Name
Date
Chapter 4 Diagnostic Test
STUDENT BOOK PAGES 192–243
1. Solve the following linear equations.
a) 3x ⫹ 2 ⫽ ⫺2x ⫹ 8
2. Factor the following expressions.
a) 8x 3 ⫺ 27
b) x 5 ⫺ 16x
b)
2x ⫹ 1
3x ⫺ 1
⫽
3
2
c) x 3 ⫹ 6x 2 ⫺ 7x
d) x 2 (x ⫹ 2) ⫺ (x ⫹ 2)
3. Given the graph of the function f (x) ⫽ x 2 ⫹ 4x ⫺ 5 shown below, determine the roots of x 2 ⫹ 4x ⫺ 5 ⫽ 0.
Copyright © 2009 by Nelson Education Ltd.
4
2
–6 –4 –2 0
–2
–4
–6
–8
–10
y
x
2
4. Determine the roots of each of the following quadratic equations.
a) x 2 ⫹ 8x ⫹ 15 ⫽ 0
b) 2x 2 ⫺ 5x ⫽ 12
Chapter 4 Diagnostic Test
1
Chapter 4 Diagnostic Test Answers
6
5
b) x ⫽ 1
1. a) x ⫽
2. a) (2x ⫺ 3)(4x 2 ⫹ 6x ⫹ 9)
b) x(x 2 ⫹ 4)(x ⫹ 2)(x ⫺ 2)
c) x(x ⫹ 7)(x ⫺ 1)
d) (x ⫹ 1)(x ⫺ 1)(x ⫹ 2)
3. x ⫽ ⫺5, 1
4. a) x ⫽ ⫺5, ⫺3
3
b) x ⫽ ⫺ , 4
2
If students have difficulty with the questions on the Diagnostic Test, it may be necessary to review the
following topics:
• Solving linear equations.
• Factoring polynomials, including special forms (sum and difference of cubes, difference of squares).
• Determining the roots of a quadratic equation from a graph.
Copyright © 2009 by Nelson Education Ltd.
• Determining the roots of a quadratic equation algebraically.
2
Advanced Functions: Chapter 4 Diagnostic Test Answers
Lesson 4.1 Extra Practice
STUDENT BOOK PAGES 196–206
1. Use the factor theorem to determine the factors of the
following functions.
a) f (x) ⫽ x 3 ⫹ 2x 2 ⫺ 5x ⫺ 6
b) g(x) ⫽ x 3 ⫹ 2x 2 ⫺ 81x ⫺ 162
c) h(x) ⫽ 2x 3 ⫹ 10x 2 ⫺ 24x ⫺ 72
d) j(x) ⫽ 3x 3 ⫺ 39x 2 ⫹ 66x
e) k(x) ⫽ 12x 3 ⫹ 170x 2 ⫹ 302x ⫺ 120
f ) l(x) ⫽ 16x 4 ⫺ 8x 2 ⫹ 1
2. Determine the zeros of the following functions.
a) f (x) ⫽ x 3 ⫹ 2x 2 ⫺ 45x ⫺ 126
b) g(x) ⫽ 2x 4 ⫹ 9x 3 ⫹ 8x 2 ⫺ 9x ⫺ 10
c) h(x) ⫽ 3x 3 ⫹ 18x 2 ⫺ 48x ⫺ 288
d) j(x) ⫽ 36x 3 ⫺ 81x
e) k(x) ⫽ 2x 4 ⫺ 9x 2 ⫹ 4
f ) l(x) ⫽ 90x 4 ⫹ 625x 3 ⫺ 215x 2 ⫺ 1250x ⫹ 70
Copyright © 2009 by Nelson Education Ltd.
3. Use long division to determine if the following are
factors of the function
f (x) ⫽ 10x 4 ⫺ 17x 3 ⫺ 140x 2 ⫹ 147x.
a) 2x ⫹ 7
b) x ⫹ 1
c) 2x ⫺ 7
d) x ⫺ 1
e) 5x ⫺ 21
f) x ⫹ 3
4. Use graphing technology to determine the real roots.
Round to the nearest hundredth if necessary.
