Mathematics 30-1
Topics Covered
Polynomial, Radical, and Rational Functions
Transformations and Operations
Exponential and Logarithmic Functions
Trigonometry One
Trigonometry Two
Permutations and Combinations
A workbook and animated series by Barry Mabillard
Copyright © 2014 | www.math30.ca
This page has been left blank for correct workbook printing.
Mathematics 30-1
Formula Sheet
Trigonometry I
The Unit Circle
Trigonometry II
Note: The unit circle is
NOT included on the
official formula sheet.
Transformations
& Operations
Exponential and
Permutations &
Logarithmic Functions Combinations
Polynomial, Radical
& Rational Functions
Curriculum Alignment
Math 30-1: Alberta | Northwest Territories | Nunavut
Pre-Calculus 12: British Columbia | Yukon
Pre-Calculus 30: Saskatchewan
Pre-Calculus 40S: Manitoba
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Mathematics 30-1
Table of Contents
Unit 1: Polynomial, Radical, and Rational Functions
Lesson
Lesson
Lesson
Lesson
Lesson
Lesson
1:
2:
3:
4:
5:
6:
Polynomial Functions
Polynomial Division
Polynomial Factoring
Radical Functions
Rational Functions I
Rational Functions II
Unit 2: Transformations and Operations
Lesson
Lesson
Lesson
Lesson
Lesson
1:
2:
3:
4:
5:
Basic Transformations
Combined Transformations
Inverses
Function Operations
Function Composition
Unit 3: Exponential and Logarithmic Functions
Lesson 1: Exponential Functions
Lesson 2: Laws of Logarithms
Lesson 3: Logarithmic Functions
Unit 4: Trigonometry I
Lesson
Lesson
Lesson
Lesson
7:45 (16 days)
1:38
1:29
1:13
0:52
1:00
1:33
(3
(3
(3
(2
(2
(3
days)
days)
days)
days)
days)
days)
4:38 (11 days)
0:57
0:50
0:42
0:48
1:21
(2
(2
(2
(2
(3
days)
days)
days)
days)
days)
5:55 (12 days)
1:52 (4 days)
2:11 (4 days)
1:52 (4 days)
9:59 (17 days)
1: Degrees and Radians
2: The Unit Circle
3: Trigonometric Functions I
4: Trigonometric Functions II
Unit 5: Trigonometry II
2:22
2:15
2:24
1:58
(4
(4
(5
(4
days)
days)
days)
days)
7:05 (12 days)
Lesson 5: Trigonometric Equations
Lesson 6: Trigonometric Identities I
Lesson 7: Trigonometric Identities II
2:12 (4 days)
2:34 (4 days)
2:19 (4 days)
Unit 6: Permutations and Combinations
4:57 (10 days)
Lesson 1: Permutations
Lesson 2: Combinations
Lesson 3: The Binomial Theorem
Total Course
2:00 (4 days)
1:56 (4 days)
1:01 (2 days)
40:19 (78 days)
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Polynomial, Radical, and Rational Functions
1
2
LESSON ONE - Polynomial Functions
3
Lesson Notes
Example 1: Introduction to Polynomial Functions.
a) Given the general form of a polynomial function, P(x) = anxn + an-1xn-1 + an-2xn-2 + ... + a1x1 + a0,
the leading coefficient is ______, the degree of the polynomial is ______, and the constant term of the polynomial is ______.
For each polynomial function given below, state the leading coefficient, degree, and constant term.
i) f(x) = 3x - 2 ii) y = x3 + 2x2 - x - 1 iii) P(x) = 5
b) Determine which expressions are polynomials. Explain your reasoning.
1
i) x5 + 3 ii) 5x + 3 iii) 3 iv) 4x2 - 5x - 1 v) x2 + x - 4 vi) |x| vii) 5 x - 1 viii)
3
1
2
Example 2: End Behaviour of Polynomial Functions.
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?
i
ii
For the graphs in parts (b - e), determine the
zeros and state each zero’s multiplicity.
quadratic
quadratic
quadratic
quadratic
vi
vii
viii
quartic
quartic
quartic
Odd-Degree Polynomials
i
ii
iv
linear
cubic
vii
cubic
quartic
Example 2b
iii
vi
cubic
d) P(x) = (x - 1)3
iv
v
v
a) Define “multiplicity of a zero”.
Example 2a
iii
linear
Example 4: Multiplicity of zeros in a polynomial function.
c) P(x) = (x - 3)2
1
x+3
Even-Degree Polynomials
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.
b) P(x) = -(x + 3)(x - 1)
7 x + 2 ix)
quintic
cubic
viii
quintic
e) P(x) = (x + 1)2(x - 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:
a) P(x) = 1 (x - 5)(x + 3)
2
a) P(x) = (x - 1)2(x + 2)2
a) P(x) = -(2x - 1)(2x + 1)
b) P(x) = -x2(x + 1)
b) P(x) = x(x + 1)3(x - 2)2
b) P(x) = x(4x - 3)(3x + 2)
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Polynomial, Radical, and Rational Functions
LESSON ONE - Polynomial Functions
1
Lesson Notes
2
3
Examples 8 - 10: Determine the polynomial function corresponding to each graph. You may leave your answer in factored form.
Example 8:
Example 9:
a)
a)
b)
b)
(0, 4)
(0, -1)
(2, -6)
4,
Example 10:
a)
15
2
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)
b)
(-6, 0)
(0, -6)
(0, -9)
a) P(x) = x2 - 2x - 168
b) P(x) = x3 + 7x2 - 44x
c) P(x) = x3 - 16x2 - 144x + 1152
Example 12: Given the characteristics of a polynomial function, draw the graph and derive the function.
a) Characteristics of P(x):
b) Characteristics of P(x):
x-intercepts: (-1, 0) and (3, 0); sign of leading coefficient: (+); x-intercepts: (-3, 0), (1, 0), and (4, 0); sign of leading coefficient: (-);
polynomial degree: 3; y-intercept at: (0, -3/2)
polynomial degree: 4; relative maximum at (1, 8)
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.
x
a) Derive a polynomial function that represents the volume of the box.
x
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.
16 cm
d) What should be the side length of a corner square if the volume of the box is maximized?
3
e) For what values of x is the volume of the box greater than 200 cm ?
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?
20 cm
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.
r
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.
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4
h
1
3
3
-4
3
-5
-7
2
2
-7
2
0
Polynomial, Radical, and Rational Functions
LESSON TWO - Polynomial Division
Lesson Notes
Example 1: Divide using long division and answer the related questions.
a) x + 2 x3 + 2x2 - 5x - 6
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 theorem
by checking that the left side and right side are equivalent.
d) Another way to represent the division theorem is
R
.
D(x)
P(x)
D(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
R
P(x)
= Q(x) +
.
D(x)
D(x)
Example 2: Divide using long division. Express answers in the form
a) (3x3 - 4x2 + 2x - 1) ÷ (x + 1)
b)
x3 - 3x - 2
x-2
P(x)
R
= Q(x) +
.
D(x)
D(x)
c) (x3 - 1) ÷ (x + 2)
Example 3: Divide using synthetic division. Express answers in the form
a) (3x3 - x - 3) ÷ (x - 1)
b)
3x4 + 5x3 + 3x - 2
x+2
P(x)
R
= Q(x) +
.
D(x)
D(x)
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.
a)
x2 - 5x + 6
x-3
b) (6x - 4) ÷ (3x - 2)
c) (x4 - 16) ÷ (x2 + 4)
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.
d)
x3 + 2x2 - 3x
x-3
V = x3 + 6x2 - 7x - 60
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.
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x+4
1
-
Polynomial, Radical, and Rational Functions
LESSON TWO - Polynomial Division
Lesson Notes
3
-4
3
-5
-7
2
2
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).
Example 8: If f(x) ÷ g(x) = 4x2 + 4x - 3 -
(3, 8)
f(x)
g(x)
6
, determine f(x) and g(x).
x-1
Example 9: The Remainder Theorem
a) 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 + 1
e) Define the remainder theorem.
Example 10: The Factor Theorem
a) 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.
x4 - 2x2 + 3x - 4
c) (3x3 + 8x2 - 1) ÷ (3x - 1)
x+2
Example 12: Use the remainder theorem to find the value of k in each polynomial.
a) (x3 - 1) ÷ (x + 1)
b)
a) (kx3 - x - 3) ÷ (x - 1) Remainder = -1
c) (2x3 + 3x2 + kx - 3) ÷ (2x + 5)
b)
3x3 - 6x2 + 2x + k
x-2
Remainder = 2
d)
d)
2x4 + 3x3 - 4x - 9
2x + 3
Remainder = -3
2x3 + kx2 - x + 6
2x - 3
(2x - 3 is a factor)
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 15
P(x) = x3 + kx2 + 5
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.
Example 15: Given the graph of P(x) = x3 + kx2 + 5
and the point (2, -3), determine the value of a on
the graph.
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(4, a)
(2, -3)
x3 - 5x2 + 2x + 8
Polynomial, Radical, and Rational Functions
LESSON THREE - Polynomial Factoring
Lesson Notes
(x + 1)(x - 2)(x - 4)
Example 1: The Integral Zero Theorem
a) 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.
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 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.
Example 2: P(x) = x3 + 3x2 - x - 3.
Example 3: P(x) = 2x3 - 6x2 + x - 3
Example 4: P(x) = x3 - 3x + 2
a) P(x) has zeros of -4, 0, 0, and 1.
The graph passes through the point (-1, -3).
Example 5: P(x) = x3 - 8
Example 6: P(x) = x3 - 2x2 - x - 6
Example 7: P(x) = x4 - 16
Example 8: P(x) = x5 - 3x4 - 5x3 + 27x2 - 32x + 12
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.
Example 14: Solve each equation
algebraically and check with a
graphing calculator.
a) x3 - 3x2 - 10x + 24 = 0
b) 3x3 + 8x2 + 4x - 1 = 0
Example 13
Quadratic Formula
(0, 0)
From Math 20-1:
The roots of a quadratic
equation with the form
ax2 + bx + c = 0 can be
found with the quadratic
formula:
(b, 0)
(a, 0)
(2, 0)
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Polynomial, Radical, and Rational Functions
LESSON THREE - Polynomial Factoring
Lesson Notes
x3 - 5x2 + 2x + 8
(x + 1)(x - 2)(x - 4)
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Polynomial, Radical, and Rational Functions
y= x
LESSON FOUR - Radical Functions
Lesson Notes
Example 1: Introduction to Radical Functions
x
-1
a) Fill in the table of values for the function f(x) = x .
b) Draw f(x) =
f(x)
0
x and state the domain and range.
1
4
Example 2: Graph each function.
b) f(x) =
a) f(x) = - x
9
-x
Example 3: Graph each function.
1
b) f(x) = 2 x
a) f(x) = 2 x
c) f(x) =
2x
d) f(x) =
1x
2
Example 4: Graph each function.
a) f(x) =
x -5
b) f(x) =
x +2
c) f(x) =
x-1
d) f(x) =
x+7
Example 5: Graph each function.
a) f(x) =
x-3+2
b) f(x) = 2 x + 4
c) f(x) = - x - 3
d) f(x) =
-2x - 4
Examples 6 - 8: Graph y = f(x) and state the domain and range.
Example 6: a) y = x + 4
b) y = -(x + 2)2 + 9
Example 7: a) y = (x - 5)2 - 4
Example 8: a) y = -(x + 5)2
b) y = x2
b) y = x2 + 0.25
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.
Example 9:
x+2 =3
Example 10: x =
x+2
Set-Builder Notation
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.
{x | x ε R, x ≥ -1}
-1
Example 11: 2 x + 3 = x + 3
0
1
State the
variable.
List conditions
on the variable.
In words: “The variable is x, such that x can be
any real number with the condition that x ≥ -1”.
As a shortcut, set-builder notation can be reduced
to just the most important condition.
Example 12:
16 - x2 = 5
-1
0
1
x ≥ -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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Interval Notation
Math 30-1 students are expected to
know that domain and range can be
expressed using interval notation.
() - Round Brackets: Exclude point
from interval.
[] - Square Brackets: Include point
in interval.
Infinity ∞ always gets a round bracket.
Examples: x ≥ -5 becomes [-5, ∞);
1 < x ≤ 4 becomes (1, 4];
x ε R becomes (-∞ , ∞);
-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].
Polynomial, Radical, and Rational Functions
y= x
LESSON FOUR - Radical Functions
Lesson Notes
Example 13: Write an equation that can be used to find the
point of intersection for each pair of graphs.
a)
b)
c)
d)
(2, 3)
(2, 2)
(8, -1)
(1, 3)
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 height
of the ladder as a function of its base distance d.
b) Graph the function and state the domain and range.
Describe the ladder’s orientation when d = 0 and d = 3.
c) How far is the base of the ladder from the wall when
the top of the ladder is 5 metres above the ground?
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=
h
4.9
where t is the time in seconds.
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 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.
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.
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r
5 cm
h
Cone Volume
Polynomial, Radical, and Rational Functions
1
y=
x
LESSON FIVE - Rational Functions I
Lesson Notes
Example 1: Reciprocal of a Linear Function.
x
1
.
x
1
b) Draw the graph of the function y =
. State the domain and range.
x
c) Draw the graph of y = x in the same grid used for part (b).
1
Compare the graph of y = x to the graph of y =
.
x
d) Outline a series of steps that can be used to draw the
1
graph of y = , starting from y = x.
x
y
-2
a) Fill in the table of values for the function y =
-1
-0.5
-0.25
0
0.25
0.5
1
2
1
.
f(x)
iii. Asymptote Equation(s)
Example 2: Reciprocal of a Linear Function. Given the graph of y = f(x), draw the graph of y =
a) y = x - 5
1
b) y = - x + 2
2
i. Domain & Range of y = f(x)
ii. Domain & Range of y =
i. Domain & Range of y = f(x)
1
f(x)
ii. Domain & Range of y =
Example 3: Reciprocal of a Quadratic Function.
x
1
a) Fill in the table of values for the function y = 2
.
x -4
1
b) Draw the graph of the function y = 2
.
x -4
State the domain and range.
x
-3
iii. Asymptote Equation(s)
y
-2.05
-2
-1.95
-1
0
x
1
c) Draw the graph of y = x - 4 in the same grid used for part (b).
1
.
Compare the graph of y = x2 - 4 to the graph of y = 2
x -4
2
y
1
f(x)
y
1.95
2
2.05
3
d) Outline a series of steps that can be used to draw the
1
graph of y = 2
, starting from y = x2 - 4.
x -4
Example 4: Reciprocal of a Quadratic Function. Given the graph of y = f(x), draw the graph of y =
a)
b)
i. Domain & Range of y = f(x)
c)
d)
ii. Domain & Range of y =
e)
1
f(x)
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f)
1
f(x)
iii. Asymptote Equation(s)
.
Polynomial, Radical, and Rational Functions
1
y=
x
LESSON FIVE - Rational Functions I
Lesson Notes
Example 5: Given the graph of y =
a)
1
f(x)
, draw the graph of y = f(x).
b)
c)
d)
Example 6: For each function, determine the equations of all asymptotes. Check with a graphing calculator.
a) f(x) =
1
2x - 3
b) f(x) =
1
x2 - 2x - 24
c) f(x) =
1
6x3 - 5x2 - 4x
d) f(x) =
1
4x2 + 9
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 =
4
x
b) y =
1
-3
x
c) y =
3
x+4
d) y =
Example 8: Convert each of the following functions to the form
2
+2
x-3
.
Identify the stretches and translations, then draw the graph without using technology.
c) y =
6 - 2x
x-1
d) y =
33 - 6x
x-5
))
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?
Example 10: The illuminance of light can be described with the reciprocal-square relation
))
)
x-1
x-2
))
b) y =
)
1 - 2x
x
))
a) y =
V
P
(L)
(kPa)
P
50
45
40
0.5
35
30
1.0
25
2.0
20
5.0
10
15
5
10.0
1
2
3
4
5
6
7
8
9 10 V
,
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-square
nature of light illuminance, a screen can be moved from a baseline position to various distances from the bulb.
I
130
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?
