe 24 km downstream and return to his starting position (upstream) in 5 h. The speed of the current is 2 km/h.What is the speed of the canoe in still water?

a) Fill in the table and derive an equationthat can be used to solve this problem.b) Solve algebraically.c) Check your answer by either(i) finding the point of intersection of two functions OR (ii) finding the x-intercept(s) of a single function.

d s t

Upstream

Downstream







Example 15: The shooting percentage of a hockey player is ratio of scored goals to total shots on goal. So far this season, Laura has scored 2 goals out of 14 shots taken. Assuming Laura scores a goal with every shot from now on, how many goals will she need to have a 40% shooting percentage?a) Derive an equation that can be used to solve this problem.b) Solve algebraically.c) Check your answer by either:(i) finding the point of intersection of two functions OR (ii) finding the x-intercept(s) of a single function.

Example 16: A 300 g mixture of nuts contains peanuts and almonds. The mixture contains 35% almonds by mass. What mass of almonds must be added to this mixture so it contains 50% almonds?a) Derive an equation that can be used to solve this problem.b) Solve algebraically.c) Check your answer by either:(i) finding the point of intersection of two functions OR (ii) finding the x-intercept(s) of a single function.

Example 1: Draw the graph resulting from each transformation. Label the invariant points.

a) y = 2f(x) c) y = f(2x) d) y = f( x)12

b) y = 3f(x) d) y = f(3x) a) y = f(x)14

c) y = f( x)15


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b) y = f(x)12

Transformations and OperationsLESSON ONE - Basic Transformations

Lesson Notes

a) y = -f(x) c) x = f(y)b) y = f(-x)


a) y = -f(x) c) x = f(y)b) y = f(-x)


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Transformations and OperationsLESSON ONE - Basic TransformationsLesson Notes

a) y = f(x) + 3 c) y = f(x - 2)b) y = f(x) - 4 d) y = f(x + 3)

Example 5: Draw the graph resulting from each transformation.

a) y - 4 = f(x) b) y = f(x) - 3 d) y = f(x + 4) c) y = f(x - 5)

Example 6: Draw the graph resulting from each transformation.

b) The graph of f(x) is horizontally translated 6 units left.

a) The graph of f(x) is horizontally stretched

c) The graph of f(x) is vertically translated 4 units down.

d) The graph of f(x) is reflected in the x-axis.

Example 7: Draw the transformed graph. Write the transformation as both an equation and a mapping.

12

by a factor of .

Transformations and OperationsLESSON ONE - Basic Transformations

Lesson Notes

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Example 8: Write a sentence describing each transformation, then write the transformation equation.

a) b) c) d)

Original graph:

Transformed graph:

Example 9: Describe each transformation and derive the equation of the transformed graph. Draw the original and transformed graphs.

Transformation: y = 2f(x)Original graph: f(x) = x2 - 1

Transformation: y = f(2x)Original graph: f(x) = x2 + 1

Transformation: y = -f(x)Original graph: f(x) = x2 - 2

Transformation: y = f(-x)Original graph: f(x) = (x - 6)2

a) b)

c) d)

Example 10: Describe each transformation and derive the equation of the transformed graph. Draw the original and transformed graphs.

Transformation: y - 2 = f(x)Original graph: f(x) = x2

Transformation: y = f(x) - 4Original graph: f(x) = x2 - 4

Transformation: y = f(x - 2)Original graph: f(x) = x2

Transformation: y = f(x - 7)Original graph: f(x) = (x + 3)2

a) b)

c) d)

Example 11: What Transformation Occured?

a) The graph of y = x2 + 3 is vertically translated so it passes through the point (2, 10).Write the equation of the applied transformation. Solve graphically first, then solve algebraically.

b) The graph of y = (x + 2)2 is horizontally translated so it passes through the point (6, 9).Write the equation of the applied transformation. Solve graphically first, then solve algebraically.

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Transformations and OperationsLESSON ONE - Basic TransformationsLesson Notes

Example 12: What Transformation Occured?a) The graph of y = x2 - 2 is vertically stretched so it passes through the point (2, 6).Write the equation of the applied transformation. Solve graphically first, then solve algebraically.b) The graph of y = (x - 1)2 is transformed by the equation y = f(bx). The transformed graph passes through the point (-4, 4). Write the equation of the applied transformation. Solve graphically first, then solve algebraically.

Example 13: Sam sells bread at a farmers’ market for $5.00 per loaf.It costs $150 to rent a table for one day at the farmers’ market, and each loaf of bread costs $2.00 to produce.

Example 14: A basketball player throws a basketball.

a) Suppose the player moves 2 m closer to the hoop before making the shot. Determine the equation of the transformed graph, draw the graph, and predict the outcome of the shot.

b) If the player moves so the equation of the shot is , what is the horizontal distance from the player to the hoop?

h(d) = -19

(d + 1)2 + 4

2

d

h(d)

3

4

1

1 2 3 4 5 6 7

5

8 9-5 -4 -3 -2 -1 0

h(d) = -19

(d - 4)2 + 4The path can be modeled with .

a) Write two functions, R(n) and C(n), to represent Sam’s revenue and costs. Graph each function.b) How many loaves of bread does Sam need to sell in order to make a profit?c) The farmers’ market raises the cost of renting a table by $50 per day. Use a transformation to find the new cost function, C2(n).d) In order to compensate for the increase in rental costs, Sam will increase the price of a loaf of bread by 20%. Use a transformation to find the new revenue function, R2(n).e) Draw the transformed functions from parts (c) and (d). How many loaves of bread does Sam need to sell now in order to break even?

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Transformations and OperationsLESSON TWO - Combined Transformations

Lesson Notesy = af[b(x - h)] + k

Example 1: Combined Transformationsa) Identify each parameter in the general transformation equation: y = af[b(x - h)] + k.b) Describe the transformations in each equation:

13

i. y = f(5x)14

ii. y = 2f( x)12

iii. y = - f( x)13

iv. y = -3f(-2x)

Example 2: Draw the transformation of each graph.

a) y = 2f( x)13

d) y = - f(-x)12

b) y = f(-x)13

c) y = -f(2x)

a) Find the horizontal translation of y = f(x + 3).

b) Describe the transformations in each equation:i. y = f(x - 1) + 3 ii. y = f(x + 2) - 4 iii. y = f(x - 2) - 3 iv. y = f(x + 7) + 5

Example 3: Answer the following questions:

b) y = f(x - 3) + 7 c) y - 12 = f(x - 6) d) y + 2 = f(x + 8)a) y = f(x + 5) - 3


a) When applying transformations to a graph, should they be applied in a specific order?

b) Describe the transformations in each equation:

i. y = 2f(x + 3) + 1

Example 5: Answer the following questions:

ii. y = -f( x) - 413

iii. y = f[-(x + 2)] - 312

iv. y = -3f[-4(x - 1)] + 2

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a) y = -f(x) - 2 d) 2y - 8 = 6f(x - 2)b) y = f(- x) + 114

c) y = - f(2x) - 114



a) y = f[ (x - 1)] + 113

d) y = f(-x - 4)13

b) y = f(2x + 6) c) y = f(3x - 6) - 2

Example 8: The mapping for combined transformations is shown.a) If the point (2, 0) exists on the graph of y = f(x), find the coordinates of the new point after the transformation y = f(-2x + 4).

b) If the point (5, 4) exists on the graph of y = f(x), find the coordinates

y = f(5x - 10) + 4.12

of the new point after the transformation

c) The point (m, n) exists on the graph of y = f(x). If the transformation y = 2f(2x) + 5 is applied tothe graph, the transformed point is (4, 7). Find the values of m and n.

ii

Example 9: For each transformation description, write the transformation equation. Use mappings to draw the transformed graph.a) The graph of y = f(x) is vertically stretched by a factor of 3, reflected about the x-axis, and translated 2 units to the right.b) The graph of y = f(x) is horizontally stretched by a factor of 1/3, reflected about the x-axis, and translated 2 units left.

Example 9a Example 9b

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Example 10: Greg applies the transformation y = -2f[-2(x + 4)] - 3 to the graph below, using the transformation order rules learned in this lesson.

Stretches & Reflections:1) Vertical stretch by a scale factor of 22) Reflection about the x-axis3) Horizontal stretch by a scale factor of 1/24) Reflection about the y-axis

Translations:5) Vertical translation 3 units down6) Horizontal translation 4 units left

Greg’s Transformation Order:

Next, Colin applies the same transformation, y = -2f[-2(x + 4)] - 3, to the graph below. He tries a different transformation order, applying all the vertical transformations first, followed by all the horizontal transformations.

According to the transformation order rules we have been using in this lesson (stretches & reflections first, translations last), Colin should obtain the wrong graph.However, Colin obtains the same graph as Greg! How is this possible?

Vertical Transformations:1) Vertical stretch by a scale factor of 22) Reflection about the x-axis3) Vertical translation 3 units down.

Horizontal Transformations:4) Horizontal stretch by a scale factor of 1/25) Reflection about the y-axis6) Horizontal translation 4 units left

Colin’s Transformation Order:

Original graph:

Transformed graph:

Original graph:

Transformed graph:

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Example 11: The goal of the video game Space Rocks is to pilot a spaceship through an asteroid field without colliding with any of the asteroids.

a) If the spaceship avoids the asteroid by navigating to the position shown, describe the transformation.

Original position of ship

Final position of ship

b) Describe a transformation that will let the spaceship pass through the asteroids.

c) The spaceship acquires a power-up that gives it greater speed, but at the same time doubles its width. What transformation is shown in the graph?

d) The spaceship acquires two power-ups. The first power-up halves the original width of the spaceship, making it easier to dodge asteroids. The second power-up is a left wing cannon. What transformation describes the spaceship’s new size and position?

e) The transformations in parts (a - d) may not be written using y = af[b(x - h)] + k.Give two reasons why.

a) Given the graph of y = 2x + 4, draw the graph of the inverse. What is the equation of the line of symmetry?b) Find the inverse function algebraically.

Example 1: Inverse Functions.

Example 3: For each graph, draw the inverse. How should the domain of the original graph be restricted so the inverse is a function?

a) b) c) d)

a) b)

a) b)

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Transformations and OperationsLESSON THREE - Inverses

Lesson Notesf-1(x)

Example 2: For each graph, answer parts (i - iv).

i. Draw the graph of the inverse.ii. State the domain and range of the original graph.iii. State the domain and range of the inverse graph.iv. Can the inverse be represented with f-1(x)?

Example 4: Find the inverse of each linear function algebraically. Draw the graph of the original function and the inverse. State the domain and range of both f(x) and its inverse.

a) f(x) = x - 3 b) 12

x- - 4f(x) =

Example 5: Find the inverse of each quadratic function algebraically. Draw the graph of the original function and the inverse. Restrict the domain of f(x) so the inverse is a function.

a) f(x) = x2 - 4 b) f(x) = -(x + 3)2 + 1

Example 6: For each graph, find the equation of the inverse.

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Transformations and OperationsLESSON THREE - InversesLesson Notes

f-1(x)

a) If f(x) = 2x - 6, find the inverse function and determine the value of f-1(10).b) Given that f(x) has an inverse function f-1(x), is it true that if f(a) = b, then f-1(b) = a? c) If f-1(4) = 5, determine f(5).d) If f-1(k) = 18, determine the value of k.

Example 7: Answer the following questions.

32°0°

CelsiusThermometer

FahrenheitThermometer

Example 8: In the Celsius temperature scale, the freezing point of water is set at 0 degrees. In the Fahrenheit temperature scale, 32 degrees is the freezing point of water.

a) Determine the temperature in degrees Fahrenheit for 28 °C.

b) Derive a function, C(F), to convert degrees Fahrenheit to degrees Celsius. Does one need to understand the concept of an inverse to accomplish this?

c) Use the function C(F) from part (b) to determine the temperature in degrees Celsius for 100 °F.

d) What difficulties arise when you try to graph F(C) and C(F) on the same grid?

e) Derive F-1(C). How does F-1(C) fix the graphing problem in part (d)?

f) Graph F(C) and F-1(C) using the graph above. What does the invariant point for these two graphs represent?

The formula to convert degrees Celsius

F(C) = C + 3295

to degrees Fahrenheit is:

°C

50

100°F

-50

50-50 100-100

-100

a) h(x) = (f + g)(x)

Example 1: Given the functions f(x) and g(x), draw the graph. State the domain and range of the combined function.

b) h(x) = (f - g)(x)

f(x)

g(x)

c) h(x) = (f • g)(x) d) h(x) =

f(x)

g(x)

f(x)

g(x)

f(x)

g(x)

Example 2: Given the functions f(x) = x - 3 and g(x) = -x + 1, evaluate:

a) (f + g)(-4) b) (f - g)(6)

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Transformations and OperationsLESSON FOUR - Function Operations

Lesson Notes

(f + g)(x) (f - g)(x)

(f • g)(x)f

(x)g

c) (fg)(-1) d)gf

(5)

Example 3: Draw each combined function and state the domain and range.

a) h(x) = (f + g)(x) b) h(x) = (f - g)(x) c) h(x) = (f • g)(x) d) h(x) = (f + g + m)(x)

f(x)

g(x)

f(x)

g(x)

m(x)

f(x)

g(x)

f(x)

g(x)

a) (f + g)(x) b) (f • g)(x)

f(x)

g(x)

f(x)

g(x)

f(x)

g(x)

f(x)

g(x)

Example 4: Given the functionsand g(x) = -1, answer the following questions.

x + 4 + 12f(x) = Example 5: Given the functions f(x) = -(x - 2)2 - 4 and g(x) = 2, answer the following questions.

a) (f - g)(x) b)

Examples 4 & 5: i. Graph. ii. Derive the resultant function, h(x). iii. State the domain & range of h(x).iv. Write a transformation equation that transforms the graph of f(x) to h(x).

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Transformations and OperationsLESSON FOUR - Function OperationsLesson Notes

(f + g)(x) (f - g)(x)

(f • g)(x)f

(x)g

a) f(x) = 1 and g(x) = x

Example 6: Draw the graph of . Derive h(x) and state the domain and range.h(x) =

b) f(x) = 1 and g(x) = x - 2

f(x)

g(x)

f(x)

g(x)

d) f(x) = x + 3c) f(x) = x + 3 and g(x) = x2 + 6x + 9

f(x)g(x)

f(x)

g(x)

i. Graph. ii. Derive h(x) = (f ÷ g)(x) iii. State the domain & range of h(x).

and g(x) = x + 2

Example 7: Two rectangular lots are adjacent to each other, as shown in the diagram.a) Write a function, AL(x), for the area of the large lot.b) Write a function, AS(x), for the area of the small lot.c) If the large rectangular lot is 10 m2 larger than the small lot, use a function operation to solve for x.d) Using a function operation, determine the total area of both lots.e) Using a function operation, determine how many times bigger the large lot is than the small lot. 3x - 3

x

4x

2x - 2

Example 8: Greg wants to to rent a stand at a flea market to sell old video game cartridges. He plans to acquire games for $4 each from an online auction site, then sell them for $12 each. The cost of renting the stand is $160 for the day.a) Using function operations, derive functions for revenue R(n), expenses E(n), and profit P(n). Graph each function.b) What is Greg’s profit if he sells 52 games? c) How many games must Greg sell to break even?

r

h

slantheight

SA = πr2 + πrs

V = 13

πr2h

Example 9: The surface area and volume of a right cone are shown,where r is the radius of the circular base, h is the heightof the apex, and s is the slant height of the side of the cone.

3A particular cone has a height that is times larger than the radius.

a) Can we write the surface area and volume formulae as single-variable functions?b) Express the apex height in terms of r.c) Express the slant height in terms of r.d) Rewrite both the surface area and volume formulae so they are single-variable functions of r.e) Use a function operation to determine the surface area to volume ratio of the cone.f) If the radius of the base of the cone is 6 m, find the exact value of the surface area to volume ratio.

Example 1: Given the functions f(x) = x - 3 and g(x) = x2:x f(x)

0

1

2

3

g(f(x))a) Complete the table of values for (f ◦ g)(x).b) Complete the table of values for (g ◦ f)(x).c) Does order matter when performing a composition?d) Derive m(x) = (f ◦ g)(x).e) Derive n(x) = (g ◦ f)(x).f) Draw m(x) and n(x).

x g(x)

-3

-2

-1

0

1

f(g(x))

2

3


Example 2: Given the functions f(x) = x2 - 3 and g(x) = 2x, evaluate each of the following:

a) m(3) = (f ◦ g)(3) b) n(1) = (g ◦ f)(1)c) p(2) = (f ◦ f)(2) d) q(-4) = (g ◦ g)(-4)

Example 3: Given the functions f(x) = x2 - 3 and g(x) = 2x (these are the same functions found in Example 2), find each composite function.

a) m(x) = (f ◦ g)(x) b) n(x) = (g ◦ f)(x) c) p(x) = (f ◦ f)(x) d) q(x) = (g ◦ g)(x)e) Using the composite functions derived in parts (a - d), evaluate m(3), n(1), p(2), and q(-4).Do the results match the answers in Example 2?

Example 4: Given the functions f(x) and g(x), find each composite function. Make note of any transformations as you complete your work.

a) m(x) = (f ◦ g)(x) b) n(x) = (g ◦ f)(x)

Examples 6 & 7: Given f(x), g(x), m(x), and n(x), find each composite function and state the domain.

m(x) = |x| n(x) = x + 2xf(x) =1

xg(x) =

Example 6: a) h(x) = [g ◦ m ◦ n](x) b) h(x) = [n ◦ f ◦ n](x)

Example 7: a) h(x) = [(gg) ◦ n](x) b) h(x) = [f ◦ (n + n)](x)

Transformations and OperationsLESSON FIVE - Function Composition

Lesson Notesf ◦ g = f(g(x))

f(x) = (x + 1)2 g(x) = 3x

Example 5: Given f(x) and g(x), find the composite function m(x) = (f ◦ g)(x) and state the domain.

a) b)

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Example 10: The price of 1 L of gasoline is $1.05. On a level road, Darlene’s car uses 0.08 L of fuel for every kilometre driven.a) If Darlene drives 50 km, how much did the gas cost to fuel the trip? How many steps does it take to solve this problem (without composition)?b) Write a function, V(d), for the volume of gas consumed as a function of the distance driven.c) Write a function, M(V), for the cost of the trip as a function of gas volume.d) Using function composition, combine the functions from parts b & c into a single function, M(d), where M is the money required for the trip. Draw the graph.e) Solve the problem from part (a) again, but this time use the function derived in part (d). How many steps does the calculation take now?

Example 11: A pebble dropped in a lake creates a circular wave that travels outward at a speed of 30 cm/s.a) Use function composition to derive a function, A(t), that expresses the area of the circular wave as a function of time.b) What is the area of the circular wave after 3 seconds?c) How long does it take for the area enclosed by the circular wave to be 44100π cm2? What is the radius of the wave?

American Dollars = 1.03 × Canadian DollarsEuros = 0.77 × American DollarsJapanese Yen = 101.36 × EurosBritish Pounds = 0.0083 × Japanese Yen

$CAD

€

£¥

$USD

Example 12: The exchange rates of several currencies on a particular day are listed below:a) Write a function, a(c), that converts Canadian Dollars to American Dollars.b) Write a function, j(a), that converts American Dollars to Japanese Yen.c) Write a function, b(a), that converts American Dollars to British Pounds.d) Write a function, b(c), that converts Canadian Dollars to British Pounds.

Example 13: A drinking cup from a water fountain has the shape of an inverted cone. The cup has a height of 8 cm, and a radius of 3 cm. The water in the cup also has the shape of an inverted cone, with a radius of r and a height of h. The diagram of the drinking cup shows two right triangles: a large triangle for the entire height of the cup, and a smaller triangle for the water in the cup. The two triangles have identical angles, so they can be classified as similar triangles.a) Use similar triangle ratios to express r as a function of h.b) Derive the composite function, Vwater(h) = (Vcone ◦ r)(h), for the volume of the water in the cone.c) If the volume of water in the cone is 3π cm3, determine the height of the water.

Transformations and OperationsLESSON FIVE - Function CompositionLesson Notes

f ◦ g = f(g(x))

Example 8: Given the composite function h(x) = (f ◦ g)(x), find the component functions, f(x) and g(x). (More than one answer is possible)

a) h(x) = 2x + 2 b) h(x) =x2 - 1

1c) h(x) = (x + 1)2 - 5(x + 1) + 1

d) h(x) = x2 + 4x + 4 f) h(x) = |x|h(x) = 2e)x1

x +31

32a) f(x) = 3x - 2 and f-1(x) = b) f(x) = x - 1 and f-1(x) = 1 - x

Example 9: Two functions are inverses if (f-1 ◦ f)(x) = x. Determine if each pair of functions are inverses of each other.