a) x 3 ⫹ 2x 2 ⫺ 45x ⫺ 126 ⫽ 0
b) 4x 4 ⫺ 4x 3 ⫺ x 2 ⫹ x ⫽ 0
c) 20x 3 ⫺ 152x 2 ⫽ 363x ⫹ 270
d) 6x 4 ⫹ 3x 3 ⫺ 12x 2 ⫺ x ⫽ ⫺4
e) 12x 4 ⫽ ⫺5x 3 ⫹ x ⫺ 5
f ) 1 ⫽ 3x 4 ⫹ 5x 3 ⫺ x 2 ⫺ x ⫺ 6
5. Determine the factors of the following functions by
grouping.
a) y ⫽ 3x 3 ⫺ 6x 2 ⫺ x ⫹ 2
b) y ⫽ 4x 3 ⫹ 16x 2 ⫺ 7x ⫺ 28
c) y ⫽ x 3 ⫺ x ⫺ 9x 2 ⫹ 9
d) y ⫽ 21x 3 ⫺ 15x
e) y ⫽ 24x 3 ⫺ 40x ⫹ 9x 2 ⫺ 15
f ) y ⫽ x 4 ⫺ 8x ⫹ 7x 3 ⫺ 56
6.
2
2x2 2 1
x13
A box has a length of 2x 2 ⫺ 1 units, a width of
x ⫹ 3 units, and a height of 2 units.
a) Write a function for the volume of the box in
terms of x.
b) Determine the values of x that will produce a zero
volume.
c) What are the length and width of the box if the
volume is 204 cm3?
d) What are the length(s) and width(s) of the box if
the volume is equal to 14 cm3?
e) If the height is doubled, how does the volume
change?
7. There are two roller coasters that have cars travelling
at the same time. The heights of the coasters
are modelled during 5 seconds. The height of
the first coaster in metres is modelled by
h(t) ⫽ t 3 ⫺ 6t 2 ⫹ 28t ⫺ 10. The height of
the second coaster in metres is modelled by
h(t) ⫽ t 2 ⫹ 14t ⫺ 2.
a) Determine the real roots of the function for the
height of the first coaster.
b) Determine the real roots of the function for the
height of the second coaster.
c) At what times are the two coasters at the same
height?
d) What are the heights that correspond to the times
in part c)?
e) What is the maximum height of the first coaster
during the 5-second interval?
f ) What is the maximum height of the second coaster
during the 5-second interval?
Lesson 4.1 Extra Practice
3
Lesson 4.1 Extra Practice Answers
(x ⫺ 2) (x ⫹ 3) (x ⫹ 1)
(x ⫹ 9) (x ⫺ 9) (x ⫹ 2)
(2)(x ⫹ 2) (x ⫹ 6) (x ⫺ 3)
(3x)(x ⫺ 2) (x ⫺ 11)
(2)(2x ⫹ 5) (3x ⫺ 1) (x ⫹ 12)
(2x ⫹ 1) 2 (2x ⫺ 1) 2
2. a) x ⫽ ⫺6, ⫺3, 7
5
b) x ⫽ ⫺ , ⫺2, ⫺1, 1
2
c) x ⫽ ⫺6, ⫺4, 4
3
3
d) x ⫽ ⫺ , 0,
2
2
兹2 兹2
,2
e) x ⫽ ⫺2, ⫺
,
2
2
1
f ) x ⫽ ⫺7, ⫺兹2, , 兹2
18
3. a) yes
b) no
c) no
d) yes
e) yes
f ) no
4. a) x ⫽ ⫺6, ⫺3, 7
b) x ⫽ ⫺0.5, 0, ⫺0.5, 1
c) x ⫽ 1.5, 2.5, 3.6
d) x ⫽ ⫺1.54, ⫺0.64, 0.68, 1
e) no solutions
f ) x ⫽ ⫺2.03, 1.04
4
Advanced Functions: Lesson 4.1 Extra Practice Answers
5. a)
b)
c)
d)
e)
f)
(3x 2 ⫺ 1) (x ⫺ 2)
(4x 2 ⫺ 7) (x ⫹ 4)
(x ⫺ 1)(x ⫹ 1) (x ⫺ 9)
(3x)(7x 2 ⫺ 5)
(8x ⫹ 3) (3x 2 ⫺ 5)
(x ⫹ 7) (x ⫺ 2)(x 2 ⫹ 4x ⫹ 4)
6. a) V(x) ⫽ 4x 3 ⫹ 12x 2 ⫺ 2x ⫺ 6
兹2
b) x ⫽ ⫺3, ⫾
2
c) width ⫽ 6, length ⫽ 17
d) width ⫽ 1, length ⫽ 7 or
width ⬟ 4.16, length ⬟ 1.68,
or width ⫽ 0.85, length ⫽ 8.32
e) The volume doubles.