120
d
I
(m)
(W/m2)
110
100
90
1
80
2
60
4
70
50
40
8
30
12
10
20
1
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2
3
4
5
6
7
8
9 10 11 12
d
y=
x2 + x - 2
Polynomial, Radical, and Rational Functions
LESSON SIX - Rational Functions II
x+2
Lesson Notes
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 =
x
2
x -9
b) y =
x+2
x2 + 1
c) y =
x+4
x2 - 16
d) y =
x2 - x - 2
x3 - x2 - 2x
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 =
4x
x-2
b) y =
x2
x2 - 1
c) y =
3x2
x2 + 9
d) y =
3x2 - 3x - 18
x2 - x - 6
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.
2
x2 + 5
x2 - x - 6
y
=
y
=
b) y = x - 4x + 3
d)
c)
x-1
x+1
x-3
x
Example 4: Graph y = 2
without using the graphing feature of your calculator.
x - 16
a) y =
x2 + 5x + 4
x+4
i. Horizontal Asymptote
ii. Vertical Asymptote(s)
ii. Vertical Asymptote(s)
Example 6: Graph y =
i. Horizontal Asymptote
iv. x - intercept(s)
v. Domain and Range
2x - 6
without using the graphing feature of your calculator.
x+2
Example 5: Graph y =
i. Horizontal Asymptote
iii. y - intercept
iii. y - intercept
iv. x - intercept(s)
v. Domain and Range
x2 + 2x - 8
without using the graphing feature of your calculator.
x-1
ii. Vertical Asymptote(s) iii. y - intercept iv. x - intercept(s) v. Domain and Range
Example 7: Graph y =
vi. Oblique Asymptote
x2 - 5x + 6
without using the graphing feature of your calculator.
x-2
i. Can this rational function be simplified?
ii. Holes
iii. y - intercept
iv. x - intercept(s)
v. Domain and Range
Example 8: Find the rational function with each set of characteristics and draw the graph.
a)
b)
vertical asymptote(s)
x = -2, x = 4
horizontal asymptote
y=1
x-intercept(s)
(-3, 0) and (5, 0)
hole(s)
none
vertical asymptote(s)
x=0
horizontal asymptote
y=0
x-intercept(s)
none
hole(s)
(-1, -1)
Example 9: Find the rational function shown in each graph.
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Polynomial, Radical, and Rational Functions
LESSON SIX - Rational Functions II
y=
Lesson Notes
Example 10: Solve the rational equation
a) Solve algebraically and
check for extraneous roots.
x2 + x - 2
x+2
3x
= 4 in three different ways.
x-1
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.
Example 11: Solve the rational equation 6 - 9
= -6 in three different ways.
x
x-1
a) Solve algebraically and
check for extraneous roots.
b) Solve the equation by finding the point
of intersection of a system of functions.
Example 12: Solve the equation
a) Solve algebraically and
check for extraneous roots.
c) Solve the equation by finding the x-intercept(s)
of a single function.
x
4
6
= 2
in three different ways.
x -x-2
x-2
x+1
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.
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 equation
t
s
d
that can be used to solve this problem.
Cynthia
b) Solve algebraically.
Alan
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 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?
t
s
a) Fill in the table and derive an equation
d
that can be used to solve this problem.
Upstream
b) Solve algebraically.
Downstream
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 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.
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Transformations and Operations
LESSON ONE - Basic Transformations
Lesson Notes
Example 1: Draw the graph resulting from each transformation. Label the invariant points.
a) y = 2f(x)
b) y =
1
f(x)
2
c) y = f(2x)
d) y = f(
1
x)
2
Example 2: Draw the graph resulting from each transformation. Label the invariant points.
a) y =
1
f(x)
4
c) y = f(
b) y = 3f(x)
1
x)
5
d) y = f(3x)
Example 3: Draw the graph resulting from each transformation. Label the invariant points.
a) y = -f(x)
b) y = f(-x)
c) x = f(y)
Example 4: Draw the graph resulting from each transformation. Label the invariant points.
a) y = -f(x)
b) y = f(-x)
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c) x = f(y)
Transformations and Operations
LESSON ONE - Basic Transformations
Lesson Notes
Example 5: Draw the graph resulting from each transformation.
a) y = f(x) + 3
b) y = f(x) - 4
c) y = f(x - 2)
d) y = f(x + 3)
Example 6: Draw the graph resulting from each transformation.
a) y - 4 = f(x)
b) y = f(x) - 3
c) y = f(x - 5)
d) y = f(x + 4)
Example 7: Draw the transformed graph. Write the transformation as both an equation and a mapping.
a) The graph of f(x) is
horizontally stretched
1
by a factor of
.
2
b) The graph of f(x) is
horizontally translated
6 units left.
c) The graph of f(x) is
vertically translated
4 units down.
d) The graph of f(x) is
reflected in the x-axis.
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Transformations and Operations
LESSON ONE - Basic Transformations
Lesson Notes
Example 8: Write a sentence describing each transformation,
then write the transformation equation.
a)
b)
c)
Original graph:
Transformed graph:
d)
Example 9: Describe each transformation and derive the equation of the transformed graph.
Draw the original and transformed graphs.
a)
Original graph: f(x) = x2 - 1
Transformation: y = 2f(x)
b)
Original graph: f(x) = x2 + 1
Transformation: y = f(2x)
c)
Original graph: f(x) = x2 - 2
Transformation: y = -f(x)
d)
Original graph: f(x) = (x - 6)2
Transformation: y = f(-x)
Example 10: Describe each transformation and derive the equation of the transformed graph.
Draw the original and transformed graphs.
a)
Original graph: f(x) = x2
Transformation: y - 2 = f(x)
b)
Original graph: f(x) = x2 - 4
Transformation: y = f(x) - 4
c)
Original graph: f(x) = x2
Transformation: y = f(x - 2)
d)
Original graph: f(x) = (x + 3)2
Transformation: y = f(x - 7)
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 Operations
LESSON ONE - Basic Transformations
Lesson 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.
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?
Example 14: A basketball player throws a basketball.
1
(d - 4)2 + 4 .
9
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.
The path can be modeled with h(d) = -
1
b) If the player moves so the equation of the shot is h(d) = - (d + 1)2 + 4, what is the horizontal
9
distance from the player to the hoop?
h(d)
5
4
3
2
1
-5
-4
-3
-2
-1
0
1
2
3
4
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5
6
7
8
9
d
y = af[b(x - h)] + k
Transformations and Operations
LESSON TWO - Combined Transformations
Lesson Notes
Example 1: Combined Transformations
a) Identify each parameter in the general transformation equation: y = af[b(x - h)] + k.
b) Describe the transformations in each equation:
1 1
1
1
ii. y = 2f( x)
iii. y = - f( x)
iv. y = -3f(-2x)
i. y = f(5x)
4
2 3
3
Example 2: Draw the transformation of each graph.
1
1
a) y = 2f( x)
b) y = f(-x)
c) y = -f(2x)
3
3
d) y = -
1
f(-x)
2
Example 3: Answer the following questions:
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 4: Draw the transformation of each graph.
a) y = f(x + 5) - 3
b) y = f(x - 3) + 7
c) y - 12 = f(x - 6)
d) y + 2 = f(x + 8)
Example 5: Answer the following questions:
a) When applying transformations to a graph, should they be applied in a specific order?
b) Describe the transformations in each equation:
1
1
iii. y = f[-(x + 2)] - 3
ii. y = -f( x) - 4
i. y = 2f(x + 3) + 1
2
3
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iv. y = -3f[-4(x - 1)] + 2
Transformations and Operations
LESSON TWO - Combined Transformations
Lesson Notes
y = af[b(x - h)] + k
Example 6: Draw the transformation of each graph.
1
1
a) y = -f(x) - 2
b) y = f(- x) + 1
c) y = - f(2x) - 1
4
4
d) 2y - 8 = 6f(x - 2)
Example 7: Draw the transformation of each graph.
1
b) y = f(2x + 6)
a) y = f[ (x - 1)] + 1
c) y = f(3x - 6) - 2
3
d) y =
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).
1
f(-x - 4)
3
i
i
b) If the point (5, 4) exists on the graph of y = f(x), find the coordinates
1
of the new point after the transformation y = f(5x - 10) + 4.
2
c) The point (m, n) exists on the graph of y = f(x). If the transformation y = 2f(2x) + 5 is applied to
the graph, the transformed point is (4, 7). Find the values of m and n.
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.
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Example 9a
Example 9b
Transformations and Operations
y = af[b(x - h)] + k
LESSON TWO - Combined Transformations
Lesson Notes
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.
Original graph:
Greg’s Transformation Order:
Stretches & Reflections:
1) Vertical stretch by a scale factor of 2
2) Reflection about the x-axis
3) Horizontal stretch by a scale factor of 1/2
4) Reflection about the y-axis
Transformed graph:
Translations:
5) Vertical translation 3 units down
6) Horizontal translation 4 units left
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.
Colin’s Transformation Order:
Original graph:
Vertical Transformations:
1) Vertical stretch by a scale factor of 2
2) Reflection about the x-axis
3) Vertical translation 3 units down.
Transformed graph:
Horizontal Transformations:
4) Horizontal stretch by a scale factor of 1/2
5) Reflection about the y-axis
6) Horizontal translation 4 units left
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?
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Transformations and Operations
LESSON TWO - Combined Transformations
Lesson Notes
y = af[b(x - h)] + k
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.
b) Describe a transformation
that will let the spaceship
pass through the asteroids.
Original position of ship
Final position of ship
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.
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f-1(x)
Transformations and Operations
LESSON THREE - Inverses
Lesson Notes
Example 1: Inverse Functions.
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 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)?
a)
b)
c)
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)
d)
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.
a)
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.
1
x-4
a) f(x) = x - 3
b) f(x) = 2
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b)
Transformations and Operations
LESSON THREE - Inverses
f-1(x)
Lesson Notes
Example 7: Answer the following questions.
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 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.
The formula to convert degrees Celsius
9
to degrees Fahrenheit is: F(C) =
C + 32
5
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?
0°
32°
Celsius
Thermometer
Fahrenheit
Thermometer
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?
°F
100
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?
50
-100
-50
50
-50
-100
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100 °C
Transformations and Operations
(f - g)(x)
f
(x)
g
(f + g)(x)
(f • g)(x)
LESSON FOUR - Function Operations
Lesson Notes
Example 1: Given the functions f(x) and g(x), draw the graph.
State the domain and range of the combined function.
a) h(x) = (f + g)(x)
b) h(x) = (f - g)(x)
d) h(x) =
c) h(x) = (f • g)(x)
f(x)
f(x)
f(x)
f(x)
g(x)
g(x)
g(x)
g(x)
Example 2: Given the functions f(x) = x - 3 and g(x) = -x + 1, evaluate:
a) (f + g)(-4)
c) (fg)(-1)
b) (f - g)(6)
d)
f
(5)
g
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)
g(x)
f(x)
f(x)
g(x)
m(x)
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).
Example 4: Given the functions f(x) = 2 x + 4 + 1
and g(x) = -1, answer the following questions.
Example 5: Given the functions f(x) = -(x - 2)2 - 4 and
g(x) = 2, answer the following questions.
a) (f + g)(x)
a) (f - g)(x)
b) (f • g)(x)
f(x)
f(x)
g(x)
g(x)
b)
g(x)
g(x)
f(x)
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f(x)
Transformations and Operations
(f + g)(x)
Lesson Notes
(f • g)(x)
LESSON FOUR - Function Operations
Example 6: Draw the graph of h(x) =
(f - g)(x)
f
(x)
g
. Derive h(x) and state the domain and range.
i. Graph. ii. Derive h(x) = (f ÷ g)(x) iii. State the domain & range of h(x).
b) f(x) = 1 and g(x) = x - 2
a) f(x) = 1 and g(x) = x
c) f(x) = x + 3 and
g(x) = x2 + 6x + 9
g(x)
g(x)
g(x)
d) f(x) = x + 3
and g(x) = x + 2
g(x)
f(x)
f(x)
f(x)
f(x)
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.
4x
2x - 2
x
3x - 3
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?
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 height
of the apex, and s is the slant height of the side of the cone.
A particular cone has a height that is 3 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.
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slant
height
h
r
SA = πr2 + πrs
1
V = πr2h
3
Transformations and Operations
f ◦ g = f(g(x))
LESSON FIVE - Function Composition
Lesson Notes
Example 1a
Example 1: Given the functions f(x) = x - 3 and g(x) = x2:
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
f(g(x))
x
-3
0
-2
1
-1
2
0
3
f(x)
g(f(x))
1
2
Example 2: Given the functions f(x) = x - 3 and
g(x) = 2x, evaluate each of the following:
2
a) m(3) = (f ◦ g)(3)
c) p(2) = (f ◦ f)(2)
g(x)
Example 1b
3
b) n(1) = (g ◦ f)(1)
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)
f(x) = (x + 1)2
g(x) = 3x
b) n(x) = (g ◦ f)(x)
Example 5: Given the functions f(x) and g(x), find the composite function m(x) = (f ◦ g)(x).
b)
a)
Examples 6 & 7: Given the functions f(x), g(x), m(x), and n(x), find each composite function.
f(x) =
x
g(x) =
1
x
m(x) = |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)
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n(x) = x + 2
Transformations and Operations
LESSON FIVE - Function Composition
Lesson 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) =
d) h(x) = x2 + 4x + 4
1
x -1
c) h(x) = (x + 1)2 - 5(x + 1) + 1
2
e) h(x) = 2
1
x
f) h(x) = |x|
Example 9: Two functions are inverses if (f-1 ◦ f)(x) = x. Determine if each pair of functions
are inverses of each other.
a) f(x) = 3x - 2 and f-1(x) =
1
2
x+
3
3
b) f(x) = x - 1 and f-1(x) = 1 - x
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?
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.
American Dollars = 1.03 × Canadian Dollars
b) Write a function, j(a), that converts American Dollars to Japanese Yen.
Euros = 0.77 × American Dollars
c) Write a function, b(a), that converts American Dollars to British Pounds.
Japanese Yen = 101.36 × Euros
British Pounds = 0.0083 × Japanese Yen
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.
8 cm
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.
$CAD
$USD
€
¥
£
3 cm
r
h
Vcone =
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1
πr2h
3
Exponential and Logarithmic Functions
y=b
x
LESSON ONE - Exponential Functions
Lesson Notes
Example 1: For each exponential function:
Interval Notation
Math 30-1 students are expected to
know that domain and range can be
expressed using interval notation.
i. Create a table of values and draw the graph.
ii. State the domain, range, intercepts, and the
equation of the asymptote.
a)
 1
c) y =  
2
b)
() - Round Brackets: Exclude point
from interval.
[] - Square Brackets: Include point
in interval.
Infinity ∞ always gets a round bracket.
x
d)
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.
a)
b)
Examples: x ≥ -5 becomes [-5, ∞);
1 < x ≤ 4 becomes (1, 4];
x ε R becomes (-∞ , ∞);
-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].
c)
d)
10
100
10
(-3, 125)
100
50
(3, 64)
5
5
(-3, n)
50
(2, 16)
(-2, n)
(3, n)
(-2, 25)
(1, n)
Example 3: Draw the graph. State the domain, range, and equation of the asymptote.
a)
d)
c)
b)
Example 4: Draw the graph. State the domain, range, and equation of the asymptote.
a)
b)
c)
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d)
Exponential and Logarithmic Functions
y = bx
LESSON ONE- Exponential Functions
Lesson Notes
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.
a)
b)
10
10
(-5, n)
5
5
(-5, n)
-5
(0, -2)
-5
5
5
Example 6: Assorted Questions.
a) What is the y-intercept of f(x) = abx - 4 ?
b) The point
c) If the graph of
exists on the graph of y = a(5)x. What is the value of a?
is stretched vertically so it passes through the point
,
what is the equation of the transformed graph?
d) If the graph of y = 2x is vertically translated so it passes through the point (3, 5), what
is 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.
iv.
ii.
v.
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iii.