3 cm

8 cmr

h

V = πr2h 13cone

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Exponential and Logarithmic FunctionsLESSON ONE - Exponential Functions

Lesson Notesy = bx

Example 1: For each exponential function:

i. Create a table of values and draw the graph.ii. State the domain, range, intercepts, and the equation of the asymptote.

12

x

y =

c) d)a) b)

e) Define exponential function. Are the functions y = 0x and y = 1x considered exponential functions? What about y = (-1)x ?

Example 2: Determine the exponential function corresponding to each graph, then use the function to find the unknown. All graphs in this example have the form y = bx.

100

50

(2, 16)

(3, 64)

(-2, n)

a)10

5

(-3, n)

b)

50

100

(-3, 125)

(-2, 25)

(1, n)

c)10

5

(3, n)

d)

a) c) d)b)

Example 3: Draw the graph. State the domain, range, and equation of the asymptote.

Example 4: Draw the graph. State the domain, range, and equation of the asymptote.

a) c) d)b)




Interval Notation


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Exponential and Logarithmic FunctionsLESSON ONE- Exponential Functions

Lesson Notesy = bx

Example 6: Assorted Questions.

a) What is the y-intercept of f(x) = abx - 4 ?

c) If the graph of is stretched vertically so it passes through the point ,

what is the equation of the transformed graph?

b) The point exists on the graph of y = a(5)x. What is the value of a?

d) If the graph of y = 2x is vertically translated so it passes through the point (3, 5), whatis the equation of the transformed graph?

e) If the graph of y = 3x is vertically stretched by a scale factor of 9, can this be written as a horizontal translation?

f) Show algebraically that each pair of graphs are identical.

i. ii.

iv. v.

iii.

Example 5: Determine the exponential function corresponding to each graph, then use the function to find the unknown. Both graphs in this example have the form y = abx + k.

5

(-5, n)

10

5-5(0, -2)

5

5-5

10

(-5, n)

a) b)

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Exponential and Logarithmic FunctionsLESSON ONE - Exponential Functions

Lesson Notesy = bx

a) c)b) d)

Example 11: Solving equations where x is in the exponent. (Multiple Powers)

Example 12: Solving equations where x is in the exponent. (Radicals)

a) b) c) d)

Example 13: Solving equations where x is in the exponent. (Factoring)

a) b) c) d)

a) b) c) d)

Example 14: Solving equations where x is in the exponent. (No Common Base - Use Technology)

a) c)b) d)

Example 9: Solving equations where x is in the exponent. (Fractional Base)

Example 10: Solving equations where x is in the exponent. (Fractional Exponents)

a) c)b) d)

Example 7: Solving equations where x is in the base. (Raising Reciprocals)

a) b) c) d)

Example 8: Solving equations where x is in the exponent. (Common Base)

a) c)b) d)

e) Determine x and y: and f) Determine m and n: and

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Exponential and Logarithmic FunctionsLESSON ONE- Exponential Functions

Lesson Notesy = bx

Example 17: In 1990, a personal computer had a processor speed of 16 MHz. In 1999, a personal computer had a processor speed of 600 MHz. Based on these values, the speed of a processor increased at an average rate of 44% per year.a) Estimate the processor speed of a computer in 1994 (t = 4). How does this compare with actual processor speeds (66 MHz) that year?b) A computer that cost $2500 in 1990 depreciated at a rate of 30% per year. How much was the computer worth four years after it was purchased?

Example 18: A city with a population of 800,000 is projected to grow at an annual rate of 1.3%.a) Estimate the population of the city in 5 years.b) How many years will it take for the population to double?c) If projections are incorrect, and the city’s population decreases at an annual rate of 0.9%, estimate how many people will leave the city in 3 years. d) How many years will it take for the population to be reduced by half?

$$$

Example 19: $500 is placed in a savings account that compounds interest annually at a rate of 2.5%.a) Write a function, A(t), that relates the amount of the investment, A, with the elapsed time t.b) How much will the investment be worth in 5 years? How much interest has been received?c) Draw the graph for the first 20 years.d) How long does it take for the investment to double?e) Calculate the amount of the investment in 5 years if compounding occurs i. semi-annually, ii. monthly, and iii. daily. Summarize your results in a table.

Example 16: A bacterial culture contains 800 bacteria initially and doubles every 90 minutes.a) Write a function, B(t), that relates the quantity of bacteria, B, to the elapsed time, t.b) How many bacteria will exist in the culture after 8 hours?c) Draw the graph for the first ten hours.d) How long ago did the culture have 50 bacteria?

Example 15: A 90 mg sample of a radioactive isotope has a half-life of 5 years.a) Write a function, m(t), that relates the mass of the sample, m, to the elapsed time, t.b) What will be the mass of the sample in 6 months?c) Draw the graph for the first 20 years.d) How long will it take for the sample to have a mass of 0.1 mg?

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Exponential and Logarithmic FunctionsLESSON TWO - Laws of Logarithms

Lesson NoteslogBA = E

Example 1: Introduction to Logarithms.

a) Label the components of logBA = E and A = BE.

b) Evaluate each logarithm.

c) Which logarithm is bigger?

i. log21 or log42

ii. ori. log21 = log22 = log24 = log28 =, , ,

ii. log 1 = log 10 = log 100 = log 1000 =, , ,

Example 2: Order each set of logarithms from least to greatest.

a)

b)

(Estimate the order using benchmarks)c)

Example 3: Convert each equation from logarithmic to exponential form. (The Seven Rule)Express answers so y is isolated on the left side.

a)

e) g)

d)

f) h)

b) c)

Example 4: Convert each equation from exponential to logarithmic form. (A Base is Always a Base)Express answers with the logarithm on the left side.

a) c)

e) g)

b) d)

f) h)

Example 5: Evaluate each logarithm using change of base. (Change of Base)

a) c)

e) g)

b) d)

f) h)

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Example 6: Expand each logarithm using the product law. (Product Law)

a) c)

e) g)

b) d)

f) h)

In parts (e - h), condense each expression to a single logarithm.

a) b) d)

e) f) g) h)


c)

Example 7: Expand each logarithm using the quotient law. (Quotient Law)

Example 8: Expand each logarithm using the power law. (Power Law)

a) b)

e) g)

d)c)

f) h)


Example 9: Expand each logarithm using the appropriate logarithm rule. (Other Rules)

a) c)

e) g)

b) d)

f) h)

If 10k = 4, then 101 + 2k =a)

If logb4 = k, then logb16 = c) Write logx + 1 as a single logarithm.g)

If 3a = k, then log3k4 =b)

If log2a = h, then log4a = d) Write 3 + log2x as a single logarithm.h)

If logbh = 3 and logbk = 4,e)

then =

If logh4 = 2 and log8k = 2,f)

then log2(hk) =

Example 10: Use logarithm laws to answer each of the following questions. (Substitution Questions)

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a)

c)

b)

d)

Example 11: Solving Exponential Equations. (No Common Base)

a) b) c) d)

Example 12: Solving Exponential Equations. (No Common Base)

a) b) c) d)

a) b)

c) d)

Example 13: Solving Logarithmic Equations. (One Solution)

a)

c) d)

b)

Example 14: Solving Logarithmic Equations. (Multiple Solutions)

Example 15: Solving Logarithmic Equations. (Multiple Solutions)

a) Evaluate. b) Condense.

h) Condense.g) If loga3 = x and loga4 = 12,

then loga122 =

(express answer in terms of x.)

c) Solve. d) Evaluate.

e) Write as a logarithm. f) Show that:

Example 16: Assorted Mix I

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e) Solve. f) Solve. h) Solve.g) Condense.

b) Evaluate.a) Evaluate. d) Solve.c) What is one-third of 3234 ?

e) Evaluate. g) Solve. h) If xy = 8, then5log2x + 5log2y =

f) Condense.

Example 17: Assorted Mix II

a) Evaluate. c) Condense.b) Solve. d) Solve.

Example 18: Assorted Mix III

f) Evaluate. h) Condense.e) Condense. g) Show that:

Example 19: Assorted Mix IV

b) Condense.a) Solve. c) Solve. d) Condense.

g) Evaluate.e) Evaluate. f) Solve. h) Condense.

Example 20: Assorted Mix V

a) Solve. c) Evaluate.b) Solve.

d) Condense.

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Exponential and Logarithmic FunctionsLESSON THREE - Logarithmic Functions

Lesson Notes

f) Are y = log1x, y = log0x, and y = log-2xlogarithmic functions? What about ?

10g) Define logarithmic function.h) How can y = log2x be graphed in a calculator?

y = logbx

Examples 2 - 6: Draw each of the following graphs without technology.State the domain, range, and asymptote equation.

a) b) c) d)Example 2:

a) b) c) d)

a) Draw the graph of f(x) = 2x.b) Draw the inverse of f(x).c) Show algebraically that the inverse of f(x) = 2x is f-1(x) = log2x.d) State the domain, range, intercepts, and asymptotes of both graphs.e) Determine the value of: i. log20.5, ii. log21, iii. log22, iv. log27

Example 1: Logarithmic Functions

Domain

Range

x-intercept

y-intercept

AsymptoteEquation

y = 2x y = log2x

5

10

-5

-5 5

Example 3:

Example 4: a) b) c) d)



Example 7: Exponential Equations. Solve each equation by (i) finding a common base (if possible), (ii) using logarithms, and (iii) graphing.

b)a) c)

Example 8: Logarithmic Equations. Solve each equation by (i) using logarithm laws, and (ii) graphing.

a) b) c)

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Exponential and Logarithmic FunctionsLESSON THREE- Logarithmic Functions

Lesson Notesy = logbx

a) The graph of y = log3x can be transformed to the graph of y = log3(9x) by either a stretch or a translation. What are the two transformation equations?

c) A vertical translation is applied to the graph of y = log3x so the image has an x-intercept of (9, 0). What is the transformation equation?

b) If the point (4, 1) exists on the graph of y = log4x, what is the point after the transformation y = log4(2x + 6)?

d) What is the point of intersection of f(x) = log2x and g(x) = log2(x + 3) - 2?

e) What is the x-intercept of y = alogb(kx)?

Example 9: Assorted Mix I

e) What is the domain of f(x) = logx(6 – x)?

a) The graph of y = logbx passes through the point (8, 2). What is the value of b?

c) What is the equation of the asymptote for y = log3(3x – 8)?

b) What are the x- and y-intercepts of y = log2(x + 4)?

d) The point (27, 3) lies on the graph of y = logbx. If the point (4, k) exists on the graph of y = bx, then what is the value of k?

Example 10: Assorted Mix II

Example 11: Assorted Mix III

c) What is the inverse of f(x) = 3x + 4?

b) If the point (a, 0) exists on the graph of f(x), and the point (0, a) exists on the graph of g(x), what is the transformation equation?

d) If the graph of f(x) = log4x is transformed by the equation y = f(3x – 12) + 2, what is the new domain of the graph?

a) What is the equation of the reflection line for the graphs of f(x) = bx and ?

e) The point (k, 3) exists on the inverse of y = 2x. What is the value of k?

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Exponential and Logarithmic FunctionsLESSON THREE - Logarithmic Functions


Example 12: The strength of an earthquake is calculated using Richter’s formula, where M is the magnitude of the earthquake (unitless), A is the seismograph amplitude of the earthquake being measured (m), and A0 is the seismograph amplitude of a threshold earthquake (10-6 m).

g) What is the magnitude of an earthquake with triple the seismograph amplitude of a magnitude 5.0 earthquake?h) What is the magnitude of an earthquake with one-fourth the seismograph amplitude of a magnitude 6.0 earthquake?

e) What is the ratio of seismograph amplitudes for earthquakes with magnitudes of 5.0 and 6.0?

which compares two amplitudes directly without requiring A0. Derive this formula.

f) Show that an equivalent form of the equation is:

a) An earthquake has a seismograph amplitude of 10-2 m. What is the magnitude of the earthquake?b) The magnitude of an earthquake is 5.0 on the Richter scale. What is the seismograph amplitude of this earthquake?c) Two earthquakes have magnitudes of 4.0 and 5.5. Calculate the seismograph amplitude ratio for the two earthquakes.d) The calculation in part (c) required multiple steps because we are comparing each amplitude with A0, instead of comparing the two amplitudes to each other. It is possible to derive the formula:

Example 13: The loudness of a sound is measured in decibels, and can be calculated using the formula shown, where L is the perceived loudness of the sound (dB), I is the intensity of the sound being measured (W/m2), and I0 is the intensity of sound at the threshold of human hearing (10-12 W/m2).

a) The sound intensity of a person speaking in a conversation is 10-6 W/m2. What is the perceived loudness?b) A rock concert has a loudness of 110 dB. What is the sound intensity?c) Two sounds have decibel measurements of 85 dB and 105 dB. Calculate the intensity ratio for the two sounds.d) The calculation in part (c) required multiple steps because we are comparing each sound with I0, instead of comparing the two sounds to each other. It is possible to derive the formula:

which compares two sounds directly without requiring I0. Derive this formula.

continued on next page...

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Exponential and Logarithmic FunctionsLESSON THREE- Logarithmic Functions


g) What is the loudness of a sound twice as intense as 20 dB?h) What is the loudness of a sound half as intense as 40 dB?

e) How many times more intense is 40 dB than 20 dB?

f) Show that an equivalent form of the equation is:

Example 14: The pH of a solution can be measured with the formula shown, where [H+] is the concentration of hydrogen ions in the solution (mol/L). Solutions with a pH less than 7 are acidic, and solutions with a pH greater than 7 are basic.

a) What is the pH of a solution with a hydrogen ion concentration of 10-4 mol/L? Is this solution acidic or basic?b) What is the hydrogen ion concentration of a solution with a pH of 11?c) Two acids have pH values of 3.0 and 6.0. Calculate the hydrogen ion ratio for the two acids.d) The calculation in part (c) required multiple steps. Derive the formulae (on right) that can be used to compare the two acids directly.e) What is the pH of a solution 1000 times more acidic than a solution with a pH of 5?f) What is the pH of a solution with one-tenth the acidity of a solution with a pH of 4?g) How many times more acidic is a solution with a pH of 2 than a solution with a pH of 4?

and

Example 15: In music, a chromatic scale divides an octave into 12 equally-spaced pitches. An octave contains 1200 cents (a unit of measure for musical intervals), and each pitch in the chromatic scale is 100 cents apart. The relationship between cents and note frequency is given by the formula shown.

a) How many cents are in the interval between A (440 Hz) and B (494 Hz)?b) There are 100 cents between F# and G. If the frequency of F# is 740 Hz, what is the frequency of G?c) How many cents separate two notes, where one note is double the frequency of the other note?

♯ ♯♯ ♯ ♯

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TrigonometryLESSON ONE - Degrees and Radians

Lesson Notes

Example 1: Define each term or phrase and draw a sample angle.

a) Angle in standard position. Draw a standard position angle, θ.

c) Reference angle. Find the reference angle of θ = 150°.

210°×π

180°= 7π

6

b) Positive and negative angles. Draw θ = 120° and θ = -120°.

d) Co-terminal angles. Draw the first positive co-terminal angle of 60°. e) Principal angle. Find the principal angle of θ = 420°.f) General form of co-terminal angles. Find the first fourpositive and negative co-terminal angles of θ = 45°.

Example 2: Three Angle Types: Degrees, Radians, and Revolutions.

a) i. Define degrees. Draw θ = 1°. ii. Define radians. Draw θ = 1 rad. iii. Define revolutions. Draw θ = 1 rev.

b) Use conversion multipliers to answer the questions and fill in the reference chart.i. 23° = __ rad ii. 23° = __ rev iii. 2.6 = __° iv. 2.6 = __ rev v. 0.75 rev = __° vi. 0.75 rev = __rad

Conversion Multiplier Reference Chart (Example 2)

degree radian revolution

degree

radian

revolution

c) Contrast the decimal approximation of a radian with the exact value of a radian.i. 45° = _____ rad (decimal approximation). ii. 45° = _____ rad (exact value).

Example 3: Convert each angle to the requested form. Round all decimals to the nearest hundredth.a) convert 175° to an approximate radian decimal. b) convert 210° to an exact-value radian.c) convert 120° to an exact-value revolution. d) convert 2.5 to degrees. e) convert 3π/2 to degrees.f) write 3π/2 as an approximate radian decimal. g) convert π/2 to an exact-value revolution.h) convert 0.5 rev to degrees. i) convert 3 rev to radians.

Example 4: The diagram shows commonly used degrees. Find exact-value radians that correspond to each degree. When complete, memorize the diagram.a) Method One: Find all exact-value radians using a conversion multiplier.b) Method Two: Use a shortcut (counting radians).

30° =

45° =

60° =

0° =

330° =

315° =300° =

90° =

= 270°

= 120°

= 135°

= 150°

= 180°

= 210°

= 225°= 240°

360° =Example 5: Draw each of the following angles in standard position. State the reference angle.a) 210° b) -260° c) 5.3 d) -5π/4 e) 12π/7

Example 6: Draw each of the following angles in standard position. State the principal and reference angles.a) 930° b) -855° c) 9 d) -10π/3

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Example 10: In addition to the three primary trigonometric ratios (sinθ, cosθ, and tanθ), there are three reciprocal ratios (cscθ, secθ, and cotθ). Given a triangle with side lengths of x and y, and a hypotenuse of length r, the six trigonometric ratios are as follows:

a) If the point P(-5, 12) exists on the terminal arm of an angle θ in standard position, determine the exact values of all six trigonometric ratios. State the reference angle and the standard position angle.

sinθ =yr

cosθ =xr

tanθ =yx

cscθ =

secθ =

cotθ =x

yr

θ

ry

rx

xy

1cosθ

1sinθ

1tanθ

=

=

=

b) If the point P(2, -3) exists on the terminal arm of an angle θ in standard position, determine the exact values of all six trigonometric ratios. State the reference angle and the standard position angle.

Example 11: Determine the sign of each trigonometric ratio in each quadrant.

a) sinθ b) cosθ c) tanθ d) cscθ e) secθ f) cotθ

g) How do the quadrant signs of the reciprocal trigonometric ratios (cscθ, secθ, and cotθ) compare to the quadrant signs of the primary trigonometric ratios (sinθ, cosθ, and tanθ)?

Example 12: Given the following conditions, find the quadrant(s) where the angle θ could potentially exist.

a) i. sinθ < 0 ii. cosθ > 0 iii. tanθ > 0

b) i. sinθ > 0 and cosθ > 0 ii. secθ > 0 and tanθ < 0 iii. cscθ < 0 and cotθ > 0

c) i. sinθ < 0 and cscθ = 1/2 ii. cosθ = - /2 and cscθ < 0 iii. secθ > 0 and tanθ = 1

Example 13: Given one trigonometric ratio, find the exact values of the other five trigonometric ratios. State the reference angle and the standard position angle, to the nearest hundredth of a radian.

a) b)

Example 14: Given one trigonometric ratio, find the exact values of the other five trigonometric ratios. State the reference angle and the standard position angle, to the nearest hundredth of a degree.

b)a)

TrigonometryLESSON ONE - Degrees and RadiansLesson Notes

210°×π

180°=

7π6

Example 7: For each angle, find all co-terminal angles within the stated domain.a) 60°, Domain: -360° ≤ θ < 1080° b) -495°, Domain: -1080° ≤ θ < 720°c) 11.78, Domain: -2π ≤ θ < 4π d) 8π/3, Domain: -13π/2 ≤ θ < 37π/5

Example 8: For each angle, use estimation to find the principal angle.a) 1893° b) -437.24 c) 912π/15 d) 95π/6

Example 9: Use the general form of co-terminal angles to find the specified angle.a) θp = 300° (Find θc, 3 rotations CC) b) θp = 2π/5 (Find θc, 14 rotations C)c) θc = -4300° (Find n and θp) d) θc = 32π/3 (Find n and θp)

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TrigonometryLESSON ONE - Degrees and Radians

Lesson Notes210°×

π180°

=7π6

b) Given the point P(-4, 3), Mark tries to find the reference angle using a sine ratio, Jordan tries to find it using a cosine ratio, and Dylan tries to find it using a tangent ratio. Why does each person get a different result from their calculator?

Mark’s Calculation of θ (using sine)

Jordan’s Calculation of θ (using cosine)

Dylan’s Calculation of θ (using tan)

sinθ =35

θ = 36.87°

cosθ =-45

θ = 143.13°

tanθ =3-4

θ = -36.87°

a) When you solve a trigonometric equation in your calculator, the answer you get for θ can seem unexpected. Complete the following chart to learn how the calculator processes your attempt to solve for θ.