7. a) t ⬟ 0.39
b) t ⬟ ⫺14.14, 0.14
c) t ⫽ 1, 2, 4 s
d) h ⫽ 13, 30, 70 m
e) h ⫽ 105 m
f ) h ⫽ 93 m
Copyright © 2009 by Nelson Education Ltd.
1. a)
b)
c)
d)
e)
f)
Lesson 4.2 Extra Practice
STUDENT BOOK PAGES 207–215
1. Choose the correct solution in set notation for the
following inequalities.
a) 9x ⫺ 4 ⬍ (7 ⫹ 8x) ⫺ 4
{x苸R 冷 x ⱕ 7 } or {x苸R 冷 x ⬍ 7}
b) 3(x ⫺ 2) ⱖ 6
{x苸R 冷 x ⱖ 4 } or {x苸R 冷 x ⱖ 0}
c) 2(6x ⫺ 1) ⬍⫺2(15 ⫺ x)
14
e x苸R x ⬍ ⫺ f or {x苸R冷 x ⬎⫺2}
5
d) ⫺3(x ⫹ 5) ⬎ 17 ⫺ x
{x苸R 冷 x ⬎⫺16 } or {x苸R 冷 x ⬍⫺16}
e) ⫺18 ⬍ 6(2x ⫺ 1) ⫹ 3(x ⫹ 1) ⬍ 42
{x苸R 冷 ⫺5 ⬍ x ⬍ 15 } or
{x苸R 冷 ⫺1 ⬍ x ⬍ 3 }
f ) 18 ⱕ ⫺3(x ⫺ 7) ⱕ 27
{x苸R 冷 ⫺2 ⱕ x ⱕ 1 } or
{x苸R 冷 ⫺2 ⱖ x ⱖ ⫺1 }
Copyright © 2009 by Nelson Education Ltd.
冟
2. Solve the following inequalities algebraically.
Express your solutions in set notation.
a) 2(7 ⫹ x) ⬍ 12
b) ⫺2(x ⫹ 2) ⱖ (x ⫹ 2)(x ⫹ 2)
c) 5x ⱖ 4 ⫺ 3x
d) 17 ⬍ 3(x ⫹ 2) ⫺ 1 ⬍ 44
e) ⫺1 ⱕ ⫺3(x ⫹ 3) ⫺ 9x ⱕ 9
x⫺4
ⱕ1
f) 0 ⱕ
5
3. Solve the following inequalities algebraically.
Express your solutions in interval notation.
a) 3x ⫺ 4 ⬍ 11
b) 17 ⱖ 2x ⫺ 5
c) x ⫺ 4 ⬎ ⫺3x ⫹ 8
d) 12 ⱕ ⫺6x ⫹ 3(x ⫺ 1) ⱕ 18
e) ⫺8 ⬍ 2x ⫺ 4 ⬍ 4
3
f ) ⫺1 ⱕ x ⫺ 4 ⬍ 8
4
4. Determine if ⫺1 is a solution for the following
inequalities.
a) ⫺3x ⫹ 4 ⬎ 7
1
b) ⫺6 ⱕ (7x ⫹ 3)
2
c) ⫺3x 2 ⫹ 18x ⫺ 4 ⱖ 21x ⫺ 4
17
3x ⫺ 2
5
⬍⫺
d) ⫺ ⱕ
4
4
4
3
x
e) ⱖ ⫺ 3 ⱖ ⫺2
2
2
7
7
f ) 35 ⱖ ⫺ x ⱖ
2
2
5. Solve the inequalities algebraically and illustrate your
solution on a number line.
a) ⫺2x ⫹ 3 ⱕ 15
1
18
b) 3ax ⫹ b ⬍
5
5
1
c) ⫺12 ⱖ ⫺ x ⫺ 12
4
7
x⫹7
11
ⱕ
d) ⱕ
4
4
4
1
x
1
17
e) ⫺ ⱖ ⫹ ⱖ ⫺
12
3
4
12
f ) 18 ⱕ (3x ⫹ 4) ⫺ (6x ⫹ 2) ⬍ 21
6. Using the graph of the inequality provided, answer
the following. Given f (x) ⫽ 6x ⫺ 4 and
g(x) ⫽ ⫺x ⫺ 3:
g(x)
2
1
–4 –3 –2 –1 0
–1
–2
–3
–4
–5
y
f(x)
x
1 2 3
a) Write the inequality for f (x) ⱖ g(x).
b) Solve the inequality algebraically.
c) Determine if ⫺1 is a solution.
d) Determine if 1 is a solution.
e) Write the solution in set notation.
f ) Write the solution in interval notation.