Exponential and Logarithmic Functions
y=b
x
LESSON ONE - Exponential Functions
Lesson Notes
Example 7: Solving equations where x is in the base. (Raising Reciprocals)
a)
c)
b)
d)
Example 8: Solving equations where x is in the exponent. (Common Base)
a)
c)
b)
e) Determine x and y:
d)
f) Determine m and n:
and
and
Example 9: Solving equations where x is in the exponent. (Fractional Base)
c)
b)
a)
d)
Example 10: Solving equations where x is in the exponent. (Fractional Exponents)
a)
c)
b)
d)
Example 11: Solving equations where x is in the exponent. (Multiple Powers)
d)
c)
b)
a)
Example 12: Solving equations where x is in the exponent. (Radicals)
a)
c)
b)
d)
Example 13: Solving equations where x is in the exponent. (Factoring)
c)
b)
a)
d)
Example 14: Solving equations where x is in the exponent. (No Common Base - Use Technology)
a)
b)
c)
d)
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Exponential and Logarithmic Functions
LESSON ONE- Exponential Functions
Lesson Notes
y = bx
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?
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 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: $500 is placed in a savings account that compounds interest
annually at a rate of 2.5%.
a) Write a function, A(t), that relates the amount of the
investment, A, with the elapsed time t.
b) How much will the investment be worth in 5 years?
How much interest has been received?
c) Draw the graph for the first 20 years.
d) How long does it take for the investment to double?
e) Calculate the amount of the investment in 5 years if compounding occurs
i. semi-annually, ii. monthly, and iii. daily. Summarize your results in a table.
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$
$ $
Exponential and Logarithmic Functions
logBA = E
LESSON TWO - Laws of Logarithms
Lesson Notes
Example 1: Introduction to Logarithms.
a) Label the components of logBA = E and A = BE.
c) Which logarithm is bigger?
b) Evaluate each logarithm.
i. log21 or log42
i. log21 =
ii. log 1 =
, log22 =
, log 10 =
, log24 =
, log28 =
, log 100 =
ii.
or
, 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)
b)
e)
f)
c)
d)
h)
g)
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)
e)
c)
b)
d)
g)
f)
h)
Example 5: Evaluate each logarithm using change of base. (Change of Base)
a)
e)
b)
f)
c)
g)
d)
h)
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Exponential and Logarithmic Functions
logBA = E
LESSON TWO - Laws of Logarithms
Lesson Notes
Example 6: Expand each logarithm using the product law. (Product Law)
b)
a)
d)
c)
In parts (e - h), condense each expression to a single logarithm.
h)
g)
f)
e)
Example 7: Expand each logarithm using the quotient law. (Quotient Law)
b)
a)
c)
d)
In parts (e - h), condense each expression to a single logarithm.
e)
f)
g)
h)
Example 8: Expand each logarithm using the power law. (Power Law)
b)
a)
d)
c)
In parts (e - h), condense each expression to a single logarithm.
e)
g)
f)
h)
Example 9: Expand each logarithm using the appropriate logarithm rule. (Other Rules)
a)
b)
c)
g)
f)
e)
d)
h)
Example 10: Use logarithm laws to answer each of the following questions. (Substitution Questions)
a) If 10k = 4, then 101 + 2k =
b) If 3a = k, then log3k4 =
f) If logh4 = 2 and log8k = 2,
then log2(hk) =
c) If logb4 = k, then logb16 =
g) Write logx + 1 as a single logarithm.
d) If log2a = h, then log4a =
h) Write 3 + log2x as a single logarithm.
e) If logbh = 3 and logbk = 4,
then
=
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Exponential and Logarithmic Functions
logBA = E
LESSON TWO - Laws of Logarithms
Lesson Notes
Example 11: Solving Exponential Equations. (No Common Base)
a)
c)
b)
d)
Example 12: Solving Exponential Equations. (No Common Base)
a)
c)
b)
d)
Example 13: Solving Logarithmic Equations. (One Solution)
a)
b)
c)
d)
Example 14: Solving Logarithmic Equations. (Multiple Solutions)
a)
b)
c)
d)
Example 15: Solving Logarithmic Equations. (Multiple Solutions)
a)
b)
c)
d)
Example 16: Assorted Mix I
a) Evaluate.
c) Solve.
b) Condense.
e) Write as a logarithm.
f) Show that:
d) Evaluate.
g) If loga3 = x and loga4 = 12,
then loga122 =
(express answer in
terms of x.)
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h) Condense.
Exponential and Logarithmic Functions
logBA = E
LESSON TWO - Laws of Logarithms
Lesson Notes
Example 17: Assorted Mix II
a) Evaluate.
e) Evaluate.
b) Evaluate.
f) Condense.
d) Solve.
c) What is onethird of 3234 ?
g) Solve.
h) If xy = 8, then
5log2x + 5log2y =
Example 18: Assorted Mix III
a) Evaluate.
e) Condense.
b) Solve.
c) Condense.
f) Evaluate.
g) Show that:
d) Solve.
h) Condense.
Example 19: Assorted Mix IV
b) Condense.
a) Solve.
e) Evaluate.
f) Solve.
c) Solve.
d) Condense.
g) Evaluate.
h) Condense.
Example 20: Assorted Mix V
a) Solve.
d) Condense.
b) Solve.
e) Solve.
f) Solve.
c) Evaluate.
g) Condense.
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h) Solve.
Exponential and Logarithmic Functions
y = logbx
LESSON THREE - Logarithmic Functions
Lesson Notes
Example 1: Logarithmic Functions
10
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
y = log2x
y = 2x
Domain
Range
5
x-intercept
y-intercept
Asymptote
Equation
-5
5
f) Are y = log1x, y = log0x, and y = log-2x
logarithmic functions? What about
?
10
g) Define logarithmic function.
h) How can y = log2x be graphed in a calculator?
-5
Examples 2 - 6: Draw each of the following graphs without technology.
State the domain, range, and asymptote equation.
Example 2: a)
b)
Example 3: a)
c)
b)
Example 6: a)
d)
c)
b)
Example 5: a)
d)
c)
b)
Example 4: a)
d)
c)
d)
c)
b)
d)
Example 7: Exponential Equations. Solve each equation by (i) finding a common base (if possible),
(ii) using logarithms, and (iii) graphing.
a)
b)
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 Functions
LESSON THREE- Logarithmic Functions
Lesson Notes
y = logbx
Example 9: Assorted Mix I
a) The graph of y = logbx passes through the point (8, 2). What is the value of b?
b) What are the x- and y-intercepts of y = log2(x + 4)?
c) What is the equation of the asymptote for y = log3(3x – 8)?
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?
e) What is the domain of f(x) = logx(6 – x)?
Example 10: Assorted Mix II
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?
b) If the point (4, 1) exists on the graph of y = log4x, what is the point after
the transformation y = log4(2x + 6)?
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?
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 11: Assorted Mix III
a) What is the equation of the reflection line for the graphs of f(x) = bx and
?
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?
c) What is the inverse of f(x) = 3x + 4?
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?
e) The point (k, 3) exists on the inverse of y = 2x. What is the value of k?
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y = logbx
Exponential and Logarithmic Functions
LESSON THREE - Logarithmic Functions
Lesson Notes
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).
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:
which compares two amplitudes directly without requiring A0.
Derive this formula.
e) What is the ratio of seismograph amplitudes for earthquakes with magnitudes of 5.0 and 6.0?
f) Show that an equivalent form of the equation is:
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?
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 Functions
y = logbx
LESSON THREE- Logarithmic Functions
Lesson Notes
e) How many times more intense is 40 dB than 20 dB?
f) Show that an equivalent form of the equation is:
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?
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.
and
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?
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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♯
♯
♯
♯
♯
π
7π
=
210°×
180°
6
Trigonometry
LESSON 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.
b) Positive and negative angles. Draw θ = 120° and θ = -120°.
c) Reference angle. Find the reference angle of θ = 150°.
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 four
positive and negative co-terminal angles of θ = 45°.
Conversion Multiplier Reference Chart (Example 2)
degree
radian
revolution
degree
radian
revolution
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
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.
90° =
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).
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
= 120°
60° =
= 135°
45° =
= 150°
30° =
0° =
360° =
= 180°
= 210°
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
330° =
= 225°
= 240°
315° =
300° =
= 270°
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Trigonometry
LESSON ONE - Degrees and Radians
210° ×
Lesson Notes
π
7π
=
180° 6
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)
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:
r
θ
y
x
sinθ =
y
r
cscθ =
1
r
=
sinθ
y
cosθ =
x
r
secθ =
1
r
=
cosθ
x
tanθ =
y
x
cotθ =
1
x
=
tanθ
y
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.
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.
a)
b)
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Trigonometry
π
7π
=
210° ×
180° 6
LESSON ONE - Degrees and Radians
Lesson Notes
Example 15: Calculating θ with a calculator.
If the angle θ could exist in
either quadrant ___ or ___ ...
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.
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?
The calculator always
picks quadrant
I or II
I or III
I or IV
II or III
II or IV
III or IV
Mark’s Calculation
of θ (using sine)
sinθ =
Jordan’s Calculation
of θ (using cosine)
3
5
cosθ =
θ = 36.87°
Dylan’s Calculation
of θ (using tan)
-4
5
tanθ =
θ = 143.13°
3
-4
θ = -36.87°
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.
b)
c)
d)
1.23π cm
5π cm
e)
6 cm
3 cm
r
θ
5 cm
π
2
6 cm
153°
n
a
Example 17: Area of a circle sector.
r2θ
a) Derive the formula for the area of a circle sector, A =
. (b - e) Find the area of each shaded region.
2
b)
c)
d)
9 cm
4 cm
3 cm
7π
6
e)
240°
120°
60°
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2π
3
6 cm 3 cm
Trigonometry
LESSON ONE - Degrees and Radians
Lesson Notes
Example 18: The formula for angular speed is
210° ×
π
7π
=
180° 6
, where ω (Greek: Omega)
∆θ
is 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.
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.
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.
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340 km
6370 km
Trigonometry
LESSON TWO - The Unit Circle
(cosθ, sinθ)
Lesson Notes
Example 1: Introduction to Circle Equations.
10
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
-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)
-10
-10
10
-10
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?
1
-1
-1
i.
ii.
, quadrant II.
iii. (-1, y)
iv.
, cosθ > 0.
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.
a) What are some useful tips to memorize the unit circle?
b) Draw the unit circle from memory.
Example 3: Use the unit circle to find the exact
value of each expression.
3π
a) sin π b) cos 180° c) cos
4
6
e) sin 0
f) cos
π
2
g) sin
4π
3
d) sin
11π
6
h) cos -120°
Example 4: Use the unit circle to find the exact
value of each expression.
e) sin
5π
2
f) -sin
9π
4
13π
6
d) cos
2π
3
g) cos2 (-840°) h) cos
7π
3
a) cos 420° b) -cos 3π c) sin
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Trigonometry
LESSON TWO - The Unit Circle
(cosθ, sinθ)
Lesson Notes
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.
Example 6: Find the exact values of tanθ and cotθ
in the first quadrant.
secθ =
1
cosθ
cscθ =
1
sinθ
tanθ =
sinθ
cosθ
cotθ =
1
cosθ
=
tanθ sinθ
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
3π
2
c) csc
π
3
d) csc
3π
4
e) tan
π
6
f) -tan
5π
4
g) cot2(270°)
h) cot
Example 9: Find the exact value of each expression.
a)
c)
b)
d)
Example 10: Find the exact value of each expression.
a)
c)
b)
d)
Example 11: Find the exact value of each expression.
a)
d)
c)
b)
Example 12: Verify each trigonometric statement with a calculator.
Note: Every question in this example has already been seen earlier in the lesson.
a)
e)
b)
c)
f)
d)
g)
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h)
2
5π
6
Trigonometry
LESSON TWO - The Unit Circle
(cosθ, sinθ)
Lesson Notes
Example 13: Coordinate Relationships on the Unit Circle
a) What is meant when you are asked to find
on the unit circle?
b) Find one positive and one negative angle such that P(θ) =
c) How does a half-rotation around the unit circle change the coordinates?
π
If θ = , find the coordinates of the point halfway around the unit circle.
6
d) How does a quarter-rotation around the unit circle change the coordinates?
2π
If θ =
, find the coordinates of the point a quarter-revolution (clockwise) around the unit circle.
3
e) What are the coordinates of P(3)? Express coordinates to four decimal places.
Example 14: Circumference and Arc Length of the Unit Circle
a) What is the circumference of the unit circle?
b) How is the central angle of the unit circle related to
its corresponding arc length?
θ
A
Diagram for
Example 14 (d).
c) If a point on the terminal arm rotates from P(θ) = (1, 0)
to P(θ) =
, what is the arc length?
θ
B
d) What is the arc length from point A to point B
on the unit circle?
θ
Example 15: Domain and Range of the Unit Circle
a) Is sinθ = 2 possible? Explain, using the unit circle as a reference.
b) Which trigonometric ratios are restricted to a range of -1 ≤ y ≤ 1?
Which trigonometric ratios exist outside that range?
c) If
exists on the unit circle, how can the unit circle
be used to find cosθ? How many values for cosθ are possible?
Chart for Example 15 (b).
Range
cosθ & sinθ
cscθ & secθ
tanθ & cotθ
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?
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Number Line
Trigonometry
LESSON TWO - The Unit Circle
(cosθ, sinθ)
Lesson Notes
Example 16: Unit Circle Proofs
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 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:
x’ = x cos θ - y sin θ
y’ = x sin θ + y cos θ
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.
a) If a particular pixel with coordinates of (250, 100) is rotated by π , what are the new
6
coordinates? Round coordinates to the nearest whole pixel.
5π
b) If a particular pixel has the coordinates (640, 480) after a rotation of
, what were the
4
original coordinates? Round coordinates to the nearest whole pixel.
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 observation
x
deck is h =
.
cotθA - cotθB
θA
θB
b) If θA =
, θB =
, and x = 212.92 m,
how high is the observation deck above the
ground, to the nearest metre?
h
B
A
x
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Trigonometry
y = asinb(θ - c) + d
LESSON THREE - Trigonometric Functions I
Lesson Notes
Example 1: Label all tick marks in the following grids and state the coordinates of each point.
a)
b)
y
y
20
5
2π
0
π
π
2π
2π
θ
0
π
-5
π
2π
θ
4π
θ
-20
c)
d)
y
y
12
40
0
8π
8π
0
4π
θ
-40
-12
Example 2: Exploring the graph of y = sinθ.
a) Draw y = sinθ. b) State the amplitude. c) State the period.
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.
Example 3: Exploring the graph of y = cosθ.
a) Draw y = cosθ. b) State the amplitude. c) State the period.
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.
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.
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.
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Trigonometry
y = asinb(θ - c) + d
LESSON THREE - Trigonometric Functions I
Lesson Notes
Example 5: The a Parameter. Graph each function over the domain 0 ≤ θ ≤ 2π.
a) y = 3sinθ
b) y = -2cosθ
c) y =
1
sinθ
2
d) y =
5
cosθ
2
Example 6: The a Parameter. Determine the trigonometric function corresponding to each graph.
a) write a sine function.
b) write a sine function.
8
c) write a cosine function.
28
d) write a cosine function.
1
5
(π, 41 )
0
2π
-8
0
0
2π
-28
0
2π
-1
2π
-5
Example 7: The d Parameter. Graph each function over the domain 0 ≤ θ ≤ 2π.
a) y = sinθ - 2
b) y = cosθ + 4
c) y = -
1
sinθ + 2
2
d) y =
1
1
cosθ 2
2
Example 8: The d Parameter. Determine the trigonometric function corresponding to each graph.
a) write a sine function.
b) write a cosine function.
4
c) write a cosine function.
35
0
2π
-4
d) write a sine function.
32
0
0
2π
-35
4
2π
-32
-4
Example 9: The b Parameter. Graph each function over the stated domain.
a) y = cos2θ
c) y = cos
1
θ
3
(0 ≤ θ ≤ 2π)
b) y = sin3θ
(0 ≤ θ ≤ 6π)
d) y = sin
1
θ
5
(0 ≤ θ ≤ 2π)
(0 ≤ θ ≤ 10π)
Example 10: The b Parameter. Graph each function over the stated domain.
a) y = -sin(3θ)
c) y = 2cos
(-2π ≤ θ ≤ 2π)
1
θ-1
2
(-2π ≤ θ ≤ 2π)
b) y = 4cos2θ + 6
d) y = sin
4
θ
3
0
(-2π ≤ θ ≤ 2π)
(0 ≤ θ ≤ 6π)
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2π
Trigonometry
y = asinb(θ - c) + d
LESSON THREE - Trigonometric Functions I
Lesson Notes
Example 11: The b Parameter. Determine the trigonometric function corresponding to each graph.
a) write a cosine function.
b) write a sine function.