Example 15: Calculating θ with a calculator. If the angle θ could exist in either quadrant ___ or ___ ...

The calculator alwayspicks quadrant

I or III or IIII or IVII or IIIII or IVIII or IV

5π cm

6 cmn

Example 16: The formula for arc length is a = rθ, where a is the arc length, θ is the central angle in radians, and r is the radius of the circle. The radius and arc length must have the same units.a) Derive the formula for arc length, a = rθ. (b - e) Solve for the unknown.

1.23π cm

r π2

a

5 cm

153°

θ3 cm

6 cm

6 cm 3 cm

2π3240°3 cm

9 cm

120°

60°

4 cm

7π6

Example 17: Area of a circle sector.a) Derive the formula for the area of a circle sector,

r2θ2

A = . (b - e) Find the area of each shaded region.

b) c) d) e)

b) c) d) e)

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Example 18: The formula for angular speed is , where ω (Greek: Omega)

∆θis the angular speed, ∆θ is the change in angle, and ∆T is the change in time. Calculate the requested quantity in each scenario. Round all decimals to the nearest hundredth.a) A bicycle wheel makes 100 complete revolutions in 1 minute. Calculate the angular speed in degrees per second.b) A Ferris wheel rotates 1020° in 4.5 minutes. Calculate the angular speed in radians per second.c) The moon orbits Earth once every 27 days. Calculate the angular speed in revolutions per second.If the average distance from the Earth to the moon is 384 400 km, how far does the moon travel in one second? d) A cooling fan rotates with an angular speed of 4200 rpm. What is the speed in rps?e) A bike is ridden at a speed of 20 km/h, and each wheel has a diameter of 68 cm. Calculate the angular speed of one of the bicycle wheels and express the answer using revolutions per second.

a) Calculate the angular speed of the satellite. Express the answer as an exact value, in radians/second.b) How many kilometres does the satellite travel in one minute? Round the answer to the nearest hundredth of a kilometre.

340 km

6370 km

Example 19: A satellite orbiting Earth 340 km above the surface makes one complete revolution every 90 minutes. The radius of Earth is approximately 6370 km.

TrigonometryLESSON ONE - Degrees and RadiansLesson Notes

210°×π

180°=

7π6

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TrigonometryLESSON TWO - The Unit Circle

Lesson Notes

a) A circle centered at the origin can be represented by the relation x2 + y2 = r2, where r is the radius of the circle. Draw each circle: i. x2 + y2 = 4 ii. x2 + y2 = 49

Example 1: Introduction to Circle Equations.

(cosθ, sinθ)

-10

-10

10

10-10

-10

10

10

b) A circle centered at the origin with a radius of 1 has the equation x2 + y2 = 1. This special circle is called the unit circle. Draw the unit circle and determine if each point exists on the circumference of the unit circle: i. (0.6, 0.8) ii. (0.5, 0.5)

1

1

-1

-1

c) Using the equation of the unit circle, x2 + y2 = 1, find the unknown coordinate of each point. Is there more than one unique answer?

iii. (-1, y)i. ii. , quadrant II. , cosθ > 0.iv.

Example 2: The following diagram is called the unit circle. Commonly used angles are shown as radians, and their exact-value coordinates are in brackets. Take a few moments to memorize this diagram. When you are done, use the blank unit circle on the next page to practice drawing the unit circle from memory.

b) Draw the unit circle from memory.

a) What are some useful tips to memorize the unit circle?

Example 3: Use the unit circle to find the exact value of each expression.

a) sin b) cos 180° c) cos 3π4

d) sin 11π6

e) sin 0 π2f) cos g) sin 4π

3 h) cos -120°

π6

Example 4: Use the unit circle to find the exact value of each expression.

a) cos 420° b) -cos 3π c) sin 13π6

d) cos 2π3

9π4f) -sin g) cos2 (-840°) h) cos 7π

3e) sin 5π2

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TrigonometryLESSON TWO - The Unit CircleLesson Notes

(cosθ, sinθ)

Example 5: The unit circle contains values for cosθ and sinθ only. The other four trigonometric ratios can be obtained using the identities on the right. Find the exact values of secθ and cscθ in the first quadrant.

tanθ =sinθcosθ

cotθ =cosθsinθ

secθ =1

cosθcscθ =

1sinθ

=1

tanθExample 6: Find the exact values of tanθ and cotθ in the first quadrant.

Example 7: Use symmetry to fill in quadrants II, III, and IV for secθ, cscθ, tanθ, and cotθ.

Example 8: Find the exact value of each expression.

a) sec 120° b) sec c) csc π3

d) csc 3π4 e) tan 5π

4f) -tan g) cot2(270°) h) cotπ6

5π6

3π2

b) c) d) a)



a) b) c) d)


a) c) d)b)

a) c)

f)

b)

g) h)e)

d) 2

Example 12: Verify each trigonometric statement with a calculator. Note: Every question in this example has already been seen earlier in the lesson.

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TrigonometryLESSON TWO - The Unit Circle

Lesson Notes(cosθ, sinθ)

a) What is meant when you are asked to find on the unit circle?

c) How does a half-rotation around the unit circle change the coordinates?

d) How does a quarter-rotation around the unit circle change the coordinates?2π3

b) Find one positive and one negative angle such that P(θ) =

e) What are the coordinates of P(3)? Express coordinates to four decimal places.

If θ = , find the coordinates of the point halfway around the unit circle.π6

If θ = , find the coordinates of the point a quarter-revolution (clockwise) around the unit circle.

Example 13: Coordinate Relationships on the Unit Circle

b) How is the central angle of the unit circle related to its corresponding arc length?

a) What is the circumference of the unit circle?

c) If a point on the terminal arm rotates from P(θ) = (1, 0)

P(θ) = , what is the arc length?

d) What is the arc length from point A to point B on the unit circle?

Example 14: Circumference and Arc Length of the Unit Circle

to θ

θ

θ

A

B

Diagram forExample 14 (d).

b) Which trigonometric ratios are restricted to a range of -1 ≤ y ≤ 1? Which trigonometric ratios exist outside that range?

Example 15: Domain and Range of the Unit Circle

a) Is sinθ = 2 possible? Explain, using the unit circle as a reference.

d) If exists on the unit circle, how can the equation of the

unit circle be used to find sinθ? How many values for sinθ are possible?

e) If cosθ = 0, and 0 ≤ θ < π, how many values for sinθ are possible?

c) If exists on the unit circle, how can the unit circle

be used to find cosθ? How many values for cosθ are possible?

cosθ & sinθ

cscθ & secθ

tanθ & cotθ

Range Number Line

Chart for Example 15 (b).

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TrigonometryLESSON TWO - The Unit CircleLesson Notes

(cosθ, sinθ)

θA

x

h

θB

A B

Example 18: From the observation deck of the Calgary Tower, an observer has to tilt their head θA down to see point A, and θB down to see point B.

a) Show that the height of the observationx

cotθA - cotθB

deck is h = .

ground, to the nearest metre?

b) If θA = , θB = , and x = 212.92 m,

how high is the observation deck above the

a) Use the Pythagorean Theorem to prove that the equation of the unit circle is x2 + y2 = 1.

b) Prove that the point where the terminal arm intersects the unit circle, P(θ), has coordinates of (cosθ, sinθ).

c) If the point θ exists on the terminal arm of a unit circle, find the exact values

of the six trigonometric ratios. State the reference angle and standard position angle to the nearest hundredth of a degree.

Example 16: Unit Circle Proofs

Example 17: In a video game, the graphic of a butterfly needs to be rotated. To make the butterfly graphic rotate, the programmer uses the equations:

to transform each pixel of the graphic from its original coordinates, (x, y), to its new coordinates, (x’, y’). Pixels may have positive or negative coordinates.

x’ = x cos θ - y sin θ

y’ = x sin θ + y cos θ

b) If a particular pixel has the coordinates (640, 480) after a rotation of , what were the original coordinates? Round coordinates to the nearest whole pixel.

5π4

a) If a particular pixel with coordinates of (250, 100) is rotated by , what are the new coordinates? Round coordinates to the nearest whole pixel.

π6

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TrigonometryLESSON THREE - Trigonometric Functions I

Lesson Notes

a) b)

0 π 2ππ2π θ

y

5

-5

0 π 2ππ2π

20

-20

θ

y

0 4π4π

40

-40

θ

y

0 8π8π θ

y

-12

12

c) d)

Example 1: Label all tick marks in the following grids and state the coordinates of each point.

Example 2: Exploring the graph of y = sinθ.

d) State the horizontal displacement (phase shift). e) State the vertical displacement.f) State the θ-intercepts. Write your answer using a general form expression.g) State the y-intercept. h) State the domain and range.

a) Draw y = sinθ. b) State the amplitude. c) State the period.

Example 3: Exploring the graph of y = cosθ.


a) Draw y = cosθ. b) State the amplitude. c) State the period.

Example 4: Exploring the graph of y = tanθ.


a) Draw y = tanθ. b) Is it correct to say a tangent graph has an amplitude? c) State the period.

y = asinb(θ - c) + d

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Lesson Notesy = asinb(θ - c) + d

d) y = cosθ -12

12

a) y = sinθ - 2 b) y = cosθ + 4 c) y = sinθ + 212

-

Example 9: The b Parameter. Graph each function over the stated domain.

Example 10: The b Parameter. Graph each function over the stated domain.

c) y = 2cos θ - 1 (-2π ≤ θ ≤ 2π)

b) y = 4cos2θ + 6a) y = -sin(3θ) (-2π ≤ θ ≤ 2π) (-2π ≤ θ ≤ 2π)

d) y = sin θ43

(0 ≤ θ ≤ 6π)12

a) y = cos2θ b) y = sin3θ(0 ≤ θ ≤ 2π) (0 ≤ θ ≤ 2π)

c) y = cos θ d) y = sin θ13

15

(0 ≤ θ ≤ 6π) (0 ≤ θ ≤ 10π)

Example 5: The a Parameter. Graph each function over the domain 0 ≤ θ ≤ 2π.

a) y = 3sinθ b) y = -2cosθ c) y = sinθ12

d) y = cosθ52

Example 7: The d Parameter. Graph each function over the domain 0 ≤ θ ≤ 2π.

c) write a cosine function.a) write a sine function. b) write a sine function. d) write a cosine function.

0

-28

28

2π0

-5

5

14

π,( )2π

0

-1

1

2π0

-8

8

2π

Example 6: The a Parameter. Determine the trigonometric function corresponding to each graph.

c) write a cosine function.a) write a sine function. b) write a cosine function. d) write a sine function.

Example 8: The d Parameter. Determine the trigonometric function corresponding to each graph.

0

-35

35

2π0

-4

4

2π0

-32

32

2π0

-4

4

2π

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Example 12: The c Parameter. Graph each function over the stated domain.

d) (-2π ≤ θ ≤ 2π)

a) (-4π ≤ θ ≤ 4π) b) (-4π ≤ θ ≤ 4π)

c) (-2π ≤ θ ≤ 2π)

Example 13: The c Parameter. Graph each function over the stated domain.

d) (-2π ≤ θ ≤ 2π)

a) 2πθπ2

b) (-2π ≤ θ ≤ 6π)

c) (-π ≤ θ ≤ 4π)

0

-6

6

4π2π

-4

4

8π-8π 4π-4π2πππ2

-1

1

-1

1

2ππ-π

Example 15: a, b, c, & d Parameters. Graph each function over the stated domain.

0

-1

1

2ππ0

-2

2

9π6π3π0

-4

4

12π6π0

-2

2

2ππ

a) (0 ≤ θ ≤ 6π) b) (0 ≤ θ ≤ 2π)

d) (0 ≤ θ ≤ 2π)c) (0 ≤ θ ≤ 2π)3-

c) write a cosine function.a) write a cosine function. b) write a sine function. d) write a sine function.

Example 11: The b Parameter. Determine the trigonometric function corresponding to each graph.

c) write a sine function.a) write a cosine function. b) write a sine function. d) write a cosine function.

Example 14: The c Parameter. Determine the trigonometric function corresponding to each graph.

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-2

2

4π2ππ 3π-2π -π 2π-2π

-12

12

π-π

Example 17: Exploring the graph of y = secθ.

a) Draw y = secθ. b) State the period. c) State the domain and range.d) Write the general equation of the asymptotes.

e) Given the graph of f(θ) = cosθ, draw y = .f(θ)1

θ

y

Example 18: Exploring the graph of y = cscθ.

a) Draw y = cscθ. b) State the period. c) State the domain and range.d) Write the general equation of the asymptotes.

e) Given the graph of f(θ) = sinθ, draw y = .f(θ)1

θ

y

Example 19: Exploring the graph of y = cotθ.

a) Draw y = cotθ. b) State the period. c) State the domain and range.d) Write the general equation of the asymptotes.

e) Given the graph of f(θ) = tanθ, draw y = .f(θ)1

θ

y

0

-3

3

2ππ

y = secθ

0

-3

3

y = cscθ

2ππ

a) b) c) d)

0

-3

3

2ππ

y = secθ

0

-3

3

2ππ

y = cotθ

Example 20: Graph each function over the domain 0 ≤ θ ≤ 2π. State the new domain and range.

2π-2π

2π-2π

2π-2π

3

-3

3

-3

3

-3

Example 16: a, b, c, & d. Determine the trigonometric function corresponding to each graph.a) write a cosine function. b) write a cosine function.

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TrigonometryLESSON FOUR - Trigonometric Functions II

Lesson Notes

h(t)

t

Example 5: Determine the trigonometric function corresponding to each graph.

0

-10

10

168

a) write a cosine function.

-5

5

168-4

b) write a sine function.

0

10

25

-10

(8, 9)

(16, -3)

c) write a cosine function. d) write a sine function.

0

300

2400

-300

(1425, 150)

(300, -50)

c) If the range of y = 3cosθ + d is [-4, k], determine the values of d and k.b) Find the range of

4.

If the amplitude of each graph is quadrupled, determine the new points of intersection.

e) The graphs of f(θ) and g(θ) intersect at the points

and

Example 6: a) If the transformation g(θ) - 3 = f(2θ) is applied to the graph of f(θ) = sinθ, find the new range.

d) State the range of f(θ) - 2 = msin(2θ) + n.

.

Example 1: Trigonometric Functions of Angles

a) b)

ii. Graph this function using technology. ii. Graph this function using technology.

i. Graph: (0 ≤ θ < 3π) i. Graph: (0º ≤ θ < 540º)

Example 2: Trigonometric Functions of Real Numbers.

i. Graph:i. Graph:

c) What are three differences between trigonometric functions of angles and trigonometric functions of real numbers?

a) b)

ii. Graph this function using technology. ii. Graph this function using technology.

Example 3: Determine the view window for each function and sketch each graph.

b)a)

b)a)

Example 4: Determine the view window for each function and sketch each graph.

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Lesson Notes

h(t)

t

a) If the point lies on the graph of , find the value of a.

b) Find the y-intercept of .

c) The graphs of f(θ) and g(θ) intersect at the point (m, n). Find the value of f(m) + g(m).

d) The graph of f(θ) = kcosθ is transformed to the graph of g(θ) = bcosθ by a vertical stretch about the x-axis.

If the point

of g(θ), state the vertical stretch factor.

exists on the graph

π2

π 3π2

2π

f(θ)

g(θ)

k

b

Graph for Example 7d

f(θ)

g(θ)(m, n)

m

n

Graph for Example 7c

Example 8: The graph shows the height of a pendulum bob as a function of time. One cycle of a pendulum consists of two swings - a right swing and a left swing.

12 cm

4 cm

8 cm

2 s1 s 3 s 4 s0 cm

t

h(t)

ground level

Graph for Example 8a) Write a function that describes the height of the pendulum bob as a function of time.b) If the period of the pendulum is halved, how will this change the parameters in the function you wrote in part (a)?c) If the pendulum is lowered so its lowest point is 2 cm above the ground, how will this change the parameters in the function you wrote in part (a)?

Example 7:

Example 9: A wind turbine has blades that are 30 m long. An observer notes that one blade makes 12 complete rotations (clockwise) every minute. The highest point of the blade during the rotation is 105 m.

a) Using Point A as the starting point of the graph, draw the height of the blade over two rotations.b) Write a function that corresponds to the graph.c) Do we get a different graph if the wind turbine rotates counterclockwise?

A

Example 10: A person is watching a helicopter ascend from a distance 150 m away from the takeoff point.

a) Write a function, h(θ), that expresses the height as a function of the angle of elevation. Assume the height of the person is negligible.b) Draw the graph, using an appropriate domain.c) Explain how the shape of the graph relates to the motion of the helicopter.

θ

h

150 m

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Lesson Notes

h(t)

t

a) Draw the graph for two full oscillations of the mass.b) Write a sine function that gives the height of the mass above the ground as a function of time.c) Calculate the height of the mass after 1.2 seconds. Round your answer to the nearest hundredth.d) In one oscillation, how many seconds is the mass lower than 3.2 m? Round your answer to the nearest hundredth.

Example 11: A mass is attached to a spring 4 m above the ground and allowed to oscillate from its equilibrium position. The lowest position of the mass is 2.8 m above the ground, and it takes 1 s for one complete oscillation.

a) Draw the graph for two full rotations of the Ferris wheel.b) Write a cosine function that gives the height of the rider as a function of time.c) Calculate the height of the rider after 1.6 rotations of the Ferris wheel.Round your answer to the nearest hundredth.d) In one rotation, how many seconds is the rider higher than 26 m? Round your answer to the nearest hundredth.

Example 12: A Ferris wheel with a radius of 15 m rotates once every 100 seconds. Riders board the Ferris wheel using a platform 1 m above the ground.

Example 13: The following table shows the number of daylight hours in Grande Prairie.

December 216h, 46m

March 2112h, 17m

June 2117h, 49m

September 2112h, 17m

December 216h, 46m

a) Convert each date and time to a number that can be used for graphing.

b) Draw the graph for one complete cycle (winter solstice to winter solstice).c) Write a cosine function that relates the number of daylight hours, d, to the day number, n.d) How many daylight hours are there on May 2? Round your answer to the nearest hundredth.e) In one year, approximately how many days have more than 17 daylight hours?Round your answer to the nearest day.

December 21 = March 21 = June 21 = September 21 = December 21 =Day Number

Daylight Hours 6h, 46m = 12h, 17m = 17h, 49m = 12h, 17m = 12h, 46m =

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Lesson Notes

h(t)

t

Example 17: Shane is on a Ferris wheel, and his height

Tim, a baseball player, can throw a baseball with a speed of 20 m/s. If Tim throws a ball directly upwards, the height can be determined by the equation hball(t) = -4.905t2 + 20t + 1.

If Tim throws the baseball 15 seconds after the ride begins, when are Shane and the ball at the same height?

can be described by the equation .

Example 14: The highest tides in the world occur between New Brunswick and Nova Scotia, in the Bay of Fundy. Each day, there are two low tides and two high tides. The chart below contains tidal height data that was collected over a 24-hour period.

a) Convert each time to a decimal hour.b) Graph the height of the tide for one full cycle (low tide to low tide).c) Write a cosine function that relates the height of the water to the elapsed time.d) What is the height of the water at 6:09 AM? Round your answer to the nearest hundredth.e) For what percentage of the day is the height of the water greater than 11 m?Round your answer to the nearest tenth.

Bay ofFundy

Bay ofFundy

Time Decimal Hour Height of Water (m)

Low Tide

Low Tide

High Tide

High Tide

2:12 AM

8:12 AM

2:12 PM

8:12 PM

3.48

13.32

3.48

13.32

a) Graph the population of owls and mice over six years.b) Describe how the graph shows the relationship between owl and mouse populations.

Example 15: A wooded region has an ecosystem that supports both owls and mice. Owl and mice populations vary over time according to the equations:

Owl population: Mouse population:

where O is the population of owls, M is the population of mice, and t is the time in years.

Example 16: The angle of elevation between the 6:00 position and the 12:00 position of a historical building’s clock, as measured from an observer

The observer also knows that he is standing 424 m away from the clock, and his eyes are at the same height as the base of the clock.The radius of the clock is the same as the length of the minute hand.

If the height of the minute hand’s tip is measured relative to the bottom of the clock, what is the height of the tip at 5:08,to the nearest tenth of a metre?

π444

π444

.standing on a hill, is

424 m

Note: Actual tides at the Bay of Fundy are 6 hours and 13 minutes apart due to daily changes in the position of the moon.

In this example, we willuse 6 hours for simplicity.