Lesson 4.2 Extra Practice
5
Lesson 4.2 Extra Practice Answers
冟
14
f
5
d) {x苸R 冷 x ⬍⫺16 }
e) {x苸R 冷 ⫺1 ⬍ x ⬍ 3 }
f ) {x苸R 冷 ⫺2 ⱕ x ⱕ 1 }
c) e x苸R x ⬍ ⫺
2. a) {x苸R 冷 x ⬍⫺1 }
b) {x苸R 冷 ⫺4 ⱕ x ⱕ ⫺2 }
1
c) e x苸R x ⱖ f
2
d) {x苸R 冷 4 ⬍ x ⬍ 13 }
3
2
e) e x苸R ⫺ ⱕ x ⱕ ⫺ f
2
3
f ) {x苸R 冷 4 ⱕ x ⱕ 9 }
冟
冟
3. a) x苸(⫺q, 5)
b) x苸(⫺q, 11)
c) x苸(3, q)
d) x苸3⫺7, ⫺5 4
e) x苸(⫺2, 4)
f ) x苸34, 16)
4. a) x ⬍⫺1, Not a solution
15
b) x ⱖ ⫺ , Solution
7
c) ⫺1 ⱕ x ⱕ 0, Solution
d) ⫺5 ⱕ x ⬍ ⫺1, Not a solution
e) 2 ⱕ x ⱕ 9, Not a solution
f ) ⫺10 ⱕ x ⱕ ⫺1, Solution
6
Advanced Functions: Lesson 4.2 Extra Practice Answers
5. a)
b)
c)
d)
e)
f)
–6 –5 –4 –3 –2 –1 0 1 2 3 4
–5 –4 –3 –2 –1 0 1 2 3 4 5
–5 –4 –3 –2 –1 0 1 2 3 4 5
–5 –4 –3 –2 –1 0 1 2 3 4 5
–5 –4 –3 –2 –1 0 1 2 3 4 5
–7
–6
–5
–4
–3
6. a) 6x ⫺ 4 ⱖ ⫺x ⫺ 3
1
b) x ⱖ
7
c) Not a solution
d) Solution
1
e) e x苸R x ⱖ f
7
1
f ) x苸 c , qb
7
冟
Copyright © 2009 by Nelson Education Ltd.
1. a) {x苸R 冷 x ⬍ 7 }
b) {x苸R 冷 x ⱖ 4 }
Chapter 4 Mid-Chapter Review Extra Practice
STUDENT BOOK PAGES 216–218
1. Use graphing technology to determine the real roots
of the following. Round to the nearest hundredth.
a) y ⫽ x 3 ⫹ 2x 2 ⫺ 5x ⫺ 4
b) y ⫽ 2x 3 ⫺ 6x ⫺ 1
c) y ⫽ ⫺x 4 ⫹ 3x 3 ⫺ 5x 2 ⫹ 4
d) y ⫽ 6x 5 ⫺ 2x 3 ⫹ 1
e) y ⫽ x 4 ⫺ 8x 3 ⫹ 2x 2 ⫹ x ⫺ 3
f) y ⫽ x 8 ⫹ x ⫹ 5
2. Consider the following functions.
2
1
Copyright © 2009 by Nelson Education Ltd.
–2 –1 0
–1
–2
–3
–4
–5
y
1
2 3 4 5 x
f(x)
g(x)
a) Determine the equations for f (x) and g(x).
b) Solve the inequality f (x) ⬍ g(x) by examining
the graph.
c) Confirm your solution by solving the inequality
algebraically.
d) Solve the inequality g(x) ⱖ f (x) by examining
the graph.
e) Confirm your solution by solving the inequality
algebraically.
f) Write your solution to part e) in set and interval
notation.