2
c) write a cosine function.
0
π
2π
-2
d) write a sine function.
4
2
0
3π
6π
9π
1
0
6π
12π
-4
-2
0
π
2π
-1
Example 12: The c Parameter. Graph each function over the stated domain.
a)
(-4π ≤ θ ≤ 4π)
b)
c)
(-2π ≤ θ ≤ 2π)
d)
(-4π ≤ θ ≤ 4π)
(-2π ≤ θ ≤ 2π)
Example 13: The c Parameter. Graph each function over the stated domain.
π
2
a)
c)
2π
b)
(-π ≤ θ ≤ 4π)
d)
θ
(-2π ≤ θ ≤ 6π)
(-2π ≤ θ ≤ 2π)
Example 14: The c Parameter. Determine the trigonometric function corresponding to each graph.
a) write a cosine function.
b) write a sine function.
π
2π
-π
π
2π
0
4
2π
4π
-8π
-4π
-6
-1
-1
d) write a cosine function.
6
1
1
π
2
c) write a sine function.
4π
-4
Example 15: a, b, c, & d Parameters. Graph each function over the stated domain.
a)
c)
(0 ≤ θ ≤ 6π)
-3
(0 ≤ θ ≤ 2π)
b)
d)
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(0 ≤ θ ≤ 2π)
(0 ≤ θ ≤ 2π)
8π
Trigonometry
y = asinb(θ - c) + d
LESSON THREE - Trigonometric Functions I
Lesson Notes
Example 16: a, b, c, & d. Determine the trigonometric function corresponding to each graph.
a) write a cosine function.
b) write a cosine function.
2
-2π
12
-π
π
2π
3π
4π
-2π
-2
-π
π
2π
-12
y
Example 17: Exploring the graph of y = secθ.
3
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 =
-2π
2π θ
1
.
f(θ)
-3
y
Example 18: Exploring the graph of y = cscθ.
3
a) Draw y = cscθ. b) State the period. c) State the domain and range.
d) Write the general equation of the asymptotes.
-2π
2π θ
1
.
f(θ)
e) Given the graph of f(θ) = sinθ, draw y =
-3
y
Example 19: Exploring the graph of y = cotθ.
3
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 =
-2π
2π θ
1
.
f(θ)
-3
Example 20: Graph each function over the domain 0 ≤ θ ≤ 2π. State the new domain and range.
a)
3
0
-3
c)
b)
3
π
2π
y = secθ
0
-3
d)
3
π
2π
y = secθ
0
-3
3
π
2π
y = cscθ
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0
-3
π
2π
y = cotθ
Trigonometry
h(t)
LESSON FOUR - Trigonometric Functions II
Lesson Notes
t
Example 1: Trigonometric Functions of Angles
(0 ≤ θ < 3π)
a) i. Graph:
(0º ≤ θ < 540º)
b) i. Graph:
ii. Graph this function using technology.
ii. Graph this function using technology.
Example 2: Trigonometric Functions of Real Numbers.
a) i. Graph:
b) i. Graph:
ii. Graph this function using technology.
ii. Graph this function using technology.
c) What are three differences between trigonometric functions of angles and trigonometric
functions of real numbers?
Example 3: Determine the view window for each function and sketch each graph.
a)
b)
Example 4: Determine the view window for each function and sketch each graph.
b)
a)
Example 5: Determine the trigonometric function corresponding to each graph.
a) write a cosine function.
b) write a sine function.
10
c) write a cosine function.
10
5
d) write a sine function.
300
(8, 9)
(1425, 150)
0
8
16
-4
8
16
0
25
0
(16, -3)
-10
-5
2400
(300, -50)
-10
-300
Example 6: a) If the transformation g(θ) - 3 = f(2θ) is applied to the graph of f(θ) = sinθ, find the new range.
b) Find the range of
4
.
c) If the range of y = 3cosθ + d is [-4, k],
determine the values of d and k.
e) The graphs of f(θ) and g(θ) intersect at the points
and
.
If the amplitude of each graph is quadrupled, determine the new points of intersection.
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d) State the range of
f(θ) - 2 = msin(2θ) + n.
Trigonometry
h(t)
LESSON FOUR - Trigonometric Functions II
Lesson Notes
t
Example 7: a) If the point
lies on the graph of
b) Find the y-intercept of
, find the value of a.
Graph for Example 7c
.
(m, n)
g(θ)
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.
n
f(θ)
m
Graph for Example 7d
k
f(θ)
g(θ)
b
If the point
exists on the graph
of g(θ), state the vertical stretch factor.
π
2
π
3π
2
2π
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.
a) 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)?
h(t)
Graph for Example 8
12 cm
8 cm
4 cm
ground level
0 cm
1s
2s
3s
4s
A
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?
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.
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h
θ
150 m
t
Trigonometry
h(t)
LESSON FOUR - Trigonometric Functions II
Lesson Notes
t
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 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 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.
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 13: The following table shows the number of daylight hours in Grande Prairie.
December 21
6h, 46m
March 21
12h, 17m
June 21
17h, 49m
September 21 December 21
12h, 17m
6h, 46m
a) Convert each date and time to a number that can be used for graphing.
Day Number December 21 =
March 21 =
June 21 =
September 21 =
December 21 =
Daylight Hours 6h, 46m =
12h, 17m =
17h, 49m =
12h, 17m =
12h, 46m =
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.
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Trigonometry
h(t)
LESSON FOUR - Trigonometric Functions II
Lesson Notes
t
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.
Time
Decimal Hour
Height of Water (m)
a) Convert each time to a decimal hour.
High Tide 8:12 AM
13.32
b) Graph the height of the tide for one
Low Tide 2:12 PM
3.48
full cycle (low tide to low tide).
13.32
High Tide 8:12 PM
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.
Low Tide
Bay of
Fundy
Bay of
Fundy
3.48
2:12 AM
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 will
use 6 hours for simplicity.
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.
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 16: The angle of elevation between the 6:00 position and the 12:00 position
π .
of a historical building’s clock, as measured from an observer standing on a hill, is
444
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?
Example 17: Shane is on a Ferris wheel, and his height
can be described by the equation
.
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?
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π
444
424
m
Trigonometry
LESSON FIVE - Trigonometric Equations
Lesson Notes
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.
a)
b)
c)
d) tan2θ = 1
0
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
f) tanθ = undefined
e) tanθ
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)
c)
b)
Example 4: Primary Ratios. Find all angles in the domain 0 ≤ θ ≤ 2π that satisfy the given equation.
Solve equations graphically with θ-intercepts.
a) sinθ = 1
b) cosθ =
-
Example 5: Primary Ratios. Solve
0 ≤ θ ≤ 2π
b) non-graphically,
using the unit circle.
a) non-graphically, using the
cos-1 feature of a calculator.
Example 6: Primary Ratios. Solve sinθ = -0.30
a) non-graphically, using the
cos-1 feature of a calculator.
c) graphically, using
point(s) of intersection.
d) graphically,
using θ-intercepts.
c) graphically, using
point(s) of intersection.
d) graphically,
using θ-intercepts.
θεR
b) non-graphically,
using the unit circle.
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)
b)
c)
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.
a)
θ
b)
c)
θ
θ
d) secθ = -1 e)
θ
f)
θ
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)
c)
b)
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.
a)
θ
b)
θ
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Trigonometry
LESSON FIVE - Trigonometric Equations
Lesson Notes
Example 11: Reciprocal Ratios. Solve cscθ = -2
0 ≤ θ ≤ 2π
b) non-graphically,
using the unit circle.
a) non-graphically, using the
cos-1 feature of a calculator.
c) graphically, using
point(s) of intersection.
Example 12: Reciprocal Ratios. Solve secθ = -2.3662
a) non-graphically, using the
cos-1 feature of a calculator.
b) non-graphically,
using the unit circle.
d) graphically,
using θ-intercepts.
0°≤ θ ≤ 360°
c) graphically, using
point(s) of intersection.
d) graphically,
using θ-intercepts.
Example 13: First-Degree Trigonometric Equations. Find all angles in the domain 0 ≤ θ ≤ 2π that
satisfy the given equation. Write the general solution.
a) cosθ - 1 = 0
b)
c) 3tanθ - 5 = 0
θ
d) 4secθ + 3 = 3secθ + 1
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 = 4sinθ
a) 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
b) 4cos2θ - 3 = 0
c) 2cos2θ = cosθ
d) tan4θ - tan2θ = 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)
0 ≤ θ ≤ 2π
θ
Example 18: Half and Quarter Angles. Solve each equation (i) graphically, and (ii) non-graphically.
a)
θ
0 ≤ θ ≤ 4π
b)
θ
-1
0 ≤ θ ≤ 8π
Example 19: It takes the moon approximately 28 days to go through all of its phases.
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?
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)?
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d
θ
P
Trigonometry
LESSON SIX - Trigonometric Identities I
Lesson Notes
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.
Example 2: The Pythagorean Identities.
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.
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.
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.
a)
b)
d)
c)
Example 6: Pythagorean Identities. Prove that each trigonometric statement is an identity.
State the non-permissible values of x so the identity is true.
b)
a)
c)
d)
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Trigonometry
LESSON SIX- Trigonometric Identities I
Lesson 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.
a)
b)
c)
d)
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.
a)
b)
c)
d)
Example 9: Assorted Proofs. Prove each identity. For simplicity, ignore NPV’s and graphs.
a)
b)
c)
d)
Example 10: Assorted Proofs. Prove each identity. For simplicity, ignore NPV’s and graphs.
a)
b)
c)
d)
Example 11: Assorted Proofs. Prove each identity. For simplicity, ignore NPV’s and graphs.
a)
b)
c)
d)
Example 12: Exploring the proof of
a) Prove algebraically that
b) Verify that
.
for π .
3
c) State the non-permissible values for
d) Show graphically that
.
. Are the graphs exactly the same?
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Trigonometry
LESSON SIX - Trigonometric Identities I
Lesson Notes
Example 13: Exploring the proof of
.
a) Prove algebraically that
.
for π .
3
b) Verify that
c) State the non-permissible values for
.
d) Show graphically that
. Are the graphs exactly the same?
Example 14: Exploring the proof of
a) Prove algebraically that
b) Verify that
for
π
.
2
c) State the the non-permissible values for
.
. Are the graphs exactly the same?
d) Show graphically that
Example 15: Equations with Identites. Solve each trigonometric equation over the domain 0 ≤ x ≤ 2π.
a)
c)
b)
d)
Example 16: Equations with Identites. Solve each trigonometric equation over the domain 0 ≤ x ≤ 2π.
a)
b)
c)
d)
Example 17: Equations with Identites. Solve each trigonometric equation over the domain 0 ≤ x ≤ 2π.
a)
b)
c)
d)
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Trigonometry
LESSON SIX- Trigonometric Identities I
Lesson Notes
Example 18: Use the Pythagorean identities to find the indicated value and
draw the corresponding triangle.
a) If the value of
find the value of cosx within the same domain.
b) If the value of
c) If cosθ =
, find the value of secA within the same domain.
7
, and cotθ < 0, find the exact value of sinθ.
7
Example 19: Trigonometric Substitution.
a) Using the triangle to the right, show that
can be expressed as
.
3
b
θ
a
Hint: Use the triangle to find a trigonometric expression equivalent to b.
b) Using the triangle to the right, show that
can be expressed as
4
.
Hint: Use the triangle to find a trigonometric expression equivalent to a.
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θ
a
Trigonometry
LESSON SEVEN - Trigonometric Identities II
Lesson Notes
Example 1: Evaluate each trigonometric sum or difference.
c)
b)
a)
e)
d)
f)
Example 2: Write each expression as a single trigonometric ratio.
a)
c)
b)
Example 3: Find the exact value of each expression.
a)
b)
d) Given the exact values of cosine and sine for 15°,
fill in the blanks for the other angles.
c)
Example 4: Find the exact value
of each expression.
a)
b)
Example 3d
c)
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,
.
Example 6: Double-angle identities.
a) Evaluate each of the following expressions using a double-angle identity.
i.
ii.
iii.
b) Express each of the following expressions using a double-angle identity.
i.
ii.
iii.
iv.
c) Write each of the following expression as a single trigonometric ratio using a double-angle identity.
i.
ii.
iii.
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iv.
Trigonometry
LESSON SEVEN- Trigonometric Identities II
Lesson Notes
Note: Variable restrictions may be
ignored for the proofs in this lesson.
Example 7: Prove each trigonometric identity.
a)
b)
c)
d)
Example 8: Prove each trigonometric identity.
a)
b)
c)
d)
Example 9: Prove each trigonometric identity.
a)
b)
c)
d)
Example 10: Prove each trigonometric identity.
a)
b)
c)
d)
Example 11: Prove each trigonometric identity.
a)
b)
d)
c)
Example 12: Prove each trigonometric identity.
b)
a)
c)
d)
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Trigonometry
LESSON SEVEN - Trigonometric Identities II
Lesson Notes
Example 13: Prove each trigonometric identity.
a)
b)
d)
c)
Example 14: Solve each trigonometric equation over the domain 0 ≤ x ≤ 2π.
a)
b)
d)
c)
Example 15: Solve each trigonometric equation over the domain 0 ≤ x ≤ 2π.
a)
b)
Diagram for
Example 18
d)
c)
A
Example 16: Solve each trigonometric equation over the domain 0 ≤ x ≤ 2π.
a)
b)
c)
d)
B
Example 17: Solve each trigonometric equation over the domain 0 ≤ x ≤ 2π.
a)
c)
b)
d)
Example 18: Trigonometric identities and geometry.
a) Show that
C
b) If A = 32° and B = 89°,
what is the value of C?
Diagram for
Example 19
x
57
176
Example 19: Trigonometric identities and geometry.
104
Solve for x. Round your answer to the nearest tenth.
B
A
153
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Trigonometry
LESSON SEVEN- Trigonometric Identities II
Lesson Notes
Example 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.
θ
a) Rewrite the function so it involves a single trigonometric identity.
d (θ ) =
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.
v i 2 sinθ cos θ
4.9
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.
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
θ
the road are included in the design.
70 m
a) If the area of the rectangle can be represented by the function
sidewalk
road width
sidewalk
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.
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.
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.
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?
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?
6
6
g(θ) = 4cosθ
f(θ) = 2cosθ
0
-6
π
2π
m(θ) = 2cosθ
0
π
2π
n(θ) = 2cos(θ - π)
-6
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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.
n!
P
=
n r
(n - r)!
Permutations and Combinations
LESSON ONE - Permutations
Lesson Notes
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.
5!
8!
a) 4!
b) 1!
c) 0!
d) (-2)!
e)
f)
3!
7!•2!
g)
n!
(n - 2)!
h)
A
B
C
(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
Round 1
designer, and database analyst. How many unique teams are possible? You can
Rock
assume that each person in the selection pool is capable of performing each task.
Metal
Punk
d) There are 13 letter tiles in a bag, and no letter is repeated. Using all of the letters
Alternative
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?
One Possible Timetable
Block
Block 1
Block 2
Block 3
Block 4
Block 5
Course
Math 30-1
Spare
Physics 30
English 30-1
Biology 30
Round 2
Round 3
Pop
Dance
Country
Blues
Folk
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?
B
A
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 Combinations
n!
P =
n r
(n - r)!
LESSON ONE - Permutations
Lesson Notes
Example 6: Constraints and Line Formations
Example 7: Constraints and Words
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 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) 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?
Example 11: Permutation Formula
b) Evaluate 12P3
a) Evaluate 4P3
c) Write
5!
3!
as a permutation.
d) Write 3! as a permutation.
Example 12: Equations with Factorials and Permutations.
a)
n!
(n - 2)!
= 5n
b) (n + 2)! = 12n!
c)
n!
10
=
P
n-1 n-3
d)
(2n + 1)!
(2n - 1)!
Example 13: Equations with Factorials and Permutations.
a) nP2 = 56
b) 6Pr = 120
c)
P = 20
n+3 2
d)
P = 2•n - 4P1
n-3 1
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= 4n + 2
P =
n r
n!
(n - r)!
n!