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Example 8: Reciprocal Ratios. Find all angles in the domain 0 ≤ θ ≤ 2π that satisfy the given equation.Write the general solution. Solve equations graphically with intersection points.

f) θe) θd) secθ = -1

TrigonometryLESSON FIVE - Trigonometric Equations

Lesson Notes

a)

Example 1: Primary Ratios. Find all angles in the domain 0 ≤ θ ≤ 2π that satisfy the given equation. Write the general solution. Solve equations non-graphically using the unit circle.

b) c) 0 d) tan2θ = 1

Example 2: Primary Ratios. Find all angles in the domain 0 ≤ θ ≤ 2π that satisfy the given equation. Write the general solution. Solve equations graphically with intersection points.

a) sinθ b) sinθ = -1 c) cosθ d) cosθ = 2 e) tanθ f) tanθ = undefined

Example 3: Primary Ratios. Find all angles in the domain 0°≤ θ ≤ 360° that satisfy the given equation. Write the general solution. Solve equations non-graphically with a calculator (degree mode).

a) b) c)

Example 4: Primary Ratios. Find all angles in the domain 0 ≤ θ ≤ 2π that satisfy the given equation.Solve equations graphically with θ-intercepts.

b) cosθ =a) sinθ = 1

Solve - 0 ≤ θ ≤ 2πExample 5: Primary Ratios.

Solve sinθ = -0.30 θ ε RExample 6: Primary Ratios.

Example 7: Reciprocal Ratios. Find all angles in the domain 0 ≤ θ ≤ 2π that satisfy the given equation. Write the general solution. Solve equations non-graphically using the unit circle.

a) c)b)

a) θ b) θ c) θ

b)a) c)

Example 9: Reciprocal Ratios. Find all angles in the domain 0°≤ θ ≤ 360° that satisfy the given equation. Write the general solution. Solve non-graphically with a calculator (degree mode).

a) non-graphically, using the cos-1 feature of a calculator.

b) non-graphically, using the unit circle.

c) graphically, usingpoint(s) of intersection.

d) graphically, using θ-intercepts.



c) graphically, usingpoint(s) of intersection.


a) θ b) θ

Example 10: Reciprocal Ratios. Find all angles in the domain 0 ≤ θ ≤ 2π that satisfy the given equation. Write the general solution. Solve equations graphically with θ-intercepts.

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TrigonometryLESSON FIVE - Trigonometric EquationsLesson Notes

Solve cscθ = -2 0 ≤ θ ≤ 2π

Solve secθ = -2.3662 0°≤ θ ≤ 360°

Example 11: Reciprocal Ratios.



c) graphically, using point(s) of intersection.


Example 12: Reciprocal Ratios.

Example 13: First-Degree Trigonometric Equations. Find all angles in the domain 0 ≤ θ ≤ 2π that satisfy the given equation. Write the general solution.

d) 4secθ + 3 = 3secθ + 1c) 3tanθ - 5 = 0a) cosθ - 1 = 0 b) θ

Example 14: First-Degree Trigonometric Equations. Find all angles in the domain 0 ≤ θ ≤ 2π that satisfy the given equation. Write the general solution.

b) 7sinθ = 4sinθa) 2sinθcosθ = cosθ c) sinθtanθ = sinθ d) tanθ + cosθtanθ = 0

Example 15: Second-Degree Trigonometric Equations. Find all angles in the domain 0 ≤ θ ≤ 2π that satisfy the given equation. Write the general solution.

a) sin2θ = 1 c) 2cos2θ = cosθ d) tan4θ - tan2θ = 0b) 4cos2θ - 3 = 0

Example 16: Second-Degree Trigonometric Equations. Find all angles in the domain 0 ≤ θ ≤ 2π that satisfy the given equation. Write the general solution.

a) 2sin2θ - sinθ - 1 = 0 b) csc2θ - 3cscθ + 2 = 0 c) 2sin3θ - 5sin2θ + 2sinθ = 0

Example 17: Double and Triple Angles. Solve each equation (i) graphically, and (ii) non-graphically.

a) θ 0 ≤ θ ≤ 2π b) θ 0 ≤ θ ≤ 2π

Example 18: Half and Quarter Angles. Solve each equation (i) graphically, and (ii) non-graphically.

a) θ 0 ≤ θ ≤ 4π b) θ 0 ≤ θ ≤ 8π-1

Example 19: It takes the moon approximately 28 days to go through all of its phases.



c) graphically, using point(s) of intersection.


Example 20: A rotating sprinkler is positioned 4 m away from the wall of a house. The wall is 8 m long. As the sprinkler rotates, the stream of water splashes the house d meters from point P.

a) Write a tangent function, d(θ), that expresses the distance where the water splashes the wall as a function of the rotation angle θ.b) Graph the function for one complete rotation of the sprinkler. Draw only the portion of the graph that actually corresponds to the wall being splashed.c) If the water splashes the wall 2.0 m north of point P, what is the angle of rotation (in degrees)?

a) Write a function, P(t), that expresses the visible percentage of the moon as a function of time. Draw the graph. b) In one cycle, for how many days is 60% or more of the moon’s surface visible?

Pθ

d

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TrigonometryLESSON SIX - Trigonometric Identities I

Lesson Notes

Example 3: Reciprocal Identities. Prove that each trigonometric statement is an identity. State the non-permissible values of x so the identity is true.

a) b)

Example 4: Reciprocal Identities. Prove that each trigonometric statement is an identity. State the non-permissible values of x so the identity is true.

a) b)

Example 5: Pythagorean Identities. Prove that each trigonometric statement is an identity. State the non-permissible values of x so the identity is true.

Example 6: Pythagorean Identities. Prove that each trigonometric statement is an identity. State the non-permissible values of x so the identity is true.

Example 1: Understanding Trigonometric Identities.

a) Why are trigonometric identities considered to be a special type of trigonometric equation?

b) Which of the following trigonometric equations are also trigonometric identities?

i. ii. iii. iv. v.

a) Using the definition of the unit circle, derive the identity sin2x + cos2x = 1. Why is sin2x + cos2x = 1 called a Pythagorean Identity?

b) Verify that sin2x + cos2x = 1 is an identity using (i) x = and (ii) x = .

c) Verify that sin2x + cos2x = 1 is an identity using a graphing calculator to draw the graph.

Example 2: The Pythagorean Identities.

d) Using the identity sin2x + cos2x = 1, derive 1 + cot2x = csc2x and tan2x + 1 = sec2x.

e) Verify that 1 + cot2x = csc2x and tan2x + 1 = sec2x are identities for x = .

f) Verify that 1 + cot2x = csc2x and tan2x + 1 = sec2x are identities graphically.

a) b)

c) d)

c) d)

a) b)

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TrigonometryLESSON SIX- Trigonometric Identities ILesson Notes

Example 7: Common Denominator Proofs. Prove that each trigonometric statement is an identity. State the non-permissible values of x so the identity is true.

Example 8: Common Denominator Proofs. Prove that each trigonometric statement is an identity. State the non-permissible values of x so the identity is true.

Example 12: Exploring the proof of

d)c)

a) b)

c) d)

a) b)

Example 9: Assorted Proofs. Prove each identity. For simplicity, ignore NPV’s and graphs.

a) b)

c) d)


a)

c) d)

b)

a) b)

c) d)


a) Prove algebraically that .

b) Verify that for π3

.

c) State the non-permissible values for .

d) Show graphically that . Are the graphs exactly the same?

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a) b)

c) d)

TrigonometryLESSON SIX - Trigonometric Identities I

Lesson Notes


a) Prove algebraically that .


.

c) State the non-permissible values for .

Example 13: Exploring the proof of .

Example 14: Exploring the proof of


a) Prove algebraically that


.

c) State the the non-permissible values for .

Example 15: Equations with Identites. Solve each trigonometric equation over the domain 0 ≤ x ≤ 2π.

a) b) c) d)


a)

c)

b)

d)


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TrigonometryLESSON SIX- Trigonometric Identities ILesson Notes

Example 18: Use the Pythagorean identities to find the indicated value and draw the corresponding triangle.

b) If the value of , find the value of secA within the same domain.

c) If find the exact value of sinθ. 7cosθ = , and cotθ < 0,

7

a) If the value of find the value of cosx within the same domain.

Example 19: Trigonometric Substitution.

3

θ

a) Using the triangle to the right, show thatb

aHint: Use the triangle to find a trigonometric expression equivalent to b.

can be expressed as .

b) Using the triangle to the right, show that

θ

4

aHint: Use the triangle to find a trigonometric expression equivalent to a.

can be expressed as .

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Example 1: Evaluate each trigonometric sum or difference.

TrigonometryLESSON SEVEN - Trigonometric Identities II

Lesson Notes

a) b) d) e) f)c)

Example 2: Write each expression as a single trigonometric ratio.

a) c)b)


a) b) c) d) Given the exact values of cosine and sine for 15°, fill in the blanks for the other angles.


b) c)a)

Example 5: Double-angle identities.

a) Prove the double-angle sine identity, sin2x = 2sinxcosx.

b) Prove the double-angle cosine identity, cos2x = cos2x - sin2x.

c) The double-angle cosine identity, cos2x = cos2x - sin2x, can be expressed as cos2x = 1 - 2sin2x or cos2x = 2cos2x - 1. Derive each identity.

d) Derive the double-angle tan identity, .

a) Evaluate each of the following expressions using a double-angle identity.

b) Express each of the following expressions using a double-angle identity.

c) Write each of the following expression as a single trigonometric ratio using a double-angle identity.

Example 6: Double-angle identities.

i.

i.

i.

ii.

ii.

ii.

iii.

iii.

iii.

iv.

iv.

Example 3d

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TrigonometryLESSON SEVEN- Trigonometric Identities II

Lesson Notes

Note: Variable restrictions may beignored for the proofs in this lesson.

Example 7: Prove each trigonometric identity.

b)a)

c) d)


a) b)

c)


a) b)

c) d)

a)

c) d)

b)



a) b)

d)c)


d)

c) d)

a) b)

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TrigonometryLESSON SEVEN - Trigonometric Identities II

Lesson Notes

b)

d)c)


a)

a)

c) d)

Example 14: Solve each trigonometric equation over the domain 0 ≤ x ≤ 2π.

b)

a) b)

c) d)


a) b)

c) d)

Solve for x. Round your answer to the nearest tenth.

a)

c)

b)


d)


b) If A = 32° and B = 89°, what is the value of C?

a) Show that

Example 18: Trigonometric identities and geometry.

A

C

B

Diagram forExample 18

Example 19: Trigonometric identities and geometry.

57

176

153

104

A

B

x

Diagram forExample 19

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TrigonometryLESSON SEVEN- Trigonometric Identities II

Lesson Notes

θExample 20: If a cannon shoots a cannonball θ degrees above the horizontal, the horizontal distance traveled by the cannonball before it hits the ground can be found with the function d(θ).The initial velocity of the cannonball is 36 m/s.

( )

2 sin cos4.9

ivd

θ θθ =a) Rewrite the function so it involves a single trigonometric identity.

b) Graph the function. Use the graph to describe the trajectory of the cannonball at the following angles: 0°, 45°, and 90°. c) If the cannonball travels a horizontal distance of 100 m, find the angle of the cannon. Solve graphically, and round your answer to the nearest tenth of a degree.

Example 21: An engineer is planning the construction of a road through a tunnel. In one possible design, the width of the road maximizes the area of a rectangle inscribed within the cross-section of the tunnel.

θ

70 m

road widthsidewalk sidewalk

a) Two sound waves are represented with f(θ) and g(θ). i. Draw the graph of y = f(θ) + g(θ) and determine the resultant wave function.ii. Is this constructive or destructive interference?iii. Will the new sound be louder or quieter than the original sound?

0

-6

6

f(θ) = 2cosθ

2ππ

g(θ) = 4cosθ

Example 22: The improper placement of speakers for a home theater system may result in a diminished sound quality at the primary viewing area. This phenomenon occurs because sound waves interact with each other in a process called interference. When two sound waves undergo interference, they combine to form a resultant sound wave that has an amplitude equal to the sum of the component sound wave amplitudes.

b) A different set of sound waves are represented with m(θ) and n(θ). i. Draw the graph of y = m(θ) + n(θ) and determine the resultant wave function. ii. Is this constructive or destructive interference?iii. Will the new sound be louder or quieter than the original sound?

m(θ) = 2cosθ

n(θ) = 2cos(θ - π)0

-6

6

2ππ

c) Two sound waves experience total destructive interference if the sum of their wave functions is zero.

Given p(θ) = sin(3θ - 3π/4) and q(θ) = sin(3θ - 7π/4), show that these waves experience total destructive interference.

If the amplitude of the resultant wave is larger than the component wave amplitudes, we say the component waves experienced constructive interference. If the amplitude of the resultant wave is smaller than the component wave amplitudes, we say the component waves experienced destructive interference.

The angle of elevation from the centre line of the road to the upper corner of the rectangle is θ. Sidewalks on either side of the road are included in the design.a) If the area of the rectangle can be represented by the function A(θ) = msin2θ, what is the value of m?b) What angle maximizes the area of the rectangular cross-section?c) For the angle that maximizes the area: (i) What is the width of the road? (ii) What is the height of the tallest vehicle that will pass through the tunnel? (iii) What is the width of one of the sidewalks?Express answers as exact values.

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Permutations and CombinationsLESSON ONE - Permutations

Lesson NotesnPr =

n!(n - r)!

A BC

Example 1: Introduction to Permutations.Three letters (A, B, and C) are taken from a set of letter tiles and arranged to form “words”. In this question, ACB counts as a word - even though it’s not an actual English word.a) Use a tree diagram to find the number of unique words.b) Use the Fundamental Counting Principle to find the number of unique words.c) Use permutation notation to find the number of unique words. Evaluate using a calculator.d) What is meant by the terms single-case permutation and multi-case permutation?e) Use permutations to find the number of ways a one-, two-, or three-letter word can be formed.

Example 2: Evaluate each of the following factorial expressions.

a) 4! b) 1! c) 0! d) (-2)!5!3!

e)n!

(n - 2)!g)f)

8!7!•2!

h)(n + 1)!(n - 1)!

Example 3: Single-Case Permutations (Repetitions NOT Allowed)

a) A Grade 12 student is taking Biology, English, Math, and Physics in her first term. If a student timetable has room for five courses (meaning the student has a spare), how many ways can she schedule her courses?b) A singing competition has three rounds. In each round, the singer has to perform one song from a particular genre. How many different ways can the performer select the genres?c) A web development team of three members is to be formed from a selection pool of 10 people. The team members will be assigned roles of programmer, graphic designer, and database analyst. How many unique teams are possible? You can assume that each person in the selection pool is capable of performing each task.d) There are 13 letter tiles in a bag, and no letter is repeated. Using all of the letters from the bag, a six-letter word, a five-letter word, and a two-letter word are made. How many ways can this be done?

Round 1 Round 2 Round 3

RockMetalPunk

Alternative

PopDance

CountryBluesFolk

Block 1Block 2Block 3Block 4Block 5

CourseMath 30-1

SparePhysics 30

English 30-1Biology 30

BlockOne Possible Timetable

Example 4: Single-Case Permutations (Repetitions NOT Allowed)

a) How many ways can the letters in the word SEE be arranged?b) How many ways can the letters in the word MISSISSAUGA be arranged?c) A multiple-choice test has 10 questions. Three questions have an answer of A, four questions have an answer of B, one question has an answer of C, and two questions have an answer of D. How many unique answer keys are possible?d) How many pathways exist from point A to point B if the only directions allowed are north and east?e) How many ways can three cars (red, green, blue) be parked in five parking stalls?f) An electrical panel has five switches. How many ways can the switches be positioned up or down if three switches must be up and two must be down? A

B

Example 5: Single-Case Permutations (Repetitions ARE Allowed)

a) There are three switches on an electrical panel. How many unique up/down sequences are there?b) How many two-letter “words” can be created using the letters A, B, C, and D?c) A coat hanger has four knobs, and each knob can be painted any color. If six different colors of paint are available, how many ways can the knobs be painted?d) A phone number in British Columbia consists of one of four area codes (236, 250, 604, and 778), followed by a 7-digit number that cannot begin with a 0 or 1. How many unique phone numbers are there?e) An identification code consists of any two letters followed by any three digits. How many identification codes can be created?

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Permutations and CombinationsLESSON ONE - PermutationsLesson Notes

nPr =n!

(n - r)!

Six people (Andrew, Brenda, Cory, Danielle, Eliza, Frank) are going to be seated in a line. How many unique lines can be formed if:a) Frank must be seated in the third chair?b) Brenda or Cory must be in the second chair, and Eliza must be in the third chair?c) Danielle can’t be at either end of the line?d) men and women alternate positions, with a woman sitting in the first chair?e) if the line starts with the pattern man-man-woman?

How many ways can you order the letters from the word TREES if:a) a vowel must be at the beginning?b) it must start with a consonant and end with a vowel?c) the R must be in the middle?d) it begins with exactly one E?e) it ends with TR?f) consonants and vowels alternate?

Example 7: Constraints and Words

Example 8: Objects ALWAYS Together

a) How many ways can 3 chemistry books, 4 math books, and 5 physics books be arranged if books on each subject must be kept together?b) How many arrangements of the word ACTIVE are there if C&E must always be together?c) How many arrangements of the word ACTIVE are there if C&E must always be together, and in the order CE?d) Six people (Andrew, Brenda, Cory, Danielle, Eliza, Frank) are going to be seated in a line. How many unique lines can be formed if Cory, Danielle, and Frank must be seated together?

Example 9: Objects NEVER Together

a) How many ways can the letters in QUEST be arranged if the vowels must never be together?b) Eight cars (3 red, 3 blue, and 2 yellow) are to be parked in a line. How many unique lines can be formed if the yellow cars must not be together? Assume that cars of each color are identical.c) How many ways can the letters in READING be arranged if the vowels must never be together?

Example 10: Multi-Case Permutations (At Least/At Most)

a) Evaluate 4P3 b) Evaluate 12P3 d) Write 3! as a permutation. nPr =n!

(n - r)!c) Write as a permutation.

5!

3!

Example 11: Permutation Formula

b) (n + 2)! = 12n!a)n!

(n - 2)!= 5n d)

(2n + 1)!

(2n - 1)!= 4n + 2c)

n!

10= n-1Pn-3

Example 12: Equations with Factorials and Permutations.

a) nP2 = 56 b) 6Pr = 120 c) n + 3P2 = 20 d) n - 3P1 = 2•n - 4P1

Example 13: Equations with Factorials and Permutations.

Example 6: Constraints and Line Formations

a) How many words (with at most three letters) can be formed from the letter tiles SUNDAY?b) How many words (with at least five letters) can be formed from the letter tiles SUNDAY?c) How many 3-digit odd numbers greater than 600 can be formed using the digits 2, 3, 4, 5, 6, and 7, if a number contains no repeating digits?d) Six vehicles (3 different brands of cars and 3 different brands of trucks) are going to be parked in a line. How many unique lines can be formed if the row starts with at least two trucks?e) Six vehicles (3 different brands of cars and 3 different brands of trucks) are going to be parked in a line. How many unique lines can be formed if trucks and cars alternate positions?

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Permutations and CombinationsLESSON TWO - Combinations

Lesson NotesnCr =

n!(n - r)!r!

Focal Flowers: Large and eye-catching flowers that draw attention to one area of the bouquet.

Roses, Peonies, Hydrangeas, Chrysanthemums, Tulips, and Lilies

Fragrant Flowers: Flowers that add a pleasant fragrance to the bouquet.

Petunia, Daffodils, Daphnes, Gardenia, Lilacs, Violets, Magnolias

Line Flowers: Tall and narrow flowers used to establish the height of a floral bouquet.

Delphiniums, Snapdragons, Bells of Ireland, Gladioli, and Liatris

Filler Flowers: Unobtrusive flowers that give depth to the bouquet.

Daisies, Baby's Breath, Wax Flowers, Solidago, and Caspia

Flower Type Examples

5♣ 6♥ 2♦

A 4♥

Example 4: Single-Case Combinations (More Sample Sets with Subdivisions)a) A committee of 5 people is to be formed from a selection pool of 12 people. If Carmen must be on the committee, how many unique committees can be formed?b) A committee of 6 people is to be formed from a selection pool of 11 people. If Grant and Helen must be on the committee, but Aaron must not be on the committee, how many committees can be formed?c) Nine students are split into three equal-sized groups to work on a collaborative assignment.How many ways can this be done? Does the sample set need to be subdivided in this question?d) From a deck of 52 cards, a 5-card hand is dealt. How many distinct 5-card hands are there if the ace of spades and two of diamonds must be in the hand?e) A lottery ticket has 6 numbers from 1-49. Duplicate numbers are not allowed, and the order of the numbers does not matter. How many different lottery tickets contain the numbers 12, 24 and 48, but exclude the numbers 30 and 40?