3. Determine the real solutions of the following.
a) (x ⫺ 2)(x ⫹ 2)x ⫽ 0
b) (x 2 ⫹ 4)(3x ⫺ 9) ⫽ 0
c) 0 ⫽ 2x 3 ⫹ 8x 2
d) x 2 ⫹ 6x ⫹ 9 ⫽ 2x 2 ⫹ 10x ⫹ 4
e) 0 ⫽ (3x 3 ⫹ 12x 2 )(x ⫺ 1)
f) ⫺16 ⫽ ⫺17x 2 ⫹ x 4
4. Use either synthetic division or long division to
determine if the following are factors of the function
f (x) ⫽ 2x 4 ⫺ x 3 ⫺ 35x 2 ⫺ 2x ⫹ 120.
a) x ⫺ 4
b) 2x ⫹ 4
c) x ⫺ 5
d) 2x ⫹ 5
e) x ⫺ 2
f) x ⫹ 3
5. Solve the following inequalities algebraically. Express
your solutions in set and interval notation.
a) 3x ⫺ 8 ⬎ 5x ⫺ 1
x
13
b) ⫹ 7 ⬍
2
2
c) x ⫹ 2x ⱖ ⫺5x ⫺ 40
d) ⫺2 ⬍ (3x ⫹ 1) ⫹ 4x ⱕ 8
3x ⫹ 5
⬎ 3
e) 6 ⬎
9
⫺x ⫹ 2
⫹6ⱕ5
f) 1 ⱕ
4
6. Use an algebraic technique to determine the factors of
the following.
a) 2x 3 ⫹ 15x 2 ⫹ 16x ⫺ 12
b) x 3 ⫹ x 2 ⫺ x ⫺ 1
c) ⫺x 3 ⫹ 5x 2 ⫹ 36x
d) ⫺3x 4 ⫺ 15x 3 ⫹ 54x 2 ⫹ 60x ⫺ 168
e) 6x 4 ⫺ 58x 3 ⫹ 22x 2 ⫹ 122x ⫹ 36
f) 3x 4 ⫹ 12x 3 ⫹ 15x 2
7. Determine if 0 is a solution for the following
inequalities.
a) 3x ⫺ 5 ⬎ ⫺8
b) ⫺32 ⱖ 3x ⫺ 8
19
7
c) x ⫹ 3 ⬎
4
4
d) ⫺4 ⬎ ⫺4x ⫺ 8 ⬎ ⫺12
e) 5 ⬍ (3x ⫹ 5) ⫹ x ⱕ 20
f) 29 ⱖ ⫺2x ⫹ 7 ⬎ 7
Chapter 4 Mid-Chapter Review Extra Practice
7
Chapter 4 Mid-Chapter Review Extra Practice Answers
1. a) x ⫽ ⫺3.18, ⫺0.68, 1.86
b) x ⫽ ⫺1.64, ⫺0.17, 1.81
c) x ⫽ ⫺0.72, 1.18
d) x ⫽ ⫺0.81
e) x ⫽ ⫺0.68, 7.73
f) no solution
2. a) f (x) ⫽ x ⫺ 5 and g(x) ⫽ ⫺x ⫹ 1
b) x ⬍ 3
c) x ⬍ 3
d) x ⱕ 3
e) x ⱕ 3
f) {x苸R 冷 x ⱕ 3 } and x苸(⫺q, 3 4
3. a) x ⫽ ⫺2, 0, 2
b) x ⫽ 3
c) x ⫽ ⫺4, 0
d) x ⫽ ⫺5, 1
e) x ⫽ ⫺4, 0, 1
f) x ⫽ ⫺4, ⫺1, 1, 4
b) {x苸R 冷 x ⬍ ⫺1} and x苸(⫺q, ⫺1)
c) {x苸R 冷 x ⱖ ⫺5} and x苸3⫺5, q)
冟
22 49
22
49
⬍x⬍
f and x苸a ,
b
e) e x苸R 冟
3
3
3 3
3
3
d) e x苸R ⫺ ⬍ x ⱕ 1 f and x苸a⫺ , 1 d
7
7
f) {x苸R 冷 6 ⱕ x ⱕ 22} and x苸36, 224
6. a)
b)
c)
d)
e)
f)
(x ⫹ 2) (x ⫹ 6) (2x ⫺ 1)
(x ⫹ 1) (x ⫺ 1) (x ⫹ 1)
(x) (x ⫹ 4) (⫺x ⫹ 9)
(3)(x ⫹ 2) (x ⫺ 2) (x ⫹ 7) (⫺x ⫹ 2)
(2)(x ⫺ 9) (x ⫺ 2) (3x ⫹ 1) (x ⫹ 1)
(3x2 )(⫺x ⫹ 5) (x ⫹ 1)
7. a) yes
b) no
c) no
d) yes
e) no
f) no
Copyright © 2009 by Nelson Education Ltd.