C
=
n r
(n - r)!r!
Permutations and Combinations
LESSON TWO - Combinations
Lesson Notes
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.
5♣
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?
6♥
A
2♦
4♥
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?
Flower Type
Examples
c) From a deck of 52 cards, a 6-card hand is dealt.
How many distinct hands are there if the hand must
Focal Flowers: Large and eye-catching flowers
Roses, Peonies, Hydrangeas,
that draw attention to one area of the bouquet.
Chrysanthemums, Tulips, and Lilies
contain 2 spades and 3 diamonds?
Fragrant Flowers: Flowers that add a
Petunia, Daffodils, Daphnes,
d) A bouquet contains four types of flowers. A florist
pleasant fragrance to the bouquet.
Gardenia, Lilacs, Violets, Magnolias
is making a bouquet that uses one type of focal flower,
Line Flowers: Tall and narrow flowers used to
Delphiniums, Snapdragons,
no fragrant flowers, three types of line flowers and
establish the height of a floral bouquet.
Bells of Ireland, Gladioli, and Liatris
all of the filler flowers. How many different bouquets
Filler Flowers: Unobtrusive flowers
Daisies, Baby's Breath, Wax Flowers,
that give depth to the bouquet.
Solidago, and Caspia
can be made?
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?
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Permutations and Combinations
n!
C =
n r
(n - r)!r!
LESSON TWO - Combinations
Lesson Notes
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?
Example 9: Combination Formula.
a) Evaluate 7C5
d) Write
6!
4!2!
c) Evaluate
b) Evaluate 3C3
as a combination.
e) Write
5!
4!
n
as a combination.
Example 10: Combination Formula. Solve for the unknown algebraically.
a) nC2 = 21
b) 4Cr = 6
d)
c)
Example 11: Combination Formula. Solve for the unknown algebraically.
a)
n
C4
C
n-2 2
=1
b)
n
Cr
C
n n-r
=1
c)
P =2×
n-1 3
n-1
C2
d)
n+1
C2 =
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1
C
×
2 n+2 3
Cr =
n!
(n - r)!r!
n!
C
=
n r
(n - r)!r!
Permutations and Combinations
LESSON TWO - Combinations
Lesson Notes
Example 12: Assorted Mix I
a) A six-character code has the pattern shown below,
and the same letter or digit may be used more than once.
Letter
Digit
Digit
Digit
Digit
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?
Letter
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?
Example 13: Assorted Mix II
a) A set of tiles contains eight letters, A - H. If two of these sets are combined, how many
ways 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 must
be kept together, and in that order?
g) A 5-card hand is dealt from a deck of 52 cards. How many hands are possible containing
at 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
C
Example 14: Assorted Mix III
B
a) 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
A
two books of each type, how many fiction books are on the bookshelf?
C
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?
Science
Math
English
Other
e) A particular college in Alberta has a list of approved pre-requisite courses.
Math 30-1
Biology 30
English 30-1
Option A
Five courses are required for admission to the college. Math 30-1 (or Math 30-2)
or
Chemistry 30
Option B
Math 30-2
Physics 30
Option C
and English 30-1 are mandatory requirements, and at least one science course
Option D
must be selected as well. How many different ways could a student select five
Option E
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?
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D
Permutations and Combinations
LESSON TWO - Combinations
Lesson Notes
n!
C =
n r
(n - r)!r!
Example 15: Assorted Mix IV
a) 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 the
same color?
d) If a 5-card hand is dealt from a deck of 52 cards, how many hands have cards that are all the
same 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
D
C
E
F
Example 16: Assorted Mix V
a) How many ways can six people be split into two
equal-sized groups?
Amino Acid
Codon(s)
b) Show that 25! + 26! = 27 × 25!
c) Five different types of fruit and six different types
Arginine (Arg)
CGU, CGC, CGA, CGG, AGA, AGG
of vegetables are available for a healthy snack tray.
Cysteine (Cys)
UGU, UGC
The snack tray is to contain two fruits and three vegetables.
Glycine (Gly)
GGU, GGC, GGA, GGG
How many different snack trays can be made if blueberries
Methionine
(Met)
AUG
or carrots must be served, but not both together?
Serine (Ser)
UCU, UCC, UCA, UCG, AGU, AGC
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.
B
A
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C
Permutations and Combinations
tk+1 = nCk(x)n-k(y)k
LESSON THREE - The Binomial Theorem
Lesson Notes
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.
First seven rows
of Pascal’s Triangle.
1
1
1
1
1
1
2
3
4
5
1
6
1
1
3
6
10
15
1
4
10
20
1
5
15
1
6
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.
i. 1
8
ii. 1
12
28
56
70
56
28
8
1
66 220 495 792 924 792 495 220
66
12
1
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
c)
A
d)
A
A
B
B
B
Example 4: The Binomial Theorem
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.
b) (x + 2)6
c) (2x - 3)4
Example 5: Expand each expression.
a) (x2 - 2y)4
b)
c)
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B
1
Permutations and Combinations
LESSON THREE - The Binomial Theorem
Lesson Notes
tk+1 = nCk(x)n-k(y)k
Example 6: Write each expression as a binomial power.
a) x4 + 4x3y + 6x2y2 + 4xy3 + y4
b) 32a5 - 240a4b + 720a3b2 - 1080a2b3 + 810ab4 - 243b5
c)
Example 7: Use the general term formula to find the
requested term in a binomial expansion.
a) Find the third term in the expansion of (x - 3)4 .
General Term
tk + 1 = nCk(x)n - k(y)k
b) Find the fifth term in the expansion of (3a3 - 2b2)8 .
6
1

c) Find the fourth term in the expansion of  x 2 -  .
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
a
. What is the value of m.
b
Example 10: Finding Specific Values.
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
b , one of the terms is 240x2. What is the value of b?
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Answer Key
Polynomial, 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
b) i) Y ii) N iii) Y iv) N v) Y vi) N vii) N viii) Y ix) N
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).
Example 5a
d) Zero: 1 (multiplicity 3). e) Zeros: -1 (multiplicity 2) and 2 (multiplicity 1).
Example 5b
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 6a
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).
Example 6b
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).
Example 7a
Example 7b
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).
Example 8:
Example 9:
Example 10:
a)
a)
a)
b)
b)
b)
3
Example 11: a) x: [-15, 15, 1], y: [-169, 87, 1]
b) x: [-12, 7, 1], y: [-192, 378, 1]
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c) x: [-12, 24, 1], y: [-1256, 2304, 1]
Answer Key
Example 12: a)
b)
Example 13:
Example 14:
Example 15:
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)
a) Pproduct(x) = x2(x + 2); Psum(x) = 3x + 2
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 inhaled
c) One breath takes 5.34 seconds to complete.
d) 64% of the breath is spent inhaling.
b) x + 2x - 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.
3
2
Interval Notation
Example 16:
Math 30-1 students are expected to
know that domain and range can be
expressed using interval notation.
Polynomial, Radical, and Rational
Functions Lesson Two: Polynomial Division
Example 1: a) Quotient: x - 5; R = 4 b) P(x): x + 2x - 5x - 6; D(x) = x + 2; Q(x) = x - 5; R = 4
c) L.S.= R.S. d) Q(x) = x2 - 5 + 4/(x + 2) e) Q(x) = x2 - 5 + 4/(x + 2)
2
3
2
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) x – 4 d) x + 5x + 12 + 36/(x - 3)
2
2
Example 7c
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
() - Round Brackets: Exclude point
from interval.
[] - Square Brackets: Include point
in interval.
Infinity ∞ always gets a round bracket.
Examples: x ≥ -5 becomes [-5, ∞);
1 < x ≤ 4 becomes (1, 4];
x ε R becomes (-∞ , ∞);
-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].
Example 9
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 = -2
e) When the polynomial P(x) is divided by
x - a, the remainder is P(a).
(1, -4)
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 = -5
Example 13: m = 4 and n = -7
Example 14: m = 4 and n = -3
Example 15: a = 5
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.
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Example 10
(1, 0)
Remainder Theorem
(R = any number)
Factor
Theorem
(R = 0)
Answer Key
Polynomial, 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) = 0
d) The x-intercepts match the zeros of the polynomial
e) 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).
Example 10: Width = 10 cm; Height = 7 cm; Length = 15 cm
b) P(x) = 2(x + 1)2(x - 2).
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 Formula
From 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 Key
Polynomial, Radical, and Rational Functions Lesson Four: Radical Functions
Example 1:
a)
f(x)
x
-1
0
1
4
9
undefined
0
1
2
3
b) Domain: x ≥ 0;
Range: y ≥ 0
Example 2:
a)
c)
b)
Interval Notation:
Domain: [0, ∞);
Range: [0, ∞)
Example 3:
a)
b)
c)
d)
Example 4:
a)
b)
c)
d)
Example 5:
a)
b)
c)
d)
Example 6:
a)
Example 7:
a)
b)
b)
ORIGINAL:
Domain: x ε R or (-∞, ∞)
Range: y ε R or (-∞, ∞)
ORIGINAL:
Domain: x ε R or (-∞, ∞)
Range: y ≤ 9 or (-∞, 9]
ORIGINAL:
Domain: x ε R or (-∞, ∞)
Range: y ≥ -4 or [-4, ∞)
ORIGINAL:
Domain: x ε R or (-∞, ∞)
Range: y ≥ 0 or [0, ∞)
TRANSFORMED:
Domain: x ≥ -4 or [-4, ∞)
Range: y ≥ 0 or [0, ∞)
TRANSFORMED:
Domain: -5 ≤ x ≤ 1 or [-5, 1]
Range: 0 ≤ y ≤ 3 or [0, 3]
TRANSFORMED:
Domain: x ≤ 3 or x ≥ 7
or (-∞, 3] U [7, ∞)
Range: y ≥ 0 or [0, ∞)
TRANSFORMED:
Domain: x ε R or (-∞, ∞)
Range: y ≥ 0 or [0, ∞)
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Answer Key
Example 14:
Example 8:
a)
ORIGINAL:
b)
Domain: x ε R or (-∞, ∞)
Range: y ≤ 0 or (-∞, 0]
ORIGINAL:
Domain: x ε R or (-∞, ∞)
Range: y ≥ 0.25 or [0.25, ∞)
TRANSFORMED:
Domain: x = -5
Range: y = 0
TRANSFORMED:
Domain: x ε R or (-∞, ∞)
Range: y ≥ 0.5 or [0.5, ∞)
a)
b) Domain: 0 ≤ d ≤ 3; Range: 0 ≤ h(d) ≤ 3
or Domain: [0, 3]; Range: [0, 3].
When d = 0, the ladder is vertical.
When d = 3, the ladder is horizontal.
c) 2 m
h(d)
5
Example 9:
a) x = 7
b)
c)
4
3
(7, 3)
(7, 0)
2
1
1
2
3
4
5
d
Example 15:
Example 10:
a) x = 2
b)
c)
a)
× original time
b) 1/2 × original time
c)
(2, 2)
(2, 0)
h
t
1
0.4517
4
0.9035
8
1.2778
t
3.0
2.5
Example 11:
a) x = -3, 1
b)
c)
2.0
1.5
1.0
(1, 4)
0.5
(-3, 0)
(-3, 0)
(1, 0)
1
2
3
4
5
6
7
8
h
Example 16:
a)
Example 12:
a) No Solution
b)
b)
c)
V(r)
60
No Solution
50
No Solution
40
30
20
10
1
Example 13:
a)
b)
c)
d)
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2
3
4
5
6
r
Answer Key
Polynomial, Radical, and Rational Functions Lesson Five: Rational Functions I
Example 1:
a)
x
y
-2
-0.5
-1
-1
-0.5
-2
-0.25
-4
0
undef.
0.25
4
0.5
2
1
1
2
0.5
b)
c)
d)
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.
Example 2:
a) Original Graph:
Domain: x ε R or (-∞, ∞);
Range: y ε R or (-∞, ∞)
b) Original Graph:
Domain: x ε R or (-∞, ∞);
Range: y ε R or (-∞, ∞)
Reciprocal Graph:
Domain: x ε R, x ≠ 5 or (-∞, 5) U (5, ∞);
Range: y ε R, y ≠ 0 or (-∞, 0) U (0, ∞)
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 = 5;
Horizontal: y = 0
Asymptote Equation(s):
Vertical: x = 4;
Horizontal: y = 0
Example 3:
a)
x
y
-3
0.20
-2
undef.
-1
-0.33
0
-0.25
1
-0.33
2
undef.
3
0.20
x
y
-2.05
4.94
-1.95
-5.06
x
y
1.95
-5.06
2.05
4.94
b)
c)
d)
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.
Example 4:
d) Original: x ε R; y ≥ 0
a) Original: x ε R; y ≥ -1
b) Original: x ε R; y ≤ 1/2
c) Original: x ε R; y ≥ -2
or D: (-∞, ∞); R: [-1, ∞).
or D: (-∞, ∞); R: (-∞, 1/2].
or D: (-∞, ∞); R: [-2, ∞).
or D: (-∞, ∞); R: [0, ∞).
Reciprocal: x ε R, x ≠ -2, 2; y ≤ -1 or y > 0
Reciprocal: x ε R, x ≠ -4, 2; y < 0 or y ≥ 2
Reciprocal: x ε R, x ≠ 4, 8; y ≤ -1/2 or y > 0
Reciprocal: x ε R, x ≠ 0; y > 0
or D: (-∞, -2) U (-2, 2) U (2, ∞); R: (-∞, -1] U (0, ∞)
or D: (-∞, -4) U (-4, 2) U (2, ∞); R: (-∞, 0) U [2, ∞)
or D: (-∞, 4) U (4, 8) U (8, ∞); R: (-∞, -1/2] U (0, ∞)
or D: (-∞, 0) U (0, ∞); R: (0, ∞)
Asymptotes: x = ±2; y = 0
Asymptotes: x = -4, x = 2; y = 0
Asymptotes: x = 4, x = 8; y = 0
Asymptotes: x = 0; y = 0
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Answer Key
Example 4 (continued):
e) Original: x ε R; y ≥ 2
f) Original: x ε R; y ≤ -1/2
or D: (-∞, ∞); R: [2, ∞).
or D: (-∞, ∞); R: (-∞, -1/2].