Example 5: Single-Case Combinations (Permutations and Combinations Together)a) How many five-letter words using letters from TRIANGLE can be made if the five-letter word must have two vowels and three consonants?b) There are 4 men and 5 women on a committee selection pool. A three-person committee consisting of President, Vice-President, and Treasurer is being formed. How many ways can exactly two men be on the committee?c) A music teacher is organizing a concert for her students. If there are six piano students and seven violin students, how many arrangements are possible if four piano students and three violin students perform in an alternating arrangement?

Example 1: Introduction to Combinations.There are four marbles on a table, and each marble is a different color (red, green, blue, and yellow). Two marbles are selected from the table at random and put in a bag.a) Is the order of the marbles, or the order of their colors, important?b) Use a tree diagram to find the number of unique color combinations for the two marbles.c) Use combination notation to find the number of unique color combinations.d) What is meant by the terms single-case combination and multi-case combination?e) How many ways can three or four marbles be chosen?

Example 2: Single-Case Combinations (Sample Sets with NO Subdivisions)a) There are five toppings available for a pizza (mushrooms, onions, pineapple, spinach, and tomatoes). If a pizza is ordered with three toppings, and no topping may be repeated, how many different pizzas can be created?b) A committee of 4 people is to be formed from a selection pool of 9 people. How many unique committees can be formed?c) How many 5-card hands can be made from a standard deck of 52 cards?d) There are 9 dots randomly placed on a circle.i. How many lines can be formed within the circle by connecting two dots?ii. How many triangles can be formed within the circle?

Example 3: Single-Case Combinations (Sample Sets with Subdivisions)a) How many 6-person committees can be formed from 11 men and 9 women if 3 men and 3 women must be on the committee? b) A crate of toy cars contains 10 working cars and 4 defective cars. How many ways can 5 cars be selected if only 3 work?c) From a deck of 52 cards, a 6-card hand is dealt.How many distinct hands are there if the hand must contain 2 spades and 3 diamonds?d) A bouquet contains four types of flowers. A florist is making a bouquet that uses one type of focal flower, no fragrant flowers, three types of line flowers and all of the filler flowers. How many different bouquets can be made?

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Permutations and CombinationsLESSON TWO - CombinationsLesson Notes

nCr =n!

(n - r)!r!

Example 6: Single-Case Combinations (Handshakes, Teams, and Shapes)a) Twelve people at a party shake hands once with everyone else in the room. How many handshakes took place?b) If each of the 8 teams in a league must play each other three times, how many games will be played? (Note: This is a multi-case combination)c) If there are 8 dots on a circle, how many quadrilaterals can be formed?d) A polygon has 6 sides. How many diagonals can be formed?

Example 7: Single-Case Combinations (Repetitions ARE Allowed)a) A jar contains quarters, loonies, and toonies. If four coins are selected from the jar, how many unique coin combinations are there?b) A bag contains marbles with four different colors (red, green, blue, and yellow).If three marbles are selected from the bag, how many unique color combinations are there?

Example 8: Multi-Case Combinations (At Least/At Most).a) A committee of 5 people is to be formed from a group of 4 men and 5 women. How many committees can be formed if at least 3 women are on the committee?b) From a deck of 52 cards, a 5-card hand is dealt. How many distinct hands can be formed if there are at most 2 queens?c) From a deck of 52 cards, a 5-card hand is dealt. How many distinct hands can be formed if there is at least 1 red card?d) A research team of 5 people is to be formed from 3 biologists, 5 chemists, 4 engineers, and 2 programmers. How many teams have exactly one chemist and at least 2 engineers?e) In how many ways can you choose one or more of 5 different candies?

a) Evaluate 7C5 b) Evaluate 3C3c) Evaluate

d) Write as a combination.6!

4!2!e) Write as a combination.

5!

4!

Example 9: Combination Formula.

nCr =n!

(n - r)!r!

a) nC2 = 21 b) 4Cr = 6 c) d)

Example 10: Combination Formula. Solve for the unknown algebraically.

Example 11: Combination Formula. Solve for the unknown algebraically.

b)nCr

nCn - r

= 1a)nC4

n - 2C2

= 1 c) n - 1P3 = 2 × n - 1C2 d) 12

× n + 2C3n + 1C2 =

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Permutations and CombinationsLESSON TWO - Combinations

Lesson NotesnCr =

n!(n - r)!r!

Example 12: Assorted Mix Ia) A six-character code has the pattern shown below, and the same letter or digit may be used more than once.

Letter LetterDigit Digit Digit Digit

How many unique codes can be created?b) If there are 2 different parkas, 5 different scarves, and 4 different tuques, how many winter outfits can be made if an outfit consists of one type of each garment?c) If a 5-card hand is dealt from a deck of 52 cards, how many hands have at most one diamond?d) If there are three cars and four motorcycles, how many ways can the vehicles park in a line such that cars and motorcycles alternate positions?e) Show that nCr =nCn - r.f) There are nine people participating in a raffle. Three $50 gift cards from the same store are to be given out as prizes. How many ways can the gift cards be awarded?g) There are nine competitors in an Olympic event. How many ways can the bronze, silver, and gold medals be awarded?h) A stir-fry dish comes with a base of rice and the choice of five toppings: broccoli, carrots, eggplant, mushrooms, and tofu. How many different stir-fry dishes can be prepared if the customer can choose zero or more toppings?

A

C

B

Example 13: Assorted Mix IIa) A set of tiles contains eight letters, A - H. If two of these sets are combined, how manyways can all the tiles be arranged? Leave your answer as an exact value.b) A pattern has five dots such that no three points are collinear. How many lines can be drawn if each dot is connected to every other dot?c) How many ways can the letters in CALGARY be arranged if L and G must be separated?d) A five-person committee is to be formed from 11 people. If Ron and Sara must be included, but Tracy must be excluded due to a conflict of interest, how many committees can be formed?e) Moving only south and east, how many unique pathways connect points A and C?f) How many ways can the letters in SASKATOON be arranged if the letters K and T mustbe kept together, and in that order?g) A 5-card hand is dealt from a deck of 52 cards. How many hands are possible containingat least three hearts?h) A healthy snack contains an assortment of four vegetables. How many ways can one or more of the vegetables be selected for eating?

A

B

D

C

Math Science English Other

Math 30-1or

Math 30-2

Biology 30Chemistry 30Physics 30

English 30-1 Option AOption BOption COption DOption E

Example 14: Assorted Mix IIIa) How many ways can the letters in EDMONTON be arranged if repetitions are not allowed? b) A bookshelf has n fiction books and six non-fiction books. If there are 150 ways to choose two books of each type, how many fiction books are on the bookshelf?c) How many different pathways exist between points A and D?d) How many numbers less than 60 can be made using only the digits 1, 5, and 8?e) A particular college in Alberta has a list of approved pre-requisite courses. Five courses are required for admission to the college. Math 30-1 (or Math 30-2) and English 30-1 are mandatory requirements, and at least one science course must be selected as well. How many different ways could a student select five courses on their college application form?f) How many ways can four bottles of different spices be arranged on a spice rack with holes for six spice bottles? g) If there are 8 rock songs and 9 pop songs available, how many unique playlists containing 3 rock songs and 2 pop songs are possible?h) A hockey team roster contains 12 forwards, 6 defencemen, and 2 goalies. During play, only six players are allowed on the ice - 3 forwards, 2 defencemen, and 1 goalie. How many different ways can the active players be selected?

STRATEGY: Organize your thoughts with these guiding questions:

1) Permutation or Combination?2) Single-Case or Multi-Case?3) Are repetitions allowed?4) What is the sample set?Are there subdivisions?5) Are there any tricks or shortcuts?

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Permutations and CombinationsLESSON TWO - CombinationsLesson Notes

nCr =n!

(n - r)!r!

Example 15: Assorted Mix IVa) A fruit mix contains blueberries, grapes, mango slices, pineapple slices, and strawberries. If six pieces of fruit are selected from the fruit mix and put on a plate, how many ways can this be done?b) How many ways can six letter blocks be arranged in a pyramid, if all of the blocks are used? c) If a 5-card hand is dealt from a deck of 52 cards, how many hands have cards that are all thesame color?d) If a 5-card hand is dealt from a deck of 52 cards, how many hands have cards that are all thesame suit?e) A multiple choice test contains 5 questions, and each question has four possible responses.How many different answer keys are possible?f) How many diagonals are there in a pentagon?g) How many ways can eight books, each covering a different subject, be arranged on a shelf such that books on biology, history, or programming are never together? h) If a 5-card hand is dealt from a deck of 52 cards, how many hands have two pairs?

A

B C

FED

Example 16: Assorted Mix Va) How many ways can six people be split into two equal-sized groups?b) Show that 25! + 26! = 27 × 25!c) Five different types of fruit and six different types of vegetables are available for a healthy snack tray. The snack tray is to contain two fruits and three vegetables. How many different snack trays can be made if blueberries or carrots must be served, but not both together?d) In genetics, a codon is a sequence of three letters that specifies a particular amino acid. A fragment of a particular protein yields the amino acid sequence: Met - Gly - Ser - Arg - Cys - Gly. How many unique codon arrangements could yield this amino acid sequence?e) In a tournament, each player plays every other player twice. If there are 56 games, how many people are in the tournament?f) The discount shelf in a bookstore has a variety of books on computers, history, music, and travel. The bookstore is running a promotion where any five books from the discount shelf can be purchased for $20. How many ways can five books be purchased?g) Show that nCr + nCr + 1 = n + 1Cr + 1.h) How many pathways are there from point A to point C, passing through point B? Each step of the pathway must be getting closer to point C.

Arginine (Arg)

Cysteine (Cys)

Glycine (Gly)

Methionine (Met)

Serine (Ser)

CGU, CGC, CGA, CGG, AGA, AGG

UGU, UGC

GGU, GGC, GGA, GGG

AUG

UCU, UCC, UCA, UCG, AGU, AGC

Amino Acid Codon(s)

A

B

C

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Permutations and CombinationsLESSON THREE - The Binomial Theorem

Lesson Notestk+1 = nCk(x)n-k(y)k

First seven rows of Pascal’s Triangle.

1

1 1

1 12

11 3 3

1 14 6 4

1 15 10 10 5

1 16 15 61520

Example 1: Pascal’s Triangle is a number pattern with useful applications in mathematics.Each row is formed by adding together adjacent numbers from the preceding row.a) Determine the eighth row of Pascal’s Triangle.b) Rewrite the first seven rows of Pascal’s Triangle, but use combination notation instead of numbers.c) Using the triangles from parts (a & b) as a reference, explain what is meant by nCk = nCn - k.

Example 2: Rows and Terms of Pascal’s Triangle.a) Given the following rows from Pascal’s Triangle, write the circled number as a combination.

1 12 66 220 495 792 924 792 495 220 66 12 1ii.

b) Use a combination to find the third term in row 22 of Pascal’s Triangle.c) Which positions in the 12th row of Pascal’s Triangle have a value of 165?d) Find the sum of the numbers in each of the first four rows of Pascal’s Triangle. Use your result to derive a function, S(n), for the sum of all numbers in the nth row of Pascal’s Triangle. What is the sum of all numbers in the eleventh row?

Example 3: Use Pascal’s Triangle to determine the number of paths from point A to point B if east and south are the only possible directions.

A

B

A

B

A

B

A

B

a) b) c) d)

b) (x + 2)6 c) (2x - 3)4

a) Define the binomial theorem and explain how it is used to expand (x + 1)3.

Expand the expressions in parts (b) and (c) using the binomial theorem.

Example 4: The Binomial Theorem

Example 5: Expand each expression.

a) (x2 - 2y)4 b) c)

1 8 28 56 70 56 28 8 1i.

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Permutations and CombinationsLESSON THREE - The Binomial TheoremLesson Notes

tk+1 = nCk(x)n-k(y)k

Example 6: Write each expression as a binomial power.

Example 7: Use the general term formula to find the requested term in a binomial expansion.

tk + 1 = nCk(x)n - k(y)k

General Term

a) Find the third term in the expansion of (x - 3)4 .

b) Find the fifth term in the expansion of (3a3 - 2b2)8 .

c) Find the fourth term in the expansion of . 6

2 1x -

x

Example 8: Finding Specific Values.a) In the expansion of (5a - 2b)9, what is the coefficient of the term containing a5 ?b) In the expansion of (4a3 + 3b3)5, what is the coefficient of the term containing b12 ?c) In the expansion of (3a - 4)8, what is the middle term?d) If there are 23 terms are in the expansion of (a - 2)3k-5, what is the value of k?

Example 9: Finding Specific Values.a) A term in the expansion of (ma - 4)5 is -5760a2. What is the value of m?b) The term -1080a2b3 occurs in the expansion of (2a - 3b)n. What is the value of n?

c) A term in the expansion of (a + m)7 is . What is the value of m. ba

a) In the expansion of , what is the constant term?

b) In the expansion of , what is the constant term?

c) In the expansion of , one of the terms is 240x2. What is the value of b?b

Example 10: Finding Specific Values.

a) x4 + 4x3y + 6x2y2 + 4xy3 + y4

b) 32a5 - 240a4b + 720a3b2 - 1080a2b3 + 810ab4 - 243b5

c)

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Answer KeyPolynomial, Radical, and Rational Functions Lesson One: Polynomial Functions

Example 1: a) Leading coefficient is an ; polynomial degree is n; constant term is a0. i) 3; 1; -2 ii) 1; 3; -1 iii) 5; 0; 5

Example 2: a) i) Even-degree polynomials with a positive leading coefficient have a trendline that matches an upright parabola. End behaviour: The graph starts in the upper-left quadrant (II) and ends in the upper-right quadrant (I).ii) Even-degree polynomials with a negative leading coefficient have a trendline that matches an upside-down parabola. End behaviour: The graph starts in the lower-left quadrant (III) and ends in the lower-right quadrant (IV).

b) i) Odd-degree polynomials with a positive leading coefficient have a trendline matching the line y = x. The end behaviour is that the graph starts in the lower-left quadrant (III) and ends in the upper-right quadrant (I). ii) Odd-degree polynomials with a negative leading coefficient have a trendline matching the line y = -x. The end behaviour is that the graph starts in the upper-left quadrant (II) and ends in the lower-right quadrant (IV).

Example 3: a) Zero of a Polynomial Function: Any value of x that satisfies the equation P(x) = 0 is called a zero of the polynomial. A polynomial can have several unique zeros, duplicate zeros, or no real zeros. i) Yes; P(-1) = 0 ii) No; P(3) ≠ 0.b) Zeros: -1, 5.c) The x-intercepts of the polynomial’s graph are -1 and 5.These are the same as the zeros of the polynomial.d) "Zero" describes a property of a function; "Root" describes a property of an equation;and "x-intercept" describes a property of a graph.

Example 4: a) Multiplicity of a Zero: The multiplicity of a zero (or root) is how many times the root appears as a solution.Zeros give an indication as to how the graph will behave near the x-intercept corresponding to the root.b) Zeros: -3 (multiplicity 1) and 1 (multiplicity 1). c) Zero: 3 (multiplicity 2).d) Zero: 1 (multiplicity 3). e) Zeros: -1 (multiplicity 2) and 2 (multiplicity 1).

Example 5: a) i) Zeros: -3 (multiplicity 1) and 5 (multiplicity 1). ii) y-intercept: (0, -7.5). iii) End behaviour: graph starts in QII, ends in QI. iv) Other points: parabola vertex (1, -8).

b) i) Zeros: -1 (multiplicity 1) and 0 (multiplicity 2). ii) y-intercept: (0, 0).iii) End behaviour: graph starts in QII, ends in QIV.iv) Other points: (-2, 4), (-0.67, -0.15), (1, -2).


Example 6: a) i) Zeros: -2 (multiplicity 2) and 1 (multiplicity 2).ii) y-intercept: (0, 4). iii) End behaviour: graph starts in QII, ends in QI.iv) Other points: (-3, 16), (-0.5, 5.0625), (2, 16).

b) i) Zeros: -1 (multiplicity 3), 0 (multiplicity 1), and 2 (multiplicity 2).ii) y-intercept: (0, 0). iii) End behaviour: graph starts in QII, ends in QI.iv) Other points: (-2, 32), (-0.3, -0.5), (1.1, 8.3), (3, 192).


Example 7: a) i) Zeros: -0.5 (multiplicity 1) and 0.5 (multiplicity 1).ii) y-intercept: (0, 1). iii) End behaviour: graph starts in QIII, ends in QIV.iv) Other points: parabola vertex (0, 1).

b) i) Zeros: -0.67 (multiplicity 1), 0 (multiplicity 1), and 0.75 (multiplicity 1).ii) y-intercept: (0, 0). iii) End behaviour: graph starts in QIII, ends in QI.iv) Other points: (-1, -7), (-0.4, 1.5), (0.4, -1.8), (1, 5).


a)

3b)

a)

b)

Example 8: Example 9: Example 10:

a)

b)

Example 11: a) x: [-15, 15, 1], y: [-169, 87, 1] b) x: [-12, 7, 1], y: [-192, 378, 1] c) x: [-12, 24, 1], y: [-1256, 2304, 1]

b) i) Y ii) N iii) Y iv) N v) Y vi) N vii) N viii) Y ix) N

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Answer Keya) b)Example 12:

Example 14:

a) Pproduct(x) = x2(x + 2); Psum(x) = 3x + 2

b) x3 + 2x2 - 3x - 11550 = 0.c) Window Settings: x: [-10, 30, 1], y: [-12320, 17160, 1]Quinn and Ralph are 22 since x = 22. Audrey is two years older, so she is 24.

a) Window Settings: x: [0, 6, 1], y: [-1.13, 1.17, 1]b) At 3.42 seconds, the maximum volume of 1.17 L is inhaledc) One breath takes 5.34 seconds to complete.d) 64% of the breath is spent inhaling.

Example 15:

Example 16:

Polynomial, Radical, and Rational Functions Lesson Two: Polynomial Division

Example 1: a) Quotient: x2 - 5; R = 4 b) P(x): x3 + 2x2 - 5x - 6; D(x) = x + 2; Q(x) = x2 - 5; R = 4c) L.S.= R.S. d) Q(x) = x2 - 5 + 4/(x + 2) e) Q(x) = x2 - 5 + 4/(x + 2)

Example 2: a) 3x2 - 7x + 9 - 10/(x + 1) b) x2 + 2x + 1 c) x2 - 2x + 4 - 9/(x + 2)

Example 3: a) 3x2 + 3x + 2 - 1/(x - 1) b) 3x3 - x2 + 2x - 1 c) 2x3 + 2x2 - 5x - 5 - 1/(x - 1)

Example 4: a) x - 2 b) 2 c) x2 – 4 d) x2 + 5x + 12 + 36/(x - 3)

Example 5: a) a = -5 b) a = -5

Example 6: The dimensions of the base are x + 5 and x - 3

Example 7: a) f(x) = 2(x + 1)(x - 2)2 b) g(x) = x + 1 c) Q(x) = 2(x – 2)2

Example 8: a) f(x) = 4x3 - 7x - 3 b) g(x) = x - 1

Example 9: a) R = -4 b) R = -4. The point (1, -4) exists on the graph. The remainder is just the y-value of the graph when x = 1.c) Both synthetic division and the remainder theorem return a result of -4 for the remainder.d) i) R = 4 ii) R = -2 iii) R = -2e) When the polynomial P(x) is divided by x - a, the remainder is P(a).

(1, -4) (1, 0)

Example 10: a) R = 0 b) R = 0. The point (1, 0) exists on the graph. The remainder is just the y-value of the graph.c) Both synthetic division and the remainder theorem return a result of 0 for the remainder.d) If P(x) is divided by x - a, and P(a) = 0, then x - a is a factor of P(x).e) When we use the remainder theorem, the result can be any real number. If we use the remainder theorem and get a result of zero, the factor theorem gives us one additional piece of information - the divisor fits evenly into the polynomial and is therefore a factor of the polynomial. Put simply, we're always using the remainder theorem, but in the special case of R = 0 we get extra information from the factor theorem.

Example 11: a) P(-1) ≠ 0, so x + 1 is not a factor.b) P(-2) ≠ 0, so x + 2 is not a factor.c) P(1/3) = 0, so 3x - 1 is a factor.d) P(-3/2) ≠ 0, so 2x + 3 is not a factor.Example 12: a) k = 3 b) k = -7 c) k = -7 d) k = -5Example 13: m = 4 and n = -7Example 14: m = 4 and n = -3Example 15: a = 5

Example 7c

Example 9Example 10

Remainder Theorem(R = any number)

Factor Theorem(R = 0)

Example 13:

a) V(x) = x(20 - 2x)(16 - 2x)b) 0 < x < 8 or (0, 8)c) Window Settings: x: [0,8, 1], y: [0, 420, 1]d) When the side length of a corner square is 2.94 cm, the volume of the box will be maximized at 420.11 cm3. e) The volume of the box is greater than 200 cm3 when 0.74 < x < 5.93.

or (0.74, 5.93)




Interval Notation


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Answer KeyPolynomial, Radical, and Rational Functions Lesson Three: Polynomial Factoring

Example 1: a) The integral factors of the constant term of a polynomial are potential zeros of the polynomial.b) Potential zeros of the polynomial are ±1 and ±3.c) The zeros of P(x) are -3 and 1 since P(-3) = 0 and P(1) = 0d) The x-intercepts match the zeros of the polynomiale) P(x) = (x + 3)(x - 1)2.

Example 2: a) P(x) = (x + 3)(x + 1)(x - 1).b) All of the factors can be found using the graph.c) Factor by grouping.

Example 3: a) P(x) = (2x2 + 1)(x - 3).b) Not all of the factors can be found using the graph.c) Factor by grouping.

Example 4: a) P(x) = (x + 2)(x – 1)2.b) All of the factors can be found using the graph.c) No.

Example 5: a) P(x) = (x2 + 2x + 4)(x - 2). b) Not all of the factors can be found using the graph.c) x3 - 8 is a difference of cubes

Example 6: a) P(x) = (x2 + x + 2)(x - 3).b) Not all of the factors can be found using the graph.c) No.

Example 7: a) P(x) = (x2 + 4)(x - 2)(x + 2). b) Not all of the factors can be found using the graph.c) x4 – 16 is a difference of squares.

Example 8: a) P(x) = (x + 3)(x - 1)2(x - 2)2.b) All of the factors can be found using the graph.c) No.

Example 9: a) P(x) = 1/2x2(x + 4)(x – 1).

b) P(x) = 2(x + 1)2(x - 2).

Example 10: Width = 10 cm; Height = 7 cm; Length = 15 cm

Example 11: -8; -7; -6

Example 12: k = 2; P(x) = (x + 3)(x - 2)(x - 6)

Example 13: a = -3 and b = -1

Example 14: a) x = -3, 2, and 4 b)

Quadratic FormulaFrom Math 20-1:The roots of a quadratic equation with the form ax2 + bx + c = 0 can be found with the quadratic formula:

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Answer KeyPolynomial, Radical, and Rational Functions Lesson Four: Radical Functions

Example 1:

x f(x)

-10149

undefined

0123

a) b) Domain: x ≥ 0;Range: y ≥ 0

Interval Notation:Domain: [0, ∞);Range: [0, ∞)

c)Example 2:a) b)

Example 3:a) b) c) d)



ORIGINAL:Domain: x ε R or (-∞, ∞)Range: y ε R or (-∞, ∞)

TRANSFORMED:Domain: x ≥ -4 or [-4, ∞)Range: y ≥ 0 or [0, ∞)

ORIGINAL:Domain: x ε R or (-∞, ∞)Range: y ≤ 9 or (-∞, 9]

TRANSFORMED:Domain: -5 ≤ x ≤ 1 or [-5, 1]Range: 0 ≤ y ≤ 3 or [0, 3]

ORIGINAL:Domain: x ε R or (-∞, ∞)Range: y ≥ -4 or [-4, ∞)

TRANSFORMED:Domain: x ≤ 3 or x ≥ 7or (-∞, 3] U [7, ∞)Range: y ≥ 0 or [0, ∞)

ORIGINAL:Domain: x ε R or (-∞, ∞)Range: y ≥ 0 or [0, ∞)

TRANSFORMED:Domain: x ε R or (-∞, ∞)Range: y ≥ 0 or [0, ∞)

Example 6:a) b)

Example 7:a) b)

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(7, 3)

(7, 0)

Example 9:a) x = 7

b) c)

Example 10:a) x = 2

b) c)

(2, 2)

(2, 0)

Example 11:a) x = -3, 1

b) c)

(-3, 0)

(1, 4)

(-3, 0) (1, 0)

Example 12:a) No Solution

b) c)

No Solution

No Solution

Example 13:

d

h(d)

1 2 3 4 5

1

2

3

4

5

Example 14:

a)b) Domain: 0 ≤ d ≤ 3; Range: 0 ≤ h(d) ≤ 3or Domain: [0, 3]; Range: [0, 3].When d = 0, the ladder is vertical. When d = 3, the ladder is horizontal.c) 2 m

h

t

1 2 3 4 5 6 7 8

0.5

1.0

1.5

2.0

2.5

3.0

× original time

h t

8

4

1

1.2778

0.9035

0.4517

Example 15:

a)b) 1/2 × original time c)

Example 16:

a)

b)

1

10

2 3 4 5 6 r

V(r)

20

30

40

50

60

Answer Key

a) b) c) d)

ORIGINAL:Domain: x ε R or (-∞, ∞)Range: y ≤ 0 or (-∞, 0]

TRANSFORMED:Domain: x = -5Range: y = 0

ORIGINAL:Domain: x ε R or (-∞, ∞)Range: y ≥ 0.25 or [0.25, ∞)

TRANSFORMED:Domain: x ε R or (-∞, ∞)Range: y ≥ 0.5 or [0.5, ∞)

Example 8:a) b)

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Answer KeyPolynomial, Radical, and Rational Functions Lesson Five: Rational Functions I

Example 1:

x y

-2

-1

-0.5

-0.25

0

0.25

0.5

1

2

-0.5

-1

-2

-4

undef.

4

2

1

0.5

Example 4:

Reciprocal: x ε R, x ≠ -2, 2; y ≤ -1 or y > 0

Asymptotes: x = ±2; y = 0

Reciprocal: x ε R, x ≠ -4, 2; y < 0 or y ≥ 2

Asymptotes: x = -4, x = 2; y = 0

b) Original: x ε R; y ≤ 1/2

Reciprocal: x ε R, x ≠ 4, 8; y ≤ -1/2 or y > 0

Asymptotes: x = 4, x = 8; y = 0

c) Original: x ε R; y ≥ -2

Reciprocal: x ε R, x ≠ 0; y > 0

Asymptotes: x = 0; y = 0

d) Original: x ε R; y ≥ 0

a) b)1. The vertical asymptote of the reciprocal graph occurs at the x-intercept of y = x.

2.The invariant points (points that are identical on both graphs) occur when y = ±1.

3. When the graph of y = x is below the x-axis, so is the reciprocal graph.When the graph of y = x is above the x-axis, so is the reciprocal graph.

c) d)

a) Original: x ε R; y ≥ -1

Example 2:

Reciprocal Graph:Domain: x ε R, x ≠ 5 or (-∞, 5) U (5, ∞); Range: y ε R, y ≠ 0 or (-∞, 0) U (0, ∞)

Asymptote Equation(s):Vertical: x = 5; Horizontal: y = 0

b) Original Graph:Domain: x ε R or (-∞, ∞); Range: y ε R or (-∞, ∞)

Reciprocal Graph:Domain: x ε R, x ≠ 4 or (-∞, 4) U (4, ∞); Range: y ε R, y ≠ 0 or (-∞, 0) U (0, ∞)

Asymptote Equation(s):Vertical: x = 4; Horizontal: y = 0

a) Original Graph:Domain: x ε R or (-∞, ∞);Range: y ε R or (-∞, ∞)

Example 3:

x y

-3

-2

-1

0

1

2

3

x y

-2.05

-1.95

x y

1.95

2.05

0.20

undef.

-0.33

-0.25

-0.33

undef.

0.20

4.94

-5.06

-5.06

4.94

b)1. The vertical asymptotes of the reciprocal graph occur at the x-intercepts of y = x2 - 4.

2. The invariant points (points that are identical in both graphs) occur when y = ±1.

3. When the graph of y = x2 - 4 is below the x-axis, so is the reciprocal graph.When the graph of y = x2 - 4 is above the x-axis, so is the reciprocal graph.

c) d)a)

or D: (-∞, ∞); R: [-1, ∞).

or D: (-∞, -2) U (-2, 2) U (2, ∞); R: (-∞, -1] U (0, ∞)

or D: (-∞, ∞); R: (-∞, 1/2].

or D: (-∞, -4) U (-4, 2) U (2, ∞); R: (-∞, 0) U [2, ∞)

or D: (-∞, ∞); R: [-2, ∞).

or D: (-∞, 4) U (4, 8) U (8, ∞); R: (-∞, -1/2] U (0, ∞)

or D: (-∞, ∞); R: [0, ∞).

or D: (-∞, 0) U (0, ∞); R: (0, ∞)

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Example 4 (continued):

Reciprocal: x ε R; -2 ≤ y < 0

Asymptotes: y = 0

f) Original: x ε R; y ≤ -1/2

Reciprocal: x ε R; 0 < y ≤ 1/2

Asymptotes: y = 0

e) Original: x ε R; y ≥ 2

Example 5:

V

P

1 2 3 4 5 6 7 8 9 10

5

10

15

20

25

30

35

40

45

50

Pressure V.S. Volume of 0.011 mol of a gas at 273.15 K

PV

0.5

1.0

2.0

5.0

10.0

(kPa)(L)

50

25

12.5

5.0

2.5

d

I

1 2 3 4 5 6

10

20

30

40

50

60

70

80

90

100

110

120

130

7 8 9 10 11 12

Illuminance V.S. Distance for a Fluorescent Bulb

Id

2

4

8

12

(W/m2)(m)

ORIGINAL

1

Example 6:

Example 7:

Example 8:


a) P(V) = nRT(1/V).

b) 1/2 × original

c) 2 × original

d) 8.3 kPa•L/mol•K

e) See table & graph

f) See table & graph

a) 1/4 × original

b) 1/9 × original

c) 4 × original

d) 16 × original

e) See table & graph

f) See table & graph

Answer Key

a) b) c) d)

a) x = 1.5; y = 0 b) x = -4, 6; y = 0 c) x = -0.5, 0, 1.33 ; y = 0 d) y = 0

a) VS: 4 b) VT: 3 down c) VS: 3; HT: 4 left d) VS: 2; HT: 3 right; VT: 2 up

a) VT: 2 down b) HT: 2 right; VT: 1 up c) VS: 4; HT: 1 right; VT: 2 down d) VS: 3; HT: 5 right; VT: 2 down

or D: (-∞, ∞); R: [2, ∞).

or D: (-∞, ∞); R: (0, 1/2]

or D: (-∞, ∞); R: (-∞, -1/2].

or D: (-∞, ∞); R: [-2, 0)

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Answer KeyPolynomial, Radical, and Rational Functions Lesson Six: Rational Functions II

Example 1:

Example 2:

Example 3:

Example 4:

i) Horizontal Asymptote: y = 2

ii) Vertical Asymptote(s): x = -2

v) Domain: x ε R, x ≠ -2; Range: y ε R, y ≠ 2

iii) y - intercept: (0, -3)iv) x - intercept(s): (3, 0)


i) Horizontal Asymptote: Noneii) Vertical Asymptote(s): x = 1

v) Domain: x ε R, x ≠ 1; Range: y ε R

iii) y - intercept: (0, 8)iv) x - intercept(s): (-4, 0), (2, 0)

vi) Oblique Asymptote: y = x + 3

Example 7:

i) y = x - 3

ii) Hole: (2, -1)

v) Domain: x ε R, x ≠ 2; Range: y ε R, y ≠ -1

iii) y - intercept: (0, -3)

iv) x - intercept(s): (3, 0)

y = xx2 - 9

y =x2 + 1x + 2 y =

x2 - 16x + 4 y =

x3 - x2 - 2xx2 - x - 2

y = 4xx - 2

y =x2 - 1

x2

y =x2 + 93x2

y =x2 - x - 6

3x2 - 3x - 18

y =x + 4

x2 + 5x + 4 y =x - 3

x2 - 4x + 3 y = x2 - x - 6x + 1

y = x2 + 5x - 1

a)

a)

a)

b)

b)

b)

c)

c)

c)

d)

d)

d)

i) Horizontal Asymptote: y = 0

ii) Vertical Asymptote(s): x = ±4

v) Domain: x ε R, x ≠ ±4; Range: y ε R

iii) y - intercept: (0, 0)iv) x - intercept(s): (0, 0)

or D: (-∞, -4) U (-4, 4) U (4, ∞); R: (-∞, ∞) or D: (-∞, -2) U (-2, ∞); R: (-∞, 2) U (2, ∞) or D: (-∞, 2) U (2, ∞); R: (-∞, -1) U (-1, ∞)or D: (-∞, 1) U (1, ∞); R: (-∞, ∞)

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Example 8:

Example 10: a) x = 4

Example 12: a) x = 1. x = 2 is an extraneous root

b) c)

(1, -3) (1, 0)

(6, 0)

c) Graphing Solution: x-intercept method.

b) Cynthia: 9 km/h; Alan: 6 km/h

d s t

Cynthia

Alan

a)

x

x + 315

10

x + 315

x10

(10, 0)(-0.4, 0)


b) Canoe speed: 10 km/h

d s t

24

24

x - 2

x + 2

24x - 2

24x + 2

Upstream

Downstream

a)

Example 13:

Example 14:

Example 15:


(6, 0)

Example 16:


a)

b) Mass of almonds required: 90 g

-300 300

-1

1

0.5

-0.5

(90, 0)

Answer Key

Sum of times equals 5 h.

Equal times.a) b)

(4, 4)

b)

(4, 0)

c)

Example 9:

a) b)

c) d)

Example 11: a) x = -1/2 and x = 2

(2, -6)(-0.5, -6)

(2, 0)(-0.5, 0)

b) c)

a)

b) Number of goals required: 6

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Answer KeyTransformations and Operations Lesson One: Basic Transformations

Example 1:


Example 3: a) b) c)

Example 4: a) b) c)



a) b) c) d)

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Example 11:

a) b)

Example 12:

a) b)

Example 13:

a) R(n) = 5nC(n) = 2n + 150b) 50 loavesc) C2(n) = 2n + 200d) R2(n) = 6ne) 50 loaves

$

n

100

200

300

400

500

20 40 60 80

(50, 250)

Example 14:

a)

$

n

100

200

300

400

500

20 40 60 80

(50, 300)

b) 12 metres

Example 13a Example 13e

Answer Key

or

or

R(n)

C(n)

R2(n)

C2(n)

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Answer KeyTransformations and Operations Lesson Two: Combined Transformations

Example 1: a) a is the vertical stretch factor.b is the reciprocal of the horizontal stretch factor.h is the horizontal displacement.k is the vertical displacement.

Example 2:

Example 3: b) i. H.T. 1 rightV.T. 3 up

a) H.T. 3 left ii. H.T. 2 leftV.T. 4 down

iii. H.T. 2 rightV.T. 3 down

iv. H.T. 7 leftV.T. 5 up

Example 4:

Example 5: b) i. V.S. 2H.T. 3 leftV.T. 1 up

a) Stretches and reflections should be applied first, in any order.Translations should be applied last, in any order.

ii. H.S. 3Reflection about x-axisV.T. 4 down

iii. V.S. 1/2Reflection about y-axisH.T. 2 left; V.T. 3 down

iv. V.S. 3; H.S. 1/4Reflection about x-axisReflection about y-axisH.T. 1 right; V.T. 2 up



Example 8:a) (1, 0)b) (3, 6)c) m = 8 and n = 1

Example 9:a) y = -3f(x – 2)b) y = -f[3(x + 2)]

Example 10:Axis-IndependenceApply all the vertical transformations together and apply all the horizontal transformations together, in either order.

Example 11:a) H.T. 8 right; V.T. 7 upb) Reflection about x-axis; H.T. 4 left; V.T. 6 downc) H.S. 2; H.T. 3 left; V.T. 7 upd) H.S. 1/2; Reflection about x & y-axis; H.T. 5 right; V.T. 7 down.e) The spaceship is not a function, and it must be translated in a specific order to avoid the asteroids.

b) i. V.S. 1/3H.S. 1/5

ii. V.S. 2H.S. 4

iii. V.S. 1/2H.S. 3Reflection about x-axis

iv. V.S. 3H.S. 1/2Reflection about x-axisReflection about y-axis

a) b) c) d)

a) b) c) d)

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Answer KeyTransformations and Operations Lesson Three: Inverses

Example 1: a) Line of Symmetry: y = x

b)

Example 2:

a) b) c) d)

Original:D: x ε RR: y ε R

Inverse:D: x ε RR: y ε R

Original:D: x ε RR: y ε R

Inverse:D: x ε RR: y ε R

The inverse is a function.

Example 3:

a) b) a) b)

Example 4:

Example 5:

a) b) a) b)

Example 6:

Example 7:

a) (10, 8)

b) True.f-1(b) = a

c) f(5) = 4

d) k = 30

Example 8:

a) 28 °C is equivalent to 82.4 °F

b)

c) 100 °F is equivalent to 37.8 °C

d) C(F) can't be graphed since its dependent variable is C, but the dependent variable on the graph's y-axis is F. This is a mismatch.

e)

f) The invariant point occurs when the temperature in degrees Fahrenheit is equal to the temperature in degrees Celsius. -40 °F is equal to -40 °C.

°C

50

100°F

-50

50-50 100-100

-100

(-40, -40)

F(C)

F-1(C)

Restrict the domain ofthe original function to-10 ≤ x ≤ -5 or -5 ≤ x ≤ 0

Restrict the domain of the original function tox ≤ 5 or x ≥ 5.

Restrict the domain ofthe original function tox ≤ 0 or x ≥ 0

Restrict the domain of the original function tox ≤ -3 or x ≥ -3.

The inverse is a function.

D: x ε R D: x ≥ 4

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Answer KeyTransformations and Operations Lesson Four: Function Operations

Example 1:

c) d)

x (f • g)(x)

-6

-3

0

3

6

-4

-8

-2

-3

DNE

x (f ÷ g)(x)

-6

-4

-2

0

2

4

6

DNE

-4

-8

-6

-4

-2

DNE

Domain: -6 ≤ x ≤ 3or [-6, 3]

Range: -8 ≤ y ≤ -2or [-8, -2]

f(x)

g(x)

f(x)

g(x)

Domain: -4 ≤ x ≤ 4or [-4, 4]

Range: -8 ≤ y ≤ -2or [-8, -2]

Example 2:

a) i. (f + g)(-4) = -2 ii. h(x) = -2; h(-4) = -2b) i. (f – g)(6) = 8 ii. h(x) = 2x – 4; h(6) = 8c) i. (fg)(-1) = -8 ii. h(x) = -x2 + 4x - 3; h(-1) = -8d) i. (f/g)(5) = -0.5 ii. h(x) = (x - 3)/(-x + 1); h(5) = -0.5

Domain: 3 < x ≤ 5 or (3, 5]Range: 0 < y ≤ 1 or (0, 1]

Domain: x ≥ -4 or [-4, ∞)Range: y ≤ 9 or (-∞, 9]

Domain: 0 < x ≤ 10 or (0, 10] Range: -10 ≤ y ≤ 0 or [-10, 0]

Domain: x > -2 or (-2, ∞)Range: y > 0 or (0, ∞)

f(x)

g(x)f(x)

g(x)

f(x)

g(x)

f(x)

g(x)

m(x)

Example 3:

Domain: x ≥ -4 or [-4, ∞)Range: y ≥ 0 or [0, ∞)Transformation: y = f(x) - 1

f(x)

g(x)

Example 4:

a)

Domain: x ≥ -4 or [-4, ∞)Range: y ≤ -1 or (-∞, -1]Transformation: y = -f(x).

f(x)

g(x)

Domain: x ε R or (-∞, ∞)Range: y ≤ -6 or (-∞, -6]Transformation: y = f(x) - 2

Example 5:

a)

Domain: x ε R or (-∞, ∞)Range: y ≤ -2 or (-∞, -2]Transformation: y = 1/2f(x)

b)b)

f(x)

g(x)

f(x)

g(x)

a)

f(x)

g(x)

x (f + g)(x)

-8

-4

-2

0

1

4

-6

-6

0

-6

-9

-9

Domain:-8 ≤ x ≤ 4or [-8, 4]

Range:-9 ≤ y ≤ 0or [-9, 0]

f(x)

g(x)

x (f - g)(x)

-9

-5

-3

0

3

6

DNE

10

9

2

5

DNE

Domain: -5 ≤ x ≤ 3;or [-5, 3]

Range:2 ≤ y ≤ 10or [2, 10]

b)

a) b) c) d)

Reminder: Math 30-1 students are expected to know that domain and range can be expressed using interval notation.