Reciprocal: x ε R; 0 < y ≤ 1/2
Reciprocal: x ε R; -2 ≤ y < 0
or D: (-∞, ∞); R: (0, 1/2]
or D: (-∞, ∞); R: [-2, 0)
Asymptotes: y = 0
Asymptotes: y = 0
Example 5:
a)
b)
c)
d)
b) x = -4, 6; y = 0
c) x = -0.5, 0, 1.33 ; y = 0
b) VT: 3 down
c) VS: 3; HT: 4 left
b) HT: 2 right; VT: 1 up
c) VS: 4; HT: 1 right; VT: 2 down
Example 6:
a) x = 1.5; y = 0
d) y = 0
Example 7:
a) VS: 4
d) VS: 2; HT: 3 right; VT: 2 up
Example 8:
a) VT: 2 down
Example 9:
c) 2 × original
d) 8.3 kPa•L/mol•K
e) See table & graph
f) See table & graph
Illuminance V.S. Distance for a Fluorescent Bulb
I
Example 10:
a) P(V) = nRT(1/V).
b) 1/2 × original
d) VS: 3; HT: 5 right; VT: 2 down
Pressure V.S. Volume of 0.011 mol
of a gas at 273.15 K
V
P
P
(L)
(kPa)
50
0.5
50
c) 4 × original
40
35
25
2.0
12.5
25
5.0
5.0
15
2.5
b) 1/9 × original
45
1.0
10.0
a) 1/4 × original
30
d) 16 × original
20
e) See table & graph
10
5
1
2
3
4
5
6
7
8
9 10 V
f) See table & graph
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130
120
d
I
(m)
(W/m )
1
2
4
ORIGINAL
8
12
2
110
100
90
80
70
60
50
40
30
20
10
1
2
3
4
5
6
7
8
9 10 11 12
d
Answer Key
Polynomial, Radical, and Rational Functions Lesson Six: Rational Functions II
Example 1:
a) y =
x
x -9
2
b) y =
x+2
x2 + 1
c)
y=
x+4
x2 - 16
d) y =
x2 - x - 2
x3 - x2 - 2x
b) y =
x2
x -1
c)
y=
3x2
x2 + 9
d) y =
3x2 - 3x - 18
x2 - x - 6
b) y =
x2 - 4x + 3
x-3
c)
y=
x2 + 5
x-1
d) y =
x2 - x - 6
x+1
Example 2:
a) y =
4x
x-2
2
Example 3:
a) y =
x2 + 5x + 4
x+4
Example 4:
Example 5:
Example 6:
Example 7:
i) Horizontal Asymptote: y = 0
ii) Vertical Asymptote(s): x = ±4
iii) y - intercept: (0, 0)
iv) x - intercept(s): (0, 0)
v) Domain: x ε R, x ≠ ±4;
Range: y ε R
i) Horizontal Asymptote: y = 2
ii) Vertical Asymptote(s): x = -2
iii) y - intercept: (0, -3)
iv) x - intercept(s): (3, 0)
v) Domain: x ε R, x ≠ -2;
Range: y ε R, y ≠ 2
i) Horizontal Asymptote: None
ii) Vertical Asymptote(s): x = 1
iii) y - intercept: (0, 8)
iv) x - intercept(s): (-4, 0), (2, 0)
v) Domain: x ε R, x ≠ 1;
Range: y ε R
i) y = x - 3
or D: (-∞, -4) U (-4, 4) U (4, ∞); R: (-∞, ∞)
or D: (-∞, -2) U (-2, ∞); R: (-∞, 2) U (2, ∞)
or D: (-∞, 1) U (1, ∞); R: (-∞, ∞)
vi) Oblique Asymptote: y = x + 3
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ii) Hole: (2, -1)
iii) y - intercept: (0, -3)
iv) x - intercept(s): (3, 0)
v) Domain: x ε R, x ≠ 2;
Range: y ε R, y ≠ -1
or D: (-∞, 2) U (2, ∞); R: (-∞, -1) U (-1, ∞)
Answer Key
Example 13:
Example 8:
a)
a)
b)
d
s
t
Cynthia
15
x+3
15
x+3
Alan
10
x
10
x
Equal times.
b) Cynthia: 9 km/h; Alan: 6 km/h
c) Graphing Solution: x-intercept method.
Example 9:
(6, 0)
a)
b)
c)
d)
Example 14:
a)
Example 10: a) x = 4
d
s
t
Upstream
24
x-2
24
x-2
Downstream
24
x+2
24
x+2
c)
b)
(4, 4)
(4, 0)
Sum of times equals 5 h.
b) Canoe speed: 10 km/h
c) Graphing Solution: x-intercept method.
Example 11: a) x = -1/2 and x = 2
b)
(-0.4, 0)
(10, 0)
c)
(-0.5, 0)
(-0.5, -6)
(2, 0)
(2, -6)
Example 12: a) x = 1. x = 2 is an extraneous root
b)
c)
Example 15:
Example 16:
a)
a)
b) Number of goals required: 6
b) Mass of almonds required: 90 g
c) Graphing Solution:
x-intercept method.
c) Graphing Solution:
x-intercept method.
1
0.5
(1, -3)
(1, 0)
(6, 0)
-300
(90, 0)
-0.5
-1
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300
Answer Key
Transformations and Operations Lesson One: Basic Transformations
a)
b)
c)
d)
Example 2: a)
b)
c)
d)
Example 3: a)
b)
c)
Example 4:
a)
b)
c)
Example 5:
a)
b)
c)
d)
Example 6:
a)
b)
c)
d)
Example 1:
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Answer Key
Example 7:
a)
b)
c)
d)
Example 8:
a)
b)
c)
d)
Example 9:
a)
b)
c)
d)
Example 10: a)
b)
c)
d)
Example 11:
a)
b)
Example 13:
a) R(n) = 5n
C(n) = 2n + 150
b) 50 loaves
c) C2(n) = 2n + 200
d) R2(n) = 6n
e) 50 loaves
$
$
Example 13a
500
Example 13e
500
R(n)
C2(n)
400
400
C(n)
300
R2(n)
(50, 300)
300
(50, 250)
200
200
100
100
or
20
Example 12:
a)
b)
Example 14:
a)
b) 12 metres
or
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40
60
80
n
20
40
60
80
n
Answer Key
Transformations 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: a)
Example 3: a) H.T. 3 left
b) i. V.S. 1/3
H.S. 1/5
b)
b) i. H.T. 1 right
V.T. 3 up
Example 4: a)
ii. V.S. 2
H.S. 4
iii. V.S. 1/2
H.S. 3
Reflection about x-axis
c)
ii. H.T. 2 left
V.T. 4 down
b)
d)
iii. H.T. 2 right
V.T. 3 down
iv. H.T. 7 left
V.T. 5 up
c)
Example 5: a) Stretches and reflections should
be applied first, in any order.
Translations should be applied last,
in any order.
b) i. V.S. 2
H.T. 3 left
V.T. 1 up
d)
ii. H.S. 3
Reflection
about x-axis
V.T. 4 down
iii. V.S. 1/2
Reflection about y-axis
H.T. 2 left; V.T. 3 down
Example 6: a)
b)
c)
d)
Example 7: a)
b)
c)
d)
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-Independence
Apply all the vertical
transformations together
and apply all the horizontal
transformations together,
in either order.
iv. V.S. 3
H.S. 1/2
Reflection about x-axis
Reflection about y-axis
iv. V.S. 3; H.S. 1/4
Reflection about x-axis
Reflection about y-axis
H.T. 1 right; V.T. 2 up
Example 11:
a) H.T. 8 right; V.T. 7 up
b) Reflection about x-axis; H.T. 4 left; V.T. 6 down
c) H.S. 2; H.T. 3 left; V.T. 7 up
d) 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.
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Answer Key
Transformations and Operations Lesson Three: Inverses
Example 1: a) Line of Symmetry: y = x
Example 2:
a)
b)
b)
Example 3:
Example 4:
a)
b)
Restrict the domain of
the original function to
-10 ≤ x ≤ -5 or -5 ≤ x ≤ 0
a)
Restrict the domain of the
original function to
x ≤ 5 or x ≥ 5.
Example 5:
b)
Restrict the domain of
the original function to
x ≤ 0 or x ≥ 0
Original:
D: x ε R
R: y ε R
Inverse:
D: x ε R
R: y ε R
Original:
D: x ε R
R: y ε R
The inverse is a function.
a)
Example 8:
a) (10, 8)
a) 28 °C is equivalent to 82.4 °F
b) True.
f-1(b) = a
b)
D: x ≥ 4
D: x ε R
°F
100
F(C)
F-1(C)
50
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)
-100
-50
50
(-40, -40)
-50
-100
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.
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Inverse:
D: x ε R
R: y ε R
The inverse is a function.
b)
Restrict the domain of the
original function to
x ≤ -3 or x ≥ -3.
Example 7:
d) k = 30
b)
Example 6:
a)
c) f(5) = 4
d)
c)
100 °C
Answer Key
Transformations and Operations Lesson Four: Function Operations
Example 1: a)
x
(f + g)(x)
-8
-6
-4
-6
-2
0
0
-6
1
-9
4
-9
b)
Domain:
-8 ≤ x ≤ 4
or [-8, 4]
f(x)
Range:
-9 ≤ y ≤ 0
or [-9, 0]
x
(f - g)(x)
-9
DNE
-5
10
-3
9
0
2
g(x)
c)
x
(f • g)(x)
-6
-4
-3
-8
Domain:
-6 ≤ x ≤ 3
or [-6, 3]
f(x)
f(x)
Range:
2 ≤ y ≤ 10
or [2, 10]
3
5
6
DNE
g(x)
d)
Range:
-8 ≤ y ≤ -2
or [-8, -2]
g(x)
Domain:
-5 ≤ x ≤ 3;
or [-5, 3]
x
(f ÷ g)(x)
-6
DNE
-4
-4
-2
-8
0
-2
3
-3
0
-6
6
DNE
2
-4
4
-2
6
DNE
Example 2:
Domain:
-4 ≤ x ≤ 4
or [-4, 4]
f(x)
Range:
-8 ≤ y ≤ -2
or [-8, -2]
g(x)
Reminder: Math 30-1 students are
expected to know that domain and
range can be expressed using
interval notation.
a) i. (f + g)(-4) = -2 ii. h(x) = -2; h(-4) = -2
b) i. (f – g)(6) = 8 ii. h(x) = 2x – 4; h(6) = 8
c) i. (fg)(-1) = -8 ii. h(x) = -x2 + 4x - 3; h(-1) = -8
d) i. (f/g)(5) = -0.5 ii. h(x) = (x - 3)/(-x + 1); h(5) = -0.5
Example 3:
a)
b)
c)
g(x)
d)
f(x)
f(x)
g(x)
g(x)
f(x)
f(x)
g(x)
Domain: 3 < x ≤ 5 or (3, 5]
Range: 0 < y ≤ 1 or (0, 1]
m(x)
Domain: x ≥ -4 or [-4, ∞)
Range: y ≤ 9 or (-∞, 9]
Example 4:
a)
Domain: 0 < x ≤ 10 or (0, 10] Domain: x > -2 or (-2, ∞)
Range: -10 ≤ y ≤ 0 or [-10, 0] Range: y > 0 or (0, ∞)
Example 5:
a)
b)
f(x)
f(x)
g(x)
g(x)
b)
g(x)
g(x)
f(x)
Domain: x ≥ -4 or [-4, ∞)
Range: y ≥ 0 or [0, ∞)
Transformation: y = f(x) - 1
Domain: x ≥ -4 or [-4, ∞)
Range: y ≤ -1 or (-∞, -1]
Transformation: y = -f(x).
Domain: x ε R or (-∞, ∞)
Range: y ≤ -6 or (-∞, -6]
Transformation: y = f(x) - 2
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f(x)
Domain: x ε R or (-∞, ∞)
Range: y ≤ -2 or (-∞, -2]
Transformation: y = 1/2f(x)
Answer Key
Example 6:
a)
c)
b)
d)
f(x)
g(x)
g(x)
g(x)
g(x)
f(x)
f(x)
f(x)
Domain: x ε R, x ≠ 0;
Range: y ε R, y ≠ 0
Domain: x ε R, x ≠ 2;
Range: y ε R, y ≠ 0
Domain: x ε R, x ≠ -3;
Range: y ε R, y ≠ 0
Domain: x ≥ -3, x ≠ -2;
Range: y ε R, y ≠ 0
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, ∞)
Example 7:
Example 8:
a) AL(x) = 8x2 – 8x
a) R(n) = 12n;
E(n) = 4n + 160;
P(n) = 8n – 160
500
b) When 52 games
are sold, the profit
is $256
200
c) Greg will break
even when he sells
20 games
-100
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 9:
R(n)
E(n)
P(n)
400
300
a) The surface area and volume formulae
have two variables, so they may not be
written as single-variable functions.
c)
b)
100
20
40
60
d)
n
f)
e)
Transformations and Operations Lesson Five: Function Composition
Example 1: a)
x
g(x) f(g(x))
b)
x
f(x)
g(f(x))
-3
9
6
0
-3
9
-2
-1
0
1
2
3
4
1
0
1
4
9
1
-2
-3
-2
1
6
1
-2
4
2
-1
1
3
0
0
b) n(1) = -4
c) p(2) = -2
c) Order matters in a
composition of functions.
f)
m(x)
d) m(x) = x2 – 3
e) n(x) = (x – 3)2
Example 2:
a) m(3) = 33
Example 3:
a) m(x) = 4x2 – 3
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.
b) n(x) = 2x2 – 6
n(x)
d) q(-4) = -16
c) p(x) = x4 – 6x2 + 6
d) q(x) = 4x
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e) All of the results match
Answer Key
Example 5:
a)
Example 6:
a) h ( x ) =
Example 7:
a)
b)
1
| x + 2|
b) h ( x ) = x + 2 + 2
b)
Example 8: a)
d)
c)
b)
f)
e)
Example 9:
Example 10:
Example 11:
Example 12:
Example 13:
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.
a) The cost of the trip is $4.20.
It took two separate calculations
to find the answer.
b) V(d) = 0.08d
c) M(V) = 1.05V
d) M(d) = 0.084d
e) Using function composition,
we were able to solve the
problem with one calculation
instead of two.
a) A(t) = 900πt2
a) a(c) = 1.03c
a)
b) A = 8100π cm2
b) j(a) = 78.0472a
c) t = 7 s; r = 210 cm
c) b(a) = 0.6478a
d) b(c) = 0.6672c
M(d)
60
50
40
30
20
10
100
200
300
400
500
600
d
www.math30.ca
b)
c) h = 4 cm
Answer Key
Exponential and Logarithmic Functions Lesson One: Exponential Functions
b)
Example 1: a)
c)
Parts (a-d):
Domain: x ε R or (-∞, ∞)
Range: y > 0 or (0, ∞)
x-intercept: None
y-intercept: (0, 1)
Asymptote: y = 0
d)
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.
Example 2: a)
;
Example 3: a)
10
;
b)
b)
5
-5
5
-5
5
-5
-5
-5
Domain: x ε R or (-∞, ∞)
Range: y > -4 or (-4, ∞)
Asymptote: y = -4
Example 5: a)
-5
5
5
-5
Domain: x ε R or (-∞, ∞)
Range: y > 0 or (0, ∞)
Asymptote: y = 0
d)
10
5
5
-5
10
5
10
5
-5
d)
Domain: x ε R or (-∞, ∞)
Range: y > 3 or (3, ∞)
Asymptote: y = 3
c)
10
5
10
;
-5
Domain: x ε R or (-∞, ∞)
Range: y > 0 or (0, ∞)
Asymptote: y = 0
b)
10
d)
-5
-5
Domain: x ε R or (-∞, ∞)
Range: y > 0 or (0, ∞)
Asymptote: y = 0
;
5
5
-5
Example 6: a)
c)
10
5
Example 4: a)
c)
5
5
-5
-5
Domain: x ε R or (-∞, ∞)
Range: y > -2 or (-2, ∞)
Asymptote: y = -2
5
-5
Domain: x ε R or (-∞, ∞)
Range: y > 0 or (0, ∞)
Asymptote: y = 0
Domain: x ε R or (-∞, ∞)
Range: y > 0 or (0, ∞)
Asymptote: y = 0
b)
b)
c)
d)
www.math30.ca
e) V.S. of 9
equals H.T.
2 units left.
f)
See Video
Answer Key
Example 7:
Example 8:
Example 9:
Example 10:
Example 11:
Example 12:
Example 13:
Example 14:
a)
b)
a)
b)
c)
d)
a)
b)
c)
a)
b)
c)
d)
a)
b) infinite
solutions
c)
d)
a)
a)
a)
b) no solution
c)
d)
c)
d)
d)
e)
b)
c)
d)
b)
c)
d)
f)
Example 15:
m
m
Example 15c
100
Example 15d
1
a)
b) 84 g
c) See Graph
50
0.5
d) 49 years
(49, 0.1)
10
Example 16:
t
20
B
30
40
800
b) 32254 bacteria
Watch Out! The graph requires hours
on the t-axis, so we can rewrite the
exponential function as:
600
50000
c) See Graph
400
d) 6 hours ago
200
5
Example 17: a)
10
P%
a) 853,370
b) 54 years
(-6, 50)
-8
0
2
t
; $600
b)
; 69 MHz
Example 18:
t
P%
Example 18b
5
1
4
0.8
3
Example 18d
(76.7, 0.5)
0.6
(53.7, 2)
c) 21406
d) 77 years
2
0.4
1
0.2
20
40
60
$
Example 19:
80
100
t
20
40
60
P%
Example 19c
1000
80
100
t
Example 19d
5
a) A (t ) = 500 (1.025 )
t
b) $565.70
Interest: $65.70
4
3
(28, 2)
500
c) See graph
2
d) 28 years
1
e) $566.14; $566.50; $566.57
As the compounding frequency
increases, there is less and less
of a monetary increase.