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f(x)

g(x)

f(x)

g(x)

f(x)

g(x)f(x)

g(x)

Domain: x ε R, x ≠ 0;Range: y ε R, y ≠ 0

Example 6:

a)

Domain: x ε R, x ≠ 2;Range: y ε R, y ≠ 0

Domain: x ε R, x ≠ -3;Range: y ε R, y ≠ 0

c)

Domain: x ≥ -3, x ≠ -2;Range: y ε R, y ≠ 0

d)b)

a) AL(x) = 8x2 – 8x

b) AS(x) = 3x2 – 3x

c) AL(x) - AS(x) = 10; x = 2

d) AL(2) + AS(2) = 22;

e) The large lot is 2.67 times larger than the small lot

Example 7:

100

200

300

400

-100

$

n

500

20 40 60

R(n)

E(n)P(n)

a) R(n) = 12n;E(n) = 4n + 160;P(n) = 8n – 160

b) When 52 games are sold, the profit is $256

c) Greg will break even when he sells 20 games


a) The surface area and volume formulae have two variables, so they may not be written as single-variable functions.

b) c) d)

e) f)

Answer Key

Transformations and Operations Lesson Five: Function Composition

Example 1:

Example 2: a) m(3) = 33 b) n(1) = -4 c) p(2) = -2 d) q(-4) = -16

Example 3: a) m(x) = 4x2 – 3 b) n(x) = 2x2 – 6 c) p(x) = x4 – 6x2 + 6 d) q(x) = 4x e) All of the results match

Example 4: a) m(x) = (3x + 1)2 b) n(x) = 3(x + 1)2

The graph of f(x) is horizontally stretched by a scale factor of 1/3.

The graph of f(x) is vertically stretched by a scale factor of 3.

n(x)

m(x)

x g(x)

-3

-2-101

f(g(x))

23

9

410149

6

1-2-3-216

x f(x)

0

1

2

3

g(f(x))

-3

-2

-1

0

9

4

1

0

d) m(x) = x2 – 3

e) n(x) = (x – 3)2

a) b) f)c) Order matters in a composition of functions.

or D: (-∞, 0) U (0, ∞); R: (-∞, 0) U (0, ∞) or D: (-∞, 2) U (2, ∞); R: (-∞, 0) U (0, ∞) or D: (-∞, -3) U (-3, ∞); R: (-∞, 0) U (0, ∞) or D: [-3, -2) U (-2, ∞); R: (-∞, 0) U (0, ∞)

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Answer KeyExample 5: a)

Example 6:

b)

a) ( ) 1

| 2|h x

x=

+b) ( ) 2 2h x x= + +

Example 7: a) b)

d

M(d)

100

10

200 300 400 500 600

20

30

40

50

60

Example 8:


a) The cost of the trip is $4.20. It took two separate calculations to find the answer.b) V(d) = 0.08dc) M(V) = 1.05Vd) M(d) = 0.084de) Using function composition, we were able to solve the problem with one calculation instead of two.

Example 11:

a) A(t) = 900πt2

b) A = 8100π cm2

c) t = 7 s; r = 210 cm

Example 12:

a) a(c) = 1.03c

b) j(a) = 78.0472a

c) b(a) = 0.6478a

d) b(c) = 0.6672c

Example 13:

a)

b)

c) h = 4 cm

a) b) c)

d) e) f)

a) (f-1 ◦ f)(x) = x, so the functions are inverses of each other.b) (f-1 ◦ f)(x) ≠ x, so the functions are NOT inverses of each other.

Domain: x ≥ 8 Domain: x ≥ 2

Domain: x ε R, x ≠ -2 Domain: x ≥ -2

Domain: x ε R, x ≠ -2 Domain: x ≥ -2

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Answer KeyExponential and Logarithmic Functions Lesson One: Exponential Functions

Example 1:

Example 2:

Example 3:

5

5-5

10

-5

5

5-5

10

-5

5

5-5

10

-5

Domain: x ε R or (-∞, ∞)Range: y > 0 or (0, ∞)Asymptote: y = 0




5

5-5

10

-5

Example 4:

5

5-5

10

-5

5

5-5

10

-5

Domain: x ε R or (-∞, ∞)Range: y > -4 or (-4, ∞)Asymptote: y = -4

Domain: x ε R or (-∞, ∞)Range: y > -2 or (-2, ∞)Asymptote: y = -2



5

5-5

10

-5

5

5-5

10

-5

Example 5: a) b)

Example 6: a) V.S. of 9equals H.T.2 units left.

See Videob) c) d) e) f)

Parts (a-d):Domain: x ε R or (-∞, ∞)Range: y > 0 or (0, ∞)x-intercept: Noney-intercept: (0, 1)Asymptote: y = 0

An exponential function is defined as y = bx, where b > 0 and b ≠ 1. When b > 1, we get exponential growth. When 0 < b < 1, we get exponential decay. Other b-values, such as -1, 0, and 1, will not form exponential functions.

a) b) c) d)

a) b) c) d)

a) b) c) d)

a) b) c) d)

; ; ; ;

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Example 7: Example 8: Example 9: Example 10: Example 11: Example 12: Example 13: Example 14:

a)b)

c)

d)

a)b)c)d)

a)b)c)

d)

a)b)c)d)

a)infinite solutions

b)

c)d)

a)

b)

c)d)

a)

b)c)d)

a)b)c)d)

no solution

m

t

100

10 20

50

m

t

1

10 50

0.5

20 30 40 60

(49, 0.1)

Example 15:

a)

b) 84 g

Example 16:

a)

b) 32254 bacteria

B

t

50000

100000

5 10

B

t

200

1000

0 2-8

400

600

800

(-6, 50)

Example 17: a) ; 69 MHz b) ; $600

Example 18:

a) 853,370

b) 54 years

c) 21406

d) 77 years

P%

t

5

20 40 60 80 100

4

3

2

1

(53.7, 2)

t

1

20 40 60 80 100

0.8

0.6

0.4

0.2

P%

(76.7, 0.5)

Example 18b

Example 19c

Example 18d

Example 19:

a)

b) $565.70Interest: $65.70

( ) ( )500 1.025t

A t =

c) See graph

d) 28 years

$

t

1000

10 20

500

P%

t20 50

1

10 30 40

2

3

4

5

(28, 2)

e) $566.14; $566.50; $566.57As the compounding frequency increases, there is less and less of a monetary increase.

Example 19d

Example 16c Example 16d

Example 15c Example 15d

c) See Graph

d) 49 years

c) See Graph

d) 6 hours ago

Answer Key

e)

f)

Watch Out! The graph requires hours on the t-axis, so we can rewrite the exponential function as:

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Answer KeyExponential and Logarithmic Functions Lesson Two: Laws of Logarithms

Example 1:

Example 2:

Example 4:

a)

b)

c)

d)

e)

f)

g)

h)

Example 5:

a)

b)

c)

d)

e)

f)

g)

h)

Example 6:

a)

b)

c)

d)

e)

f)

g)

h)

Example 7:

a)

b)

c)

d)

e)

f)

g)

h)

Example 8:

a)

b)

c)

d)

e)

f)

g)

h)

Example 9:

a)

b)

c)

d)

e)

f)

g)

h)

Example 10:

a)

b)

c)

d)

e)

f)

g)

h)

Example 17:

a)

b)

c)

d)

e)

f)

g)

h)

a) The base of the logarithm is b,a is called the argument of the logarithm,and E is the result of the logarithm.

In the exponential form, a is the result,b is the base, and E is the exponent.

b) i. 0; 1; 2; 3 ii. 0; 1; 2; 3

c) i. log42 ii.

a)

b)

c)

Example 3:

a)

b)

c)

d)

e)

f)

g)

h)

Example 11:

a)

b)

c)

d)

Example 12:

a)

b)

c)

d)

Example 13:

a)

b)

c)

d) ±

Example 14:

a)

b)

c)

d)

Example 15:

a)

b)

c)

d)

Example 16:

a)

b)

c)

d)

e)

f)

g)

h)

Example 18:

a)

b)

c)

d)

e)

f)

g)

h)

see video

Example 19:

a)

b)

c)

d)

e)

f)

g)

h)

Example 20:

a)

b)

c)

d)

e)

f)

g)

h)

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Answer KeyExponential and Logarithmic Functions Lesson Three: Logarithmic Functions

Example 1:

Example 2:

Example 3:

D: x > 0or (0, ∞)

R: y ε Ror (-∞, ∞)

A: x = 0

D: x > 0or (0, ∞)

R: y ε Ror (-∞, ∞)

A: x = 0

Example 4:

D: x > -2or (-2, ∞)

R: y ε Ror (-∞, ∞)

A: x = -2

D: x > 3or (3, ∞)

R: y ε Ror (-∞, ∞)

A: x = 3

D: x > -4or (-4, ∞)

R: y ε Ror (-∞, ∞)

A: x = -4

Example 5:

D: x > -3or (-3, ∞)

R: y ε Ror (-∞, ∞)

A: x = -3

D: x > 0or (0, ∞)

R: y ε Ror (-∞, ∞)

A: x = 0

D: x > -3or (-3, ∞)

R: y ε Ror (-∞, ∞)

A: x = -3

D: x > -2or (-2, ∞)

R: y ε Ror (-∞, ∞)

A: x = -2

D: x > 0or (0, ∞)

R: y ε Ror (-∞, ∞)

A: x = 0

D: x > 0or (0, ∞)

R: y ε Ror (-∞, ∞)

A: x = 0

D: x > 0or (0, ∞)

R: y ε Ror (-∞, ∞)

A: x = 0

D: x > 0or (0, ∞)

R: y ε Ror (-∞, ∞)

A: x = 0

D: x > 0or (0, ∞)

R: y ε Ror (-∞, ∞)

A: x = 0

D: x > 0or (0, ∞)

R: y ε Ror (-∞, ∞)

A: x = 0

D: x > 0or (0, ∞)

R: y ε Ror (-∞, ∞)

A: x = 0

a) b) c) d)

a) b) c) d)

a) b) c) d)

a) b) c) d)

Domain

Range

x-intercept

y-intercept

AsymptoteEquation

y = 2x y = log2x

x ε R

y > 0

none

(0, 1)

y = 0

x > 0

y ε R

(1, 0)

none

x = 0

5

10

-5

-5 5

f(x) = 2x

f-1(x) = log2x

a) See Graph c) See Video d) e)i) -1, ii) 0, iii) 1, iv) 2.8

b) See Graph

10

is a function.

f)y = log1x, y = log0x, and y = log-2x are not functions.

g) The logarithmic function y = logbx is the inverse of the exponential function y = bx. It is defined for all real numbers such that b>0 and x>0.

h) Graph log2x using logx/log2

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Example 6:

D: x > 0or (0, ∞)

R: y ε Ror (-∞, ∞)

A: x = 0

D: x > 1or (1, ∞)

R: y ε Ror (-∞, ∞)

A: x = 1

D: x > 2or (2, ∞)

R: y > log34or (log34, ∞)

A: none

D: x > 0or (0, ∞)

R: y ε Ror (-∞, ∞)

A: x = 0

Example 7:

(8, 262144)

(-0.60, 0.72)

No Solution

(8, 2)

(25, 12)

(4, 3)

Example 8:

a) b) c) d)

a) x = 8 b) c) No Solution

a) x = 8 b) x = 25 c) x = 4

Example 9:a)

b) (-3, 0) and (0, 2)

c)

d)

e)

Example 10:a)

b)

c)

d)

e)

Example 11:a)

b)

c)

d)

e)

Example 12:

a) 4

b) 0.1 m

c) 31.6 times stronger

d) See Video

e) 10 times stronger

f) See Video

g) 5.5

h) 5.4

Example 13:

a) 60 dB

b) 0.1 W/m2

c) 100 times more intense

d) See Video

e) 100 times more intense

f) See Video

g) 23 dB

h) 37 dB

Example 14:

a) pH = 4

b) 10-11 mol/L

c) 1000 times stronger

d) See Video

e) pH = 2

f) pH = 5

g) 100 times more acidic

Example 15:

a) 200 cents

b) 784 Hz

c) 1200 cents separate the two notes

Answer Key

(y-axis)

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Answer KeyTrigonometry Lesson One: Degrees and Radians Note: For illustrative purposes,

all diagram angles will be in degrees.

120°

-120°

θ

a) The rotation angle between the initial arm and the terminal arm is called the standard position angle.

Example 1:b) An angle is positive if we rotate the terminal arm counter-clockwise, and negative if rotated clockwise.

c) The angle formed between the terminal arm and the x-axis is called the reference angle.

d) If the terminal arm is rotated by a multiple of 360° in either direction, it will return to its original position. These angles are called co-terminal angles.

e) A principal angle is an angle that exists between 0°and 360°.

f) The general form of co-terminal angles is θc = θp + n(360°) using degrees, or θc = θp + n(2π) using radians.

420° 60°

45°,405°, 765°, 1125°, 1485°

45°, -315°, -675°, -1035°, -1395°

57.3°

1°

Example 2:

150°

30°

a) i. One degree is defined as 1/360th of a full rotation.

ii. One radian is the angle formed when the terminal arm swipes out an arc that has the same length as the terminal arm. One radian is approximately 57.3°.

iii. One revolution is defined as 360º, or 2pi. It is one complete rotation around a circle.

b) i. 0.40 rad ii. 0.06 rev iii. 148.97° iv. 0.41 rev v. 270° vi. 4.71 rad c) i. 0.79 rad ii. π/4 rad

Conversion Multiplier Reference Chart

degree radian revolution

degree

radian

revolution

π180°

π180°

2π1 rev

2π1 rev

1 rev

360°

1 rev

360°

Example 3: a) 3.05 rad b) 7π/6 rad c) 1/3 rev d) 143.24° e) 270° f) 4.71 rad g) 1/4 rev h) 180° i) 6π rad

30° =

45° =

60° =

0° =

330° =

315° =

300° =

90° =

= 270°

= 120°

= 135°

= 150°

= 180°

= 210°

= 225°

= 240°

360° =

Example 4:

210°

30°θ

-260°

80°

θ

304°

θ

56°

135°

θ45°

309°

51°

θ

Example 5:a) θr = 30° b) θr = 80° c) θr = 56°

(or 0.98 rad)d) θr = 45°(or π/4 rad)

e) θr = 51°(or 2π/7 rad)

120°

θ

60°156°

θ24°

225°

θ45°

210°

θ30°

Example 6:a) θp = 210°, θr = 30° b) θp = 225°, θr = 45° c) θp = 156°, θr = 24° d) θp = 120°, θr = 60°

(or θp = 2.72, θr = 0.42) (or θp = 2π/3, θr = π/3)

Example 7:

a) θ = 60°, θp = 60° b) θ = -495°, θp = 225°

c) θ = 675°, θp = 315° d) θ = 480°, θp = 120°

θc = -855°, -135°, 225°, 585°θc = -300°, 420°, 780° Example 8:a) θp = 93° b) θp = 148° c) θp = 144° d) θp = 330°

(or 2.58 rad) (or 4π/5 rad) (or 11π/6 rad)

Example 9:a) θc = 1380° b) θc = -138π/5 c) θc = 20° d) θc = 2π/3

60°

225°

θc = -45°, 315° θc = -960°, -600°, -240°, 120°,840°, 1200°(or θc = -0.785, 5.50)

(or θc = -16π/3,-10π/3, -4π/3, 2π/3, 14π/3, 20π/3)

120°

315°

93°

148° 144°

330°

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Example 10:a) θp = 112.62°, θr = 67.38° b) θp = 303.69°, θr = 56.31°

Example 11:a) sinθ: QI: +, QII: +, QIII: -, QIV: -b) cosθ: QI +, QII: -, QIII: -, QIV: +c) tanθ: QI +, QII: -, QIII: +, QIV: -d) cscθ: QI: +, QII: +, QIII: -, QIV: -e) secθ: QI +, QII: -, QIII: -, QIV: +f) cotθ: QI +, QII: -, QIII: +, QIV: -g) sinθ & cscθ share the same quadrant signs.cosθ & secθ share the same quadrant signs.tanθ & cotθ share the same quadrant signs

Example 12: a) i. QIII or QIV ii. QI or QIV iii. QI or QIII b) i. QI ii. QIV iii. QIII c) i. none ii. QIII iii. QI

-12

-513

22.62°202.62°

θ

73

154.62°

25.38°

θ

Example 13:a) θp = 202.62°, θr = 22.62° b) θp = 154.62°, θr = 25.38°

(or θp = 3.54 rad, θr = 0.39 rad) (or θp = 2.70 rad, θr = 0.44 rad)

Example 14:

-2

3

33.69°

326.31°

θ

4

-35

323.13°

36.87°

θ

a) θp = 323.13°, θr = 36.87° b) θp = 326.31°, θr = 33.69°

Example 15:

If the angle θ could exist in either quadrant ___ or ___ ...

The calculator alwayspicks quadrant

I or III or IIII or IVII or IIIII or IVIII or IV

IIIIIIVIV

a)

b) Each answer is different because the calculator is unawareof which quadrant the triangle is in. The calculator assumesMark’s triangle is in QI, Jordan’s triangle is in QII, and Dylan’striangle is in QIV.

Example 16:a) The arc length can be found by multiplying the circumference by the sector percentage. This gives us:a = 2πr × θ/2π = rθ.b) 13.35 cmc) 114.59°d) 2.46 cme) n = 7π/6

Example 17:a) The area of a sector can be found by multiplying the area of the full circle by the sector percentage to get the area of the sector.This gives us:a = πr2 × θ/2π = r2θ/2.b) 28π/3 cm2

c) 3π cm2

d) 81π/2 cm2

e) 15π cm2

Example 18:a) 600°/sb) 0.07 rad/sc) 1.04 kmd) 70 rev/se) 2.60 rev/s

Example 19:a) π/2700 rad/sb) 468.45 km

θ-5

1312

112.62°

67.38°2

-356.31°

θ

303.69°

Answer Key

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Answer KeyTrigonometry Lesson Two: The Unit Circle

Example 1:

-10

10

10 -10

10

10

(0.6, 0.8)

(0.5, 0.5)

a) i. ii. b) i. Yes ii. No

Example 2: See Video.

a)Example 3: b) -1 c)

d) e) 0 f) 0 g) h)

a)Example 4: b) 1 c)

d) e) -1 f) g) h)

, , , ,

, , , ,

Example 5:

a)

b)

, , , ,

, , , ,

Example 6:

a)

b)


Example 8:

a) -2 b) undefined c) d) e) f) -1 g) 0 h)

Example 9:

a) b) 1 c) d)

Example 10:

a) 1 b) c) d)

Example 11:

a) -1 b) c) undefined d) undefined


Example 13:a) P(π/3) means "point coordinates at π/3".

b) c)

d) e) P(3) = (-0.9900, 0.1411)

Example 14:a) C = 2π b) The central angle and arc length of

the unit circle are equal to each other.c) a = 2π/3 d) a = 7π/6

Example 15:a) The unit circle and the line y = 2do not intersect, so it's impossible for sinθ to equal 2.

y = 2

b)

cosθ & sinθ

cscθ & secθ

tanθ & cotθ

Range Number Line

10-1

10-1

10-1

c) d) 53.13°, 302.70° e)

Example 16:a) Inscribe a right triangle with side lengths of |x|, |y|, and a hypotenuse of 1 into the unit circle. We use absolute values because technically, a triangle must have positive side lengths. Plug these side lengths into the Pythagorean Theorem to get x2 + y2 = 1.

|x|

|y|1

b) Use basic trigonometric ratios (SOHCAHTOA)to show that x = cosθ and y = sinθ. c) θp = 167.32°, θr = 12.68°

Example 17:a) (167, 212) b) (-792, 113)

Example 18:a) See Video b) 160 m

c) i. ii.

iii. y = 0 iv.