10
20
t
t
Example 16d
1000
a)
60
50
B
Example 16c
100000
20
10
10
20
30
www.math30.ca
40
50
t
Answer Key
Exponential and Logarithmic Functions Lesson Two: Laws of Logarithms
Example 1:
Example 3:
Example 4:
Example 5:
Example 6:
a) The base of the logarithm is b,
a is called the argument of the logarithm,
and E is the result of the logarithm.
a)
a)
a)
a)
b)
b)
b)
b)
In the exponential form, a is the result,
b is the base, and E is the exponent.
c)
c)
c)
d)
d)
e)
e)
b) i. 0; 1; 2; 3
c) i. log42
c)
d)
ii. 0; 1; 2; 3
d)
e)
ii.
e)
f)
Example 2:
g)
a)
h)
f)
f)
f)
g)
g)
g)
h)
h)
h)
b)
c)
Example 7:
Example 8:
Example 9:
Example 10:
a)
a)
a)
a)
Example 11:
Example 12:
a)
a)
b)
b)
c)
b)
b)
b)
b)
c)
c)
c)
c)
d)
d)
d)
d)
e)
e)
e)
e)
f)
f)
f)
f)
Example 13:
Example 14:
Example 15:
g)
g)
g)
g)
a)
a)
a)
h)
h)
h)
h)
b)
b)
b)
c)
c)
c)
d)
d)
c)
d)
d)
±
Example 16:
Example 17:
Example 18:
Example 19:
Example 20:
a)
a)
a)
a)
a)
b)
b)
b)
b)
b)
c)
c)
d)
d)
c)
c)
d)
d)
e)
e)
f)
f)
g)
g)
g) see video
g)
h)
h)
h)
h)
e)
f)
e)
f)
www.math30.ca
c)
d)
e)
f)
g)
h)
d)
Answer Key
Exponential and Logarithmic Functions Lesson Three: Logarithmic Functions
Example 1:
a) See Graph
c) See Video
b) See Graph
10
d)
y=2
f(x) = 2x
Domain
5
-5
f (x) = log2x
-1
5
-5
x
xεR
y = log2x
x>0
Range
y>0
yεR
x-intercept
none
(1, 0)
y-intercept
(0, 1)
none
Asymptote
Equation
y=0
x=0
e)
i) -1,
ii) 0,
iii) 1,
iv) 2.8
f)
y = log1x, y = log0x,
and y = log-2x are
not functions.
is a function.
10
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
Example 2:
a)
b)
c)
d)
D: x > 0
or (0, ∞)
D: x > 0
or (0, ∞)
D: x > 0
or (0, ∞)
D: x > 0
or (0, ∞)
R: y ε R
or (-∞, ∞)
R: y ε R
or (-∞, ∞)
R: y ε R
or (-∞, ∞)
R: y ε R
or (-∞, ∞)
A: x = 0
A: x = 0
A: x = 0
A: x = 0
Example 3:
a)
D: x > 0
or (0, ∞)
b)
D: x > 0
or (0, ∞)
c)
D: x > 0
or (0, ∞)
d)
D: x > 0
or (0, ∞)
R: y ε R
or (-∞, ∞)
R: y ε R
or (-∞, ∞)
R: y ε R
or (-∞, ∞)
R: y ε R
or (-∞, ∞)
A: x = 0
A: x = 0
A: x = 0
A: x = 0
Example 4:
a)
D: x > 0
or (0, ∞)
b)
D: x > -2
or (-2, ∞)
c)
D: x > 3
or (3, ∞)
d)
D: x > -4
or (-4, ∞)
R: y ε R
or (-∞, ∞)
R: y ε R
or (-∞, ∞)
R: y ε R
or (-∞, ∞)
R: y ε R
or (-∞, ∞)
A: x = 0
A: x = -2
A: x = 3
A: x = -4
Example 5:
a)
D: x > -3
or (-3, ∞)
b)
D: x > 0
or (0, ∞)
c)
D: x > -3
or (-3, ∞)
d)
D: x > -2
or (-2, ∞)
R: y ε R
or (-∞, ∞)
R: y ε R
or (-∞, ∞)
R: y ε R
or (-∞, ∞)
R: y ε R
or (-∞, ∞)
A: x = -3
A: x = 0
A: x = -3
A: x = -2
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Answer Key
Example 6:
a)
b)
c)
d)
D: x > 0
or (0, ∞)
D: x > 1
or (1, ∞)
D: x > 2
or (2, ∞)
D: x > 0
or (0, ∞)
R: y ε R
or (-∞, ∞)
R: y ε R
or (-∞, ∞)
R: y > log34
or (log34, ∞)
R: y ε R
or (-∞, ∞)
A: x = 0
A: x = 1
A: none
A: x = 0
Example 7:
a) x = 8
b)
c) No Solution
(8, 262144)
No Solution
(-0.60, 0.72)
Example 8:
a) x = 8
(25, 12)
b) x = 25
c) x = 4
(8, 2)
(4, 3)
Example 9:
Example 10:
Example 11:
a)
a)
a)
b) (-3, 0) and (0, 2)
b)
b)
c)
c)
c)
d)
d)
d)
e)
e)
e)
Example 12:
Example 13:
Example 14:
Example 15:
a) 4
a) 60 dB
a) pH = 4
a) 200 cents
b) 0.1 m
b) 0.1 W/m2
b) 10-11 mol/L
b) 784 Hz
c) 31.6 times stronger
c) 100 times more intense
c) 1000 times stronger
c) 1200 cents separate
the two notes
d) See Video
d) See Video
d) See Video
e) 10 times stronger
e) 100 times more intense
e) pH = 2
f) See Video
f) See Video
f) pH = 5
g) 5.5
g) 23 dB
g) 100 times more acidic
h) 5.4
h) 37 dB
(y-axis)
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Answer Key
Trigonometry Lesson One: Degrees and Radians
Note: For illustrative purposes,
all diagram angles will be in degrees.
Example 1:
a) The rotation angle
between the initial
arm and the terminal
arm is called the
standard position
angle.
b) An angle is positive
if we rotate the
terminal arm counterclockwise, 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°.
420°
120°
f) The general form of
co-terminal angles is
θc = θp + n(360°) using
degrees, or θc = θp + n(2π)
using radians.
45°,
405°,
765°,
1125°,
1485°
60°
150°
45°,
-315°,
-675°,
-1035°,
-1395°
θ
30°
-120°
Example 2:
Conversion Multiplier Reference Chart
iii. One revolution is defined
a) i. One degree is ii. One radian is the angle formed when
defined as 1/360th the terminal arm swipes out an arc that as 360º, or 2pi. It is one complete
of a full rotation. has the same length as the terminal arm. rotation around a circle.
One radian is approximately 57.3°.
degree
degree
57.3°
revolution
c) i. 0.79 rad
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
Example 4:
revolution
π
1 rev
180°
360°
1 rev
180°
π
radian
1°
b) i. 0.40 rad ii. 0.06 rev iii. 148.97° iv. 0.41 rev v. 270° vi. 4.71 rad
radian
2π
360°
2π
1 rev
1 rev
ii. π/4 rad
h) 180° i) 6π rad
Example 5:
90° =
a) θr = 30°
= 120°
b) θr = 80°
c) θr = 56°
(or 0.98 rad)
60° =
= 135°
d) θr = 45°
(or π/4 rad)
-260°
45° =
135°
30° =
= 150°
45°
θ
θ
30°
0° =
θ
θ
80°
360° =
= 210°
Example 6:
a) θp = 210°, θr = 30°
315° =
b) θp = 225°, θr = 45°
300° =
= 240°
51°
309°
304°
330° =
= 225°
θ
56°
210°
= 180°
e) θr = 51°
(or 2π/7 rad)
c) θp = 156°, θr = 24°
d) θp = 120°, θr = 60°
(or θp = 2.72, θr = 0.42)
(or θp = 2π/3, θr = π/3)
120°
= 270°
24°
θ
60°
θ
θ
θ
30°
Example 7:
156°
45°
210°
a) θ = 60°, θp = 60°
b) θ = -495°, θp = 225°
θc = -300°, 420°, 780°
θc = -855°, -135°, 225°, 585°
225°
Example 8:
a) θp = 93°
60°
b) θp = 148°
c) θp = 144°
d) θp = 330°
(or 2.58 rad)
(or 4π/5 rad)
(or 11π/6 rad)
93°
148°
144°
225°
c) θ = 675°, θp = 315°
θc = -45°, 315°
(or θc = -0.785, 5.50)
d) θ = 480°, θp = 120°
θc = -960°, -600°, -240°, 120°,
840°, 1200°
120°
330°
(or θc = -16π/3,
-10π/3, -4π/3,
2π/3, 14π/3,
20π/3)
Example 9:
a) θc = 1380°
b) θc = -138π/5
315°
www.math30.ca
c) θc = 20°
d) θc = 2π/3
Answer Key
Example 10:
Example 11:
a) θp = 112.62°, θr = 67.38°
b) θp = 303.69°, θr = 56.31°
112.62°
13
12
67.38°
2
θ
θ
-5
56.31°
-3
303.69°
Example 12: a) i. QIII or QIV ii. QI or QIV iii. QI or QIII
b) i. QI ii. QIV iii. QIII
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
c) i. none ii. QIII iii. QI
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)
154.62°
3
-12
-5
θ
13
202.62°
7
25.38°
θ
22.62°
Example 14:
a) θp = 323.13°, θr = 36.87°
b) θp = 326.31°, θr = 33.69°
3
4
θ
36.87°
5
θ
-3
33.69°
323.13°
-2
326.31°
Example 15:
a)
If the angle θ could exist in
either quadrant ___ or ___ ...
The calculator always
picks quadrant
I or II
I or III
I or IV
II or III
II or IV
III or IV
I
I
I
II
IV
IV
b) Each answer is different because the calculator is unaware
of which quadrant the triangle is in. The calculator assumes
Mark’s triangle is in QI, Jordan’s triangle is in QII, and Dylan’s
triangle is in QIV.
Example 16:
Example 17:
Example 18:
Example 19:
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 cm
c) 114.59°
d) 2.46 cm
e) n = 7π/6
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
a) 600°/s
b) 0.07 rad/s
c) 1.04 km
d) 70 rev/s
e) 2.60 rev/s
a) π/2700 rad/s
b) 468.45 km
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Answer Key
Trigonometry Lesson Two: The Unit Circle
Example 1:
a) i.
c)
b) i. Yes ii. No
ii.
ii.
iii. y = 0
iv.
(0.6, 0.8)
10
10
i.
(0.5, 0.5)
-10
-10
10
10
Example 2: See Video.
Example 3: a)
d)
Example 4: a)
b) -1 c)
e) 0 f) 0 g)
d)
h)
e) -1 f)
b) 1
c)
g)
h)
Example 5:
,
a)
,
,
,
b)
,
,
,
,
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 = 2
do not intersect, so it's impossible
for sinθ to equal 2.
b)
Number Line
Range
cosθ & sinθ
Example 6:
,
a)
,
,
,
b)
,
,
cscθ & secθ
,
tanθ & cotθ
,
c)
Example 7: See Video.
-1
0
1
-1
0
1
-1
0
1
d) 53.13°, 302.70°
y=2
e)
Example 8:
a) -2 b) undefined c)
d)
e)
f) -1 g) 0 h)
Example 9:
Example 16:
a)
b) 1
c)
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.
b) Use basic trigonometric ratios (SOHCAHTOA)
to show that x = cosθ and y = sinθ.
c) θp = 167.32°, θr = 12.68°
d)
Example 10:
a) 1
b)
c)
d)
Example 11:
a) -1 b)
c) undefined d) undefined
Example 12: See Video.
Example 13:
a) P(π/3) means "point coordinates at π/3".
b)
d)
Example 17:
a) (167, 212) b) (-792, 113)
c)
e) P(3) = (-0.9900, 0.1411)
Example 18:
a) See Video b) 160 m
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1
|x|
|y|
Answer Key
Trigonometry 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)
Example 2: a) y = sinθ b) a = 1 c) P = 2π
d) c = 0 e) d = 0 f) θ = nπ, nεI g) (0, 0)
h) Domain: θ ε R, Range: -1 ≤ y ≤ 1
2π
11π
6
7π
4
5π
3
3π
2
4π
3
5π
4
7π
6
π
5π
6
3π
4
2π
3
π
2
π
3
π
4
d) (-3π, 10), (3π/2, -30), (5π/2, -20)
Example 3: a) y = cosθ b) a = 1 c) P = 2π
d) c = 0 e) d = 0 f) θ = π/2 + nπ, nεI g) (0, 1)
h) Domain: θ ε R, Range: -1 ≤ y ≤ 1
y
y
1
1
3
2
3
2
2
2
2
2
1
2
1
2
0
π
6
π
6
π
4
π
3
π
2
2π
3
3π
4
5π
6
π
7π
6
5π
4
4π
3
3π
2
5π
3
7π
4
11π
6
2π
2π
θ
11π
6
7π
4
5π
3
3π
2
4π
3
5π
4
7π
6
π
5π
6
3π
4
π
2
2π
3
π
3
π
4
0
π
6
1
2
1
2
2
2
2
2
3
2
π
6
π
4
π
3
π
2
2π
3
3π
4
5π
6
π
7π
6
5π
4
4π
3
3π
2
5π
3
7π
4
11π
6
2π
θ
π
6
π
4
π
3
π
2
2π
3
3π
4
5π
6
π
7π
6
5π
4
4π
3
3π
2
5π
3
7π
4
11π
6
2π
θ
3
2
-1
-1
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, nεI
g) (0, 0) h) Domain: θ ε R, θ ≠ π/2 + nπ, nεI, Range: y ε R
y
3
3
Example 5:
1
a)
3
3
b)
5
2π
5
11π
6
7π
4
5π
3
3π
2
4π
3
5π
4
7π
6
π
5π
6
3π
4
π
2
2π
3
π
3
π
4
π
6
3
3
-1
3
0
2π
0
2π
-3
-5
Example 7:
-5
c)
a)
d)
5
5
5
2π
0
Example 6:
2π
0
2π
-5
Example 8: a)
Example 9:
b)
a)
d)
5
0
2π
0
2π
-5
-5
-5
a)
c)
5
2π
-5
-5
d)
5
0
0
c)
b)
0
-1
π
2π
0
d)
1
1
π
-1
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2π
0
-1
5
d)
c)
b)
1
-
c)
b)
1
2π
4π
6π
0
-1
2π
4π
6π
8π
10π
Answer Key
Example 10:
Example 13:
a)
b)
a)
1
b)
1
12
1
6
-2π
-π
π
2π
-2π
-π
π
2π
π
2
π
2π
-2π
2π
4π
6π
-6
-1
-12
c)
-1
d)
-1
c)
d)
1
3
2
-2π
-π
4
2
1
π
2π
0
2π
4π
6π
1
-1
-2
-π
2π
-2π
4π
-π
π
2π
-1
-3
-1
Example 11:
a)
b)
Example 14:
d)
c)
a)
b)
c)
d)
Example 12:
a)
Example 15:
b)
1
a)
1
b)
4π
-4π
-2π
2π
4π
-4π
-2π
-1
2π
5
1
-1
0
c)
2π
4π
0
6π
π
2π
d)
1
4
-5
-1
c)
d)
2π
-2π
-π
π
2π
-2π
-π
π
6
5
-1
-4
0
π
2π
0
-6
-5
Example 16:
a)
b)
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π
2π
Answer Key
Example 17: a) y = secθ b) P = 2π
c) Domain: θ ε R, θ ≠ π/2 + nπ, nεI; Range: y ≤ -1, y ≥ 1
d) θ = π/2 + nπ, nεI
y
3
2
y
3
2
2 3
3
2π
0
2π
2π
11π
6
7π
4
5π
3
θ
3π
2
4π
3
5π
4
π
7π
6
5π
6
3π
4
2π
3
π
2
π
3
π
4
1
0
π
6
π
6
π
4
π
3
π
2
2π
3
3π
4
5π
6
π
7π
6
5π
4
4π
3
3π
2
5π
3
7π
4
11π
6
2π
θ
π
6
π
4
π
3
π
2
2π
3
3π
4
5π
6
π
7π
6
5π
4
4π
3
3π
2
5π
3
7π
4
11π
6
2π
θ
π
6
π
4
π
3
π
2
2π
3
3π
4
5π
6
π
7π
6
5π
4
4π
3
3π
2
5π
3
7π
4
11π
6
2π
θ
-1
2 3
3
2
-2
-3
-3
Example 18: a) y = cscθ b) P = 2π
c) Domain: θ ε R, θ ≠ nπ, nεI; Range: y ≤ -1, y ≥ 1
d) θ = nπ, nεI
y
3
2
y
3
2
2 3
3
2π
0
2π
2π
11π
6
7π
4
5π
3
θ
3π
2
4π
3
5π
4
π
7π
6
5π
6
3π
4
2π
3
π
2
π
3
π
4
1
0
π
6
2 3
3
-1
2
-2
-3
-3
Example 19: a) y = cotθ b) P = π
c) Domain: θ ε R, θ ≠ nπ, nεI; Range: yεR
d) θ = nπ, nεI
y
3
y
3
3
1
3
3
0
2π
2π
2π
11π
6
7π
4
θ
5π
3
3π
2
4π
3
5π
4
π
7π
6
5π
6
3π
4
2π
3
π
2
π
3
π
4
π
6
3
3
-1
3
-3
-3
Example 20:
a)
0
3
3
3
π
2π
-3
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
0
d)
c)
b)
π
2π
0
3
π
2π
-3
-3
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
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0
π
2π
-3
Domain: θ ε R, θ ≠ n(2π), nεI;
(or: θ ε R, θ ≠ ±n(2π), nεW)
Range: y ε R
Answer Key
Trigonometry Lesson Four: Trigonometric Functions II
Example 1:
Example 2:
a)
b)
a)
b)
y
y
h
y
2
2
2
2
1
1
1
1
π
2π
θ
3π
540º θ
360º
180º
15
30
45
60 t
8
-1
-1
-1
-1
-2
-2
-2
-2
Example 3:
Example 5:
a)
b)
1
2
3
4
5
6
7
8
a)
b)
y
x
-2
-4
-6
-8
-10
-12
-14
-16
-18
-20
-22
-24
-26
y
75
c)
50
25
-225
-175
-200
-125
-150
-75
-100
-25
-50
0
25
75
50
125
100
175
150
225
200
275
x
d)
250
Example 6:
a)
b)
Example 4:
c)
a)
b)
d)
e)
y
20
18
16
14
12
10
8
6
4
2
0
-2
-4
-6
-8
y
20
18
16
14
12
10
8
6
4
2
10
20
30
40
50
60
70
80
90
100 110
120
x
-3
-2
-1
Example 7:
a)
1
2
3
4
b)
5 x
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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5
16
24
x
Answer Key
Example 9:
Example 13:
a)
a) Decimal daylight hours: 6.77 h, 12.28 h, 17.82 h, 12.28 h, 6.77 h
h(t)
105
b)
75
d(n)
 2π
c) d ( n ) = −5.525cos 
( n + 11) + 12.295
 365

24
45
20
5
12
t
10
e) 64 days
d) 15.86 h
16
8
b)
c) If the wind
turbine rotates
counterclockwise,
we still get the
same graph.
-50
0
50
100
150
200
250
300
350
n
400
Example 14:
a) Decimal hours past midnight: 2.20 h, 8.20 h, 14.20 h, 20.20 h
Example 10:
a)
4
b)
h(θ)
h(t)
c)
16
d) 10.75 m
12
e) 32.3%
8
4
θ
b)
,
c) The angle of elevation
increases quickly at first,
but slows down as the
helicopter reaches greater
heights. The angle never
actually reaches 90°.
0
a)
8
12
16
20
t
24
b) See Video.
Population
16000
Example 11:
M(t)
12000
8000
300
h(t)
Owls
a)
4
Example 15:
Mice
90°
5.2
O(t)
250
200
4.0
2.8
0
1
2
3
4
5
6
7
8
Time
(years)
1
2
b)
t
c) 2.86 m
d) 0.26 s
Example 16:
2.5 m
h(t)
3.0
(8, 2.5)
1.5
Example 12:
a)
h(t)
15
31
Example 17:
15.6 s and 18.3 s
16
45
60
45
60
t
h(t)
19
1
(18.3, 13.1)
0
b)
30
25
50
75
100
125
150
175
200
t
10
c) 28.14 m d) 26.78 s
(15.6, 10.5)
1
15
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30
t
Answer Key
Trigonometry Lesson Five: Trigonometric Equations
Note: n ε I for all general solutions.
Example 1:
a)
b)
,
c)
,
d)
Example 2:
a)
b)
c)
1
1
1
2π
π
2π
π
-1
-1
-1
d) no solution
e)
f)
2
1
π
3
3
2
2
1
1
2π
π
2π
-1
-1
-2
-2
-2
-3
-3
-1
π
2π
π
2π
Example 3:
a)
b)
c)
90°
150°
30°
180°
45°
150°
30°
30°
90°
90°
0°
360°
180°
30°
0°
30°
360°
45°
45°
180°
0°
360°
210°
225°
270°
270°
270°
Example 4:
a)
3
2
b)
Intersection point(s)
of original equation
1
-1
3
3
2
θ-intercepts
1
π
2π
-1
-2
-2
-3
-3
π
2
Intersection point(s)
of original equation
1
2π
-1
3
2
π
2π
-1
-2
-2
-3
-3
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θ-intercepts
1
π
2π
Answer Key
Example 5:
a)
b)
c)
120°
60°
d)
3
3
2
2
1
1
60°
π
-1
240°
2π
π
-1
-2
-2
-3
-3
2π
Example 6:
a) 197.46° and 342.54°
b) 197.46° and 342.54°
c) 197.46° and 342.54°
The unit circle is not
useful for this question.
17.46°
d) 197.46° and 342.54°
3
3
2
2
1
1
17.46°
197.46°
342.54°
π
-1
Example 7:
a)
2π
-2
-2
-3
-3
b)
π
-1
c)
Example 8:
a) No Solution
b)
c)
2
2
2
1
1
1
π
π
2π
2π
-1
-1
-1
-2
-2
-2
d)
e)
f)
2
2
2
1
1
1
π
2π
π
2π
-1
-1
-1
-2
-2
-2
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π
2π
π
2π
2π
Answer Key
Example 9:
a)
b)
c)
90°
180°
90°
90°
120°
120°
60°
0°
60°
360°
60°
60°
60°
0°
60°
180°
60°
60°
180°
360°
240°
0°
360°
240°
270°
270°
270°
Example 10:
a) No Solution
b)
Intersection point(s)
of original equation
Intersection point(s)
of original equation
θ-intercepts
3
3
3
2
2
2
1
1
1
π
-1
2π
-1
π
2π
-1
-2
-2
-2
-3
-3
-3
θ-intercepts
3
2
1
π
2π
π
-1
2π
-2
-3
Example 11:
a)
b)
c)
d)
3
3
2
2
1
30°
210°
1
30°
330°
π
-1
2π
-1
-2
π
2π
-2
-3
-3
Example 12:
a)
b)
115°
The unit circle is not
useful for this question.
65°
65°
c)
d)
3
2
3
1
2
π
-1
245°
1
2π
-2
-1
-3
-2
-3
Example 13:
a)
b)
c)
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d)
π
2π
Answer Key
Example 14:
a)
b)
c)
d)
1
1
1
1
π
2π
π
2π
π
2π
-1
-1
-1
a)
b)
c)
d)
1
1
3
1
-1
π
2π
π
2π
Example 15:
2
1
π
2π
π
2π
π
-1
2π
-2
-1
-1
-3
a)
b)
c)
3
3
3
2
2
2
1
1
1
-1
Example 16:
π
-1
2π
π
-1
2π
π
-1
-2
-2
-2
-3
-3
-3
Example 17:
2π
Example 18:
a)
1
b)
a)
b)
1
1
1
2π
π
π
2
3π
2
π
2π
π
-1
-1
2π
-1
Example 20: a)
b) Approximately 12 days.
b) See graph. c) 0.4636 rad (or 26.6°)
d
4
2
0.50
0.4636
-4
0
7
14
21
28 t
π
2
π
2π
Example 21: See Video
8
1.00
4π
-1
Example 19: a)
Visible %
3π
3π
2
2π
θ
-8
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4π
6π
8π
Answer Key
Trigonometry Lesson Six: Trigonometric Identities I
Example 1: a)
Identity
Equation
3
2
1
Not an Identity
π
2π
-1
π
-1
-2
-3
-3
2π
v)
iv)
iii)
Not an Identity
2π
2
-2
ii)
1
3
1
π
-1
b) i)
Note: n ε I for all general solutions.
Identity
Not an Identity
Identity
3
3
3
3
2
2
2
2
1
1
1
-1
π
2π
-1
π
2π
-1
1
π
2π
π
-1
-2
-2
-2
-2
-3
-3
-3
-3
2π
Example 2:
b) Verify that the
L.S. = R.S. for
each angle.
a)
Use basic trigonometry
(SOHCAHTOA) to show
that x = cosθ and y = sinθ.
1
c) The graphs of y = sin2x + cos2x and y = 1 are the same.
1
y
θ
x
π
2π
-1
e) Verify that the
L.S. = R.S. for
each angle.
d) Divide both sides of
sin2x + cos2x = 1 by
sin2x to get 1 + cot2x = csc2x.
Divide both sides of
sin2x + cos2x = 1 by
cos2x to get tan2x + 1 = sec2x.
f) The graphs of y = 1 + cot2x
and y = csc2x are the same.
3
3
2
2
1
1
π
-1
Example 3:
a)
b)
3
1
2π
-2
-2
-3
-3
a)
1
π
π
2π
2π
,
b)
2π
-2
1
-1
-3
1
Example 4:
π
a)
3
3
2
π
2π
-1
-2
-2
-3
-3
2π
π
2π
-1
c)
1
1
π
2π
-1
b)
2
-1
π
-1
Example 5:
2
-1
The graphs of y = tan2x + 1
and y = sec2x are the same.
d)
1
π
1
2π
π
-1
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2π
-1
Answer Key
Example 6:
a)
b)
1
π
1
2π
-1
c)
d)
2
0
π
a)
b)
3
2
-1
2π
2
1
3
1
2π
-1
π
2π
π
2π
π
2π
π
2π
-2
-2
-3
-3
c)
d)
3
2
3
1
2
-1
π
2π
1
-2
-1
-3
-2
-3
Example 8:
a)
b)
3
2
2
1
3
1
-1
π
2π
-1
-2
-2
-3
-3
c)
d)
3
2
3
1
2
-1
π
2π
π
2π
1
Example 7:
π
2π
-1
1
-1
π
1
-2
-1
-3
-2
-3
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Answer Key
Example 9: See Video
Example 10: See Video
Example 11: See Video
Example 12:
Example 13:
Example 14:
a) See Video
a) See Video
a) See Video
b)
b)
b)
c)
c)
c)
d)
d)
d)
The graphs
are NOT identical.
The R.S. has holes.
1
π
2π
1
-1
-1
The graphs
are identical.
3
2
π
The graphs
are identical.
3
2
1
2π
π
-1
-2
-2
-3
-3
2π
Example 15:
a)
b)
,
3
3
2
2
1
-1
,
1
π
2π
-1
-2
-2
-3
-3
c)
π
d)
,
2π
,
1
6
4
2
-2
π
π
2π
2π
-4
-1
-6
Example 16:
a)
,
3
b)
,
10
2
1
-1
π
π
2π
2π
Note: All terms from the original
equation were collected on the
left side before graphing.
-2
-3
-10
c)
d)
,
2
1
π
-1
,
2π
π
2π
-2
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Answer Key
Example 17:
a)
b)
,
10
π
2π
Note: All terms from the
original equation were
collected on the left
side before graphing.
,
3
0
π
-3
2π
-6
Note: All terms from the
original equation were
collected on the left
side before graphing.
-9
-10
-12
c)
d)
,
,
1
3
2
1
-1
π
π
2π
2π
-2
-1
-3
Example 18:
a)
b)
7
4
c)
-2
-3
7
Example 19: See Video
www.math30.ca
Answer Key
Trigonometry Lesson Seven: Trigonometric Identities II
Note: n ε I for all general solutions.
Example 1:
a)
b)
c)
d)
e)
Example 2:
a)
Example 3:
b)
a)
c)
Example 4:
a)
b)
c)
d) See Video
Example 5: See Video
b)
c)
Example 20:
At 0°, the cannonball hits
the ground as soon as it
leaves the cannon, so the
horizontal distance is 0 m.
a)
Example 6:
a) i.
f)
b)
ii. 0 iii. undefined
b) (answers may vary)
c) (answers may vary)
i.
i.
ii.
ii.
iii.
iii.
iv.
iv.
Examples 7 - 13: Proofs. See Video.
Example 14:
Example 15:
a)
a)
b)
b)
c)
c)
d)
d)
d
132.2
At 45°, the cannonball hits
the ground at the maximum
horizontal distance, 132.2 m.
90°
180°
270°
360° θ
At 90°, the cannonball goes
straight up and down, landing
on the cannon at a horizontal
distance of 0 m
-132.2
c) θ = 24.6° and θ = 65.4°
Example 21:
a)
b)
The maximum area occurs
when θ = 45°. At this angle,
the rectangle is the top half
of a square.
A
4900
45°
c) i.
90° θ
ii.
iii.
Example 22:
Example 16:
Example 17:
a)
a)
b)
b)
c)
c)
d)
d)
Example 18: 57°
a) i.
y = f(θ) + g(θ)
6
0
b) i.
y = f(θ) + g(θ)
6
π
2π
-6
ii. The waves experience
constructive interference.
iii. The new sound will be louder
than either original sound.
0
π
-6
ii. The waves experience
destructive interference.
iii. The new sound will be quieter
than either original sound.
c) All of the terms subtract out leaving y = 0,
Example 19: 92.9
A flat line indicating no wave activity.
Example 23: See Video.
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2π
Example 24: See Video.
Answer Key
Permutations 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 + 3P3
Example 2: a) 24 b) 1 c) 1 d) (-2)! Does not exist.
e) 20 f) 4 g) n2 – n h) n2 + n
Example 3: a) 120 b) 24 c) 720 d) 13!
Example 4: a) 3 b) 415800 c) 12600 d) 20 e) 60 f) 10
Example 5: a) 8 b) 16 c) 1296 d) 32 × 106 e) 676 000
Example 6: a) 120 b) 48 c) 480 d) 108
Example 7: a) 24 b) 18 c) 12 d) 18 e) 3 f) 6
Example 8: a) 103 680 b) 240 c) 120 d) 144
Example 9: a) 72 b) 420 c) 1440
Example 10: a) 156 b) 1440 c) 20 d) 144 e) 72
Example 11: a) 24 b) 1320 c) 5P2 d) 3P2 or 3P3
Example 12: a) n = 6 b) n = 2 c) n = 5 d) n = 1
Example 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 + 4C4
Example 2: a) 10 b) 126 c) 2598960 d) 36; 84
Example 3: a) 13860 b) 720 c) 580008 d) 60
Example 4: a) 330 b) 70 c) 1680 d) 19600 e) 13244
Example 5: a) 3600 b) 180 c) 75600
Example 6: a) 66 b) 84 c) 70 d) 9
Example 7: a) 15 b) 20
Example 8: a) 81 b) 2594400 c) 2533180 d) 405 e) 31
Example 9: a) 21 b) 1 c) 6 d) 6C2 e) 5C1
Example 10: a) n = 7 b) 4C2 c) n = 5 d) n = 6
Example 11: a) n = 4 b) All n-values c) n = 4 d) n = 4
Example 12: a) 6760000 b) 40 c) 1645020 d) 144
e) See Video f) 84 g) 504 h) 32
Example 13: a) 16!/(2!)8 b) 10 c) 1800 d) 56
e) 120 f) 5040 g) 241098 h) 15
Example 14: a) 10080 b) 5 c) 8 d) 9
e) 92 f) 360 g) 241920 h) 6600
Example 15: a) 210 b) 720 c) 5148 d) 131560
e) 1024 f) 5 g) 14400 h) 123552
Example 16: a) 20 b) See Video c) 100 d) 1152
e) 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; 12C10
b) 21C2 = 210
c) k = 3 and 8, so the fourth and ninth positions have a value of 165.
d) 1024
Example 3:
a) 20
b) 120
c) 66
d) 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 nCkxn-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:
Example 7:
Example 8:
a)
a)
a)
b)
b)
c)
c)
b)
d)
c)
Example 9:
Example 10:
a)
a)
b)
3
b)
c)
c)
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