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Answer KeyTrigonometry Lesson Three: Trigonometric Functions I

Example 1:a) (-5π/6, 3), (-π/6, -4), (7π/6, 1) b) (-3π/4, -12), (π/4, 16), (7π/4, -8) c) (-6π, 8), (-2π, -8), (4π, -4) d) (-3π, 10), (3π/2, -30), (5π/2, -20)

Example 2: a) y = sinθ b) a = 1 c) P = 2πd) c = 0 e) d = 0 f) θ = nπ, nεI g) (0, 0)h) Domain: θ ε R, Range: -1 ≤ y ≤ 1

π2

3π2

0 π 2ππ2

π3π2

2π

-1

22

32

12

12

22

32

1

π6

π4

π3

3π4

2π3

5π6

7π6

5π4

4π3

5π3

7π4

11π6

π6

π4

π3

2π3

3π4

5π6

7π6

5π4

4π3

5π3

7π4

11π6 θ

y

Example 3: a) y = cosθ b) a = 1 c) P = 2πd) c = 0 e) d = 0 f) θ = π/2 + nπ, nεI g) (0, 1)h) Domain: θ ε R, Range: -1 ≤ y ≤ 1

π2

3π2

0 π 2ππ2

π3π2

2π

-1

22

32

12

12

22

32

1

π6

π4

π3

3π4

2π3

5π6

7π6

5π4

4π3

5π3

7π4

11π6

π6

π4

π3

2π3

3π4

5π6

7π6

5π4

4π3

5π3

7π4

11π6 θ

y

π 2ππ2π π6

π4

π3

3π4

2π3

5π6

7π6

5π4

4π3

5π3

7π4

11π6

π6

π4

π3

2π3

3π4

5π6

7π6

5π4

4π3

7π4

11π6 θ

y

3

-3

1

-1

3π2

π2

3π2

π2

3

33

3

33

5π3

Example 4: a) y = tanθ b) Tangent graphs do not have an amplitude. c) P = π d) c = 0 e) d = 0 f) θ = nπ, nεIg) (0, 0) h) Domain: θ ε R, θ ≠ π/2 + nπ, nεI, Range: y ε R

Example 5:

a)

b)

Example 6:

c)

d)

a) b)Example 8: -c) d) 5

c) d)

a) b)

0

-5

5

2π0

-5

5

2π

0

-5

5

2π0

-5

5

2π

0

-5

5

2π0

-5

5

2π2π0

-5

5

0

-5

5

2π

Example 7:

c) d)a) b)

Example 9:

c) d)a) b)

0

-1

1

2ππ0

-1

1

2ππ0

-1

1

6π2π 4π0

-1

1

10π8π6π4π2π

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Answer KeyExample 10:

a) b)

2π-2π π-π

-1

1

2π-2π π-π

-12

12

6

-6

c) d)

2π-2π π-π

-3

3

1

-1

-2

2

0

-1

1

6π2π 4π

a) b)

Example 11:

c) d)

4π-4π 2π-2π

-1

1

4π

-4π 2π-2π

-1

1

2π-2π π-π

-1

1

2π

-2π π-π

-4

4

Example 12:

a) b)

c) d)

2πππ2

-1

1

6π-2π 4π2π

-1

1

4π2π-π

2

1

-1

Example 13:

a) b)

c) d)

2π-2π π-π

4

a) b)

Example 14:

c) d)

0

-5

5

6π2π 4π0

-1

1

2ππ

0

-5

5

2ππ0

-6

6

2ππ

Example 15:

a) b)

c) d)

Example 16:

a) b)

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Answer Key

π2

3π2

0 π 2ππ2

π3π2

2π π6

π4

π3

3π4

2π3

5π6

7π6

5π4

4π3

5π3

7π4

11π6

π6

π4

π3

2π3

3π4

5π6

7π6

5π4

4π3

5π3

7π4

11π6 θ

y

3

2

-3

22

33

-2

233

2

1

-1

π 2ππ2π π6

π4

π3

3π4

2π3

5π6

7π6

5π4

4π3

5π3

7π4

11π6

π6

π4

π3

2π3

3π4

5π6

7π6

5π4

4π3

5π3

7π4

11π6 θ

y

3

-3

1

-1

3π2

π2

3π2

π2

3

3

33

33

π2

3π2

0 π 2ππ2

π3π2

2π π6

π4

π3

3π4

2π3

5π6

7π6

5π4

4π3

5π3

7π4

11π6

π6

π4

π3

2π3

3π4

5π6

7π6

5π4

4π3

5π3

7π4

11π6 θ

y

3

2

-3

22

33

-2

233

2

1

-1

Example 17: a) y = secθ b) P = 2π c) Domain: θ ε R, θ ≠ π/2 + nπ, nεI; Range: y ≤ -1, y ≥ 1d) θ = π/2 + nπ, nεI

Example 18: a) y = cscθ b) P = 2π c) Domain: θ ε R, θ ≠ nπ, nεI; Range: y ≤ -1, y ≥ 1d) θ = nπ, nεI

Example 19: a) y = cotθ b) P = π c) Domain: θ ε R, θ ≠ nπ, nεI; Range: yεRd) θ = nπ, nεI

Example 20:

Domain: θ ε R, θ ≠ π/2 + nπ, nεI;(or: θ ε R, θ ≠ π/2 ± nπ, nεW) Range: y ≤ -1/2, y ≥ 1/2

Domain: θ ε R, θ ≠ n(2π), nεI;(or: θ ε R, θ ≠ ±n(2π), nεW) Range: y ε R

Domain: θ ε R, θ ≠ π/4 + nπ/2, nεI;(or: θ ε R, θ ≠ π/4 ± nπ/2, nεW) Range: y ≤ -1, y ≥ 1

Domain: θ ε R, θ ≠ π/4 + nπ, nεI;(or: θ ε R, θ ≠ π/4 ± nπ, nεW) Range: y ≤ -1, y ≥ 1

0 2π2π θ

y

-3

3

0 2π2π θ

y

-3

3

0 2π2π θ

y

-3

3

a)

0

-3

3

2ππ

b)

0

-3

3

2ππ 0

-3

3

2ππ

d)c)

0

-3

3

2ππ

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Answer KeyTrigonometry Lesson Four: Trigonometric Functions II

Example 1:

-2

2

π 2π 3π θ

y

1

-1

-2

2

180º θ

y

1

-1

360º 540º

a) b)

-2

2

t

h

1

-1

15 30 45 60

-2

2

x

y

1

-1

88 16 24

Example 2:

a) b)

-26

4 8

y

x

-24-22-20-18-16-14-12-10-8-6-4-2

1 2 3 5 6 7

x-225-200 -150

-175 -125 -75 -25-100 -50

025

50 10075 125 175

150 200 250225 275

25

50

75y

Example 3:

a) b)

-8

y

x

-6-4-202468

10121416

120110100908070605040302010

1820

y

x

2

54321-1-2-3

468

101214161820

Example 4:

a) b)

Example 5:

a)

b)

c)

d)

Example 6:

a)

b)

c)

d)

5e)

Example 7:

a)

b)

c)

d)

Example 8:

a) b) The b-parameter is doubled when the period is halved. The a, c, and d parameters remain the same.

c) The d-parameter decreases by 2 units, giving us d = 4. All other parameters remain unchanged.

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Answer Key

t

h(t)

5 10

45

75

105

Example 9:

a)

b) c) If the wind turbine rotates counterclockwise, we still get the same graph.

Example 10:

a)

θ

h(θ)

90°

c) The angle of elevation increases quickly at first, but slows down as the helicopter reaches greater heights. The angle never actually reaches 90°.

b) ,

Example 11:

a)

b) c) 2.86 m

Example 12:

a)

c) 28.14 mb)

t

h(t)

5.2

4.0

2.8

1 2

t

h(t)

0 25 50 75 100 125 150 175 200

1

16

31

d) 0.26 s

Example 13:

b)

d) 15.86 h

n

d(n)

-50 0 50 100 150 200 250 300 350 400

4

8

12

16

20

24

( ) ( )2

5.525cos 11 12.295365

d n nπ = − + +

c)

a) Decimal daylight hours: 6.77 h, 12.28 h, 17.82 h, 12.28 h, 6.77 h

e) 64 days

Example 14:

b)

d) 10.75 m

a) Decimal hours past midnight: 2.20 h, 8.20 h, 14.20 h, 20.20 h

e) 32.3%

t

h(t)

4

8

12

16

0 4 8 12 16 20 24

c)

Population

Time(years)

200

250

300

8000

12000

16000

1 2 3 4 5 60

Mic

eO

wls

7 8

M(t)

O(t)

Example 15:

a) b) See Video.

Example 16: 2.5 m

Example 17: 15.6 s and 18.3 s

t

h(t)

1.5

15

3.0

30 45 60

(8, 2.5)

t

h(t)

10

15

19

30 45 60

(18.3, 13.1)

1

(15.6, 10.5)

d) 26.78 s

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Answer KeyTrigonometry Lesson Five: Trigonometric Equations

Example 1:

b) , c) d)

Note: n ε I for all general solutions.

Example 2:

-1

1

2ππ

-1

1

2ππ

-1

1

2ππ

Example 3:

360°

0°

90°

180°

270°

30°30°

30°150°

360°

0°

90°

180°

270°

30°

150°

30°

210°

360°

0°

90°

180°

270°

45°

45°

45°

225°

Intersection point(s)of original equation

-3

3

-2

-1

2

1

2ππ

θ-intercepts

-3

3

-2

-1

2

1

2ππ

-3

3

-2

-1

2

1

2ππ

Intersection point(s)of original equation

-3

3

-2

-1

2

1

2ππ

θ-intercepts

Example 4:

b)

,a)

a) b) c)

d) no solution e) f)

-2

2

2ππ

1

-1

3

-3

-2

2

2ππ

1

-1

3

-3-2

2

1

-1

2ππ

a) b) c)

a)

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60°

60°

120°

240°

-3

3

-2

-1

2

1

2ππ

-3

3

-2

-1

2

1

2ππ

Example 6:

a) 197.46° and 342.54° b) 197.46° and 342.54° c) 197.46° and 342.54° d) 197.46° and 342.54°

17.46°

197.46°

17.46°

342.54°

-3

3

-2

-1

2

1

2ππ

-3

3

-2

-1

2

1

2ππ

The unit circle is not useful for this question.

Example 7:

Example 8:

a) No Solution b) c)

d) e) f)

-2

2

1

-1

2ππ

-2

2

1

-1

2ππ

-2

2

2ππ

1

-1

-2

2

2ππ

1

-1

-2

2

2ππ

1

-1

-2

2

2ππ

1

-1

a) b) c) d)

a) b) c)

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Answer Key

Intersection point(s)of original equation θ-intercepts

Intersection point(s)of original equation θ-intercepts

Example 10:

b)a) No Solution

-3

3

-2

-1

2

1

2ππ

-3

3

-2

-1

2

1

2ππ

-3

3

-2

-1

2

1

2ππ

-3

3

-2

-1

2

1

2ππ

Example 11:

a) b) c) d)

30°

210°

30°

330°

-3

3

-2

-1

2

1

2ππ

-3

3

-2

-1

2

1

2ππ

Example 12:

a) b) c) d)

115°

245°

65°65°

The unit circle is not useful for this question.

-3

3

-2

-1

2

1

2ππ

-3

3

-2

-1

2

1

2ππ

Example 13:

a) b) d)c)

360°

0°

90°

180°

270°

60°60°

60°120°

360°

0°

90°

180°

270°

60°

60°

240°

60°

Example 9:

a) b) c)

360°

0°

90°

180°

270°

60°

60°

120°

240°

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-1

1

2ππ

-1

1

2ππ

-1

1

2ππ

-1

1

2ππ

Example 15:

a) b) c) d)

-1

1

2ππ

-1

1

2ππ

-3

3

-2

-1

2

1

2ππ

-1

1

2ππ

Example 16:

a) b) c)

-3

3

-2

-1

2

1

2ππ

-3

3

-2

-1

2

1

2ππ

-3

3

-2

-1

2

1

2ππ

Example 17:

a) b) a) b)

Example 18:

Example 19:

1.00

28147 210

0.50

t

Visible %

a)

b) Approximately 12 days.

Example 20: a)

b) See graph. c) 0.4636 rad (or 26.6°)

-8

8

2πππ2

3π2

4

-4

θ

d

0.4636

2

Example 21: See Video

a) b) c) d)

-1

1

2ππ

-1

1

π2

π 2π3π2

-1

1

4π2ππ 3π

-1

1

8π4π2π 6π

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Answer KeyTrigonometry Lesson Six: Trigonometric Identities I

Example 1:


Example 2:

Example 3:

a) b)

Example 4:

a) b) -1

1

2ππ

-1

1

2ππ

-1

1

2ππ

-1

1

2ππ

Example 5:

,a) b)

c) d)

Identity

-3

3

-2

-1

2

1

2ππ

Equation

-3

3

-2

-1

2

1

2ππ

-1

1

2ππ

-3

3

2ππ

-2

-1

2

1

-3

3

2ππ

-2

-1

2

1

-3

3

-2

-1

2

1

2ππ

-3

3

-2

-1

2

1

2ππ

a)

a)

b) i) ii) iii) iv) v)

Not an Identity Not an Identity Identity Identity

-1

1

2ππx

yθ

1Use basic trigonometry(SOHCAHTOA) to showthat x = cosθ and y = sinθ.

b) Verify that the L.S. = R.S. for each angle.

c) The graphs of y = sin2x + cos2x and y = 1 are the same.

d) Divide both sides ofsin2x + cos2x = 1 bysin2x to get 1 + cot2x = csc2x.Divide both sides ofsin2x + cos2x = 1 bycos2x to get tan2x + 1 = sec2x.

e) Verify that the L.S. = R.S. for each angle.

f) The graphs of y = 1 + cot2x and y = csc2x are the same.

-3

3

2ππ

-2

-1

2

1

-3

3

2ππ

-2

-1

2

1

The graphs of y = tan2x + 1 and y = sec2x are the same.

-3

3

2ππ

-2

-1

2

1

-1

1

2ππ

-3

3

2ππ

-2

-1

2

1

-3

3

2ππ

-2

-1

2

1

Not an Identity

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a) b)

c) d)

-1

1

2ππ

-1

1

2ππ

Example 7:

a) b)

c) d)

-3

3

2ππ

-2

-1

2

1

-3

3

2ππ

-2

-1

2

1

-3

3

2ππ

-2

-1

2

1

-3

3

2ππ

-2

-1

2

1

0

2

2ππ

1

-1

1

2ππ

Example 8:

a) b)

c) d)

-3

3

2ππ

-2

-1

2

1

-3

3

2ππ

-2

-1

2

1

-3

3

2ππ

-2

-1

2

1

-3

3

2ππ

-2

-1

2

1

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Answer KeyExample 9: See Video Example 10: See Video Example 11: See Video

Example 12:

a) See Video

b)

c)

d)

Example 13:

a) See Video

b)

c)

d)

Example 14:

a) See Video

b)

c)

d)

-1

1

2ππ

-3

3

2ππ

-2

-1

2

1

-3

3

2ππ

-2

-1

2

1

Example 15:

Example 16:

a) b)

c) d)

, ,

, ,

-3

3

2ππ

-2

-1

2

1

-3

3

2ππ

-2

-1

2

1

-6

6

2ππ

-4

-2

4

2

-1

1

2ππ

a) b)

c) d)

, ,

, ,

-3

3

2ππ

-2

-1

2

1

-10

10

2ππ

-1

1

2ππ

-2

2

2ππ

Note: All terms from the originalequation were collected on theleft side before graphing.

The graphs are identical.

The graphs are NOT identical.The R.S. has holes.

The graphs are identical.

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a) b)

c) d)

, ,

, ,

-10

10

2ππ

-3

3

2ππ

-2

-1

2

1

1

2ππ

-1

-12

-9

-6

3

02ππ

-3

Note: All terms from the original equation were collected on the left side before graphing.

Note: All terms from the original equation were collected on the left side before graphing.

Example 18:

74

-3

-2

7

a) b) c)

Example 19: See Video

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Answer KeyTrigonometry Lesson Seven: Trigonometric Identities II

Example 1:



a) b) c) d) See Video

Example 4: Example 5: See Video

Example 6:

a) i. ii. 0 iii. undefined

b) (answers may vary)

ii.

iii.

iv.

i.

c) (answers may vary)

ii.

iii.

iv.

i.

Examples 7 - 13: Proofs. See Video.

Example 14:

a)

b)

c)

d)

Example 15:

a)

b)

c)

d)

Example 16:

a)

b)

c)

d)

Example 17:

a)

b)

c)

d)

Example 18: 57°

Example 19: 92.9

Example 20:

a)

d

θ

132.2

-132.2

90° 180° 270° 360°

b)

c) θ = 24.6° and θ = 65.4°

Example 21:

a)

b)

c) i. ii. iii.

A

θ

4900

90°45°

At 0°, the cannonball hits the ground as soon as it leaves the cannon, so the horizontal distance is 0 m.

At 45°, the cannonball hits the ground at the maximum horizontal distance, 132.2 m.

At 90°, the cannonball goes straight up and down, landing on the cannon at a horizontal distance of 0 m

The maximum area occurs when θ = 45°. At this angle, the rectangle is the top half of a square.

0

-6

6

2ππ

y = f(θ) + g(θ)

ii. The waves experience constructive interference. iii. The new sound will be louder than either original sound.

0

-6

6

2ππ

y = f(θ) + g(θ)

ii. The waves experience destructive interference.iii. The new sound will be quieter than either original sound.

All of the terms subtract out leaving y = 0, A flat line indicating no wave activity.

Example 22:

a) i. b) i.

c)

Example 23: See Video. Example 24: See Video.

a) b) c) d) e) f)

a) b) c)

a) b) c)

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Answer KeyPermutations and Combinations Lesson One: Permutations

Example 1:a) Six words can be formed. b) 3 × 2 × 1 = 6 c) 3P3 d) See Video e) 3P1 + 3P2 + 3P3Example 2: a) 24 b) 1 c) 1 d) (-2)! Does not exist.e) 20 f) 4 g) n2 – n h) n2 + nExample 3: a) 120 b) 24 c) 720 d) 13!Example 4: a) 3 b) 415800 c) 12600 d) 20 e) 60 f) 10Example 5: a) 8 b) 16 c) 1296 d) 32 × 106 e) 676 000Example 6: a) 120 b) 48 c) 480 d) 108Example 7: a) 24 b) 18 c) 12 d) 18 e) 3 f) 6Example 8: a) 103 680 b) 240 c) 120 d) 144Example 9: a) 72 b) 420 c) 1440Example 10: a) 156 b) 1440 c) 20 d) 144 e) 72Example 11: a) 24 b) 1320 c) 5P2 d) 3P2 or 3P3Example 12: a) n = 6 b) n = 2 c) n = 5 d) n = 1Example 13: a) n = 8 b) r = 3 c) n = 2 d) n = 5

Permutations and Combinations Lesson Two: Combinations

Example 1:a) The order of the colors is not important.b) 6 c) 4C2 d) See Video e) 4C3 + 4C4Example 2: a) 10 b) 126 c) 2598960 d) 36; 84Example 3: a) 13860 b) 720 c) 580008 d) 60Example 4: a) 330 b) 70 c) 1680 d) 19600 e) 13244Example 5: a) 3600 b) 180 c) 75600Example 6: a) 66 b) 84 c) 70 d) 9Example 7: a) 15 b) 20Example 8: a) 81 b) 2594400 c) 2533180 d) 405 e) 31Example 9: a) 21 b) 1 c) 6 d) 6C2 e) 5C1Example 10: a) n = 7 b) 4C2 c) n = 5 d) n = 6Example 11: a) n = 4 b) All n-values c) n = 4 d) n = 4Example 12: a) 6760000 b) 40 c) 1645020 d) 144e) See Video f) 84 g) 504 h) 32Example 13: a) 16!/(2!)8 b) 10 c) 1800 d) 56e) 120 f) 5040 g) 241098 h) 15Example 14: a) 10080 b) 5 c) 8 d) 9e) 92 f) 360 g) 241920 h) 6600Example 15: a) 210 b) 720 c) 5148 d) 131560e) 1024 f) 5 g) 14400 h) 123552Example 16: a) 20 b) See Video c) 100 d) 1152e) n = 8 f) 56 g) See Video h) 36

Permutations and Combinations Lesson Three: The Binomial Theorem

Example 1:a) The eighth row of Pascal's Triangle is: 1, 7, 21, 35, 35, 21, 7, 1.b) See Video. Note that rows and term positions use a zero-based index.c) There is symmetry in each row. For example, the second position of the sixth row is equal to the second-last position of the same row.

Example 2:a) 8C0; 12C10b) 21C2 = 210c) k = 3 and 8, so the fourth and ninth positions have a value of 165.d) 1024

Example 3:a) 20b) 120c) 66d) 54

Example 4:a) The binomial theorem states that a binomial power of the form (x + y)n can be expanded into a series of terms with the form nCkx

n-kyk, where n is the exponent of the binomial (and also the zero-based row of Pascal's Triangle), and k is the zero-based term position.

b)

c)

Example 5:

a)

b)

c)

Example 6:

a)

b)

c)

Example 7:

a)

b)

c)

Example 8:

a)

b)

c)

d)

3

Example 9:

a)

b)

c)

Example 10:

a)

b)

c)

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Math 30-1: Workbook€¦ · Trigonometry One Trigonometry Two Permutations and ... A box with no lid can be made by cutting out squares from each corner of a rectangular piece of