4. a) yes
b) no
c) no
d) yes
e) yes
f) yes
冟
7
7
5. a) e x苸R x ⬍ ⫺ f and x苸a⫺q, ⫺ b
2
2
8
Advanced Functions: Chapter 4 Mid-Chapter Review Extra Practice Answers
Lesson 4.3 Extra Practice
STUDENT BOOK PAGES 219–228
1. Solve the following inequalities.
a) (x ⫺ 3)(x ⫺ 2) ⬎ 0
b) (2x ⫺ 4)(x ⫹ 1) ⱕ 0
c) (3x ⫺ 6)(x ⫹ 2)(x ⫺ 1) ⱕ 0
d) 2x(x ⫺ 1) ⫹ x(x 2 ⫺ 1) ⬎ 0
e) (x ⫹ 1)(x ⫺ 3)(x ⫺ 4) ⱖ 0
f) (3x ⫺ 9)(x ⫺ 2)x ⬍ 0
Copyright © 2009 by Nelson Education Ltd.
2. Use graphing technology and state the intervals when
f (x) ⱖ 0 in interval notation.
a) f (x) ⫽ x 3 ⫹ 2x 2 ⫺ 5x ⫺ 6
b) f (x) ⫽ x 2 ⫺ 3x ⫹ 2
c) f (x) ⫽ 2x 2 ⫹ 6x ⫺ 8
d) f (x) ⫽ 6x 4 ⫹ 28x 3 ⫺ 16x 2 ⫺ 28x ⫹ 10
e) f (x) ⫽ 4x 4 ⫺ 40x 3 ⫹ 55x 2 ⫹ 90x ⫺ 144
f) f (x) ⫽ 6x 5 ⫺ 32x 4 ⫺ 74x 3 ⫹ 28x 2
3. Given f (x) and g(x), use graphing technology to
determine the following. Write your solution in
interval notation. f (x) ⫽ x 3 ⫹ 4x 2 ⫹ x ⫺ 6 and
g(x) ⫽ x 2 ⫺ x ⫺ 6
a) f (x) ⫽ 0
b) g(x) ⫽ 0
c) f (x) ⬎ g(x)
d) f (x) ⱖ g(x)
e) g(x) ⬎ f (x)
f) g(x) ⱖ f (x)
4. Use graphing technology to solve the inequalities.
a) x 2 ⫺ 6x ⫹ 9 ⬎ 0
b) x 3 ⫹ 5x 2 ⫹ 2x ⫺ 8 ⱖ 16x ⫺ 8
c) x 2 ⫺ 9x ⫹ 20 ⱕ ⫺x 2 ⫹ 9x ⫺ 20
d) x 3 ⫺ 5x 2 ⫺ 13x ⫺ 7 ⬎ 0
e) 2x 3 ⫹ 21x 2 ⫹ 19x ⫺ 210 ⱖ 90
f) 2x 3 ⫹ 3x 2 ⫺ 8x ⫹ 3 ⱕ 0
5. Consider the following graph.
y
8
4
f (x)
x
–6 –5 –4 –3 –2 –1 0
–4
g(x)
–8
–12
–16
–20
1
2
a) Determine when f (x) ⬎ g(x). Express your
solution in interval notation.
b) Express your solution in set notation.
c) Determine if 0 is a solution to the inequality
f (x) ⬎ g(x) .
d) Determine when f (x) ⱕ g(x). Express your
solution in interval notation.
e) Express your solution in set notation.
f) Determine if 0 is a solution to the inequality
f (x) ⱕ g(x).
6. Consider the following graph.
4
3
2
1
–3 –2 –1 0
–1
–2
y
g(x)
f(x)
x
1
2 3
a) Determine when g(x) ⬎ f (x). Express your
solution in interval notation.
b) Express your solution in set notation.
c) Determine if 2 is a solution to the inequality
g(x) ⬎ f (x).
d) Determine when g(x) ⬍ f (x). Express your
solution in interval notation.
e) Express your solution in set notation.
f) Determine if 3 is a solution to the inequality
g(x) ⬍ f (x).
Lesson 4.3 Extra Practice
9
Lesson 4.3 Extra Practice Answers
1. a) x苸(⫺q, 2) and x苸(3, q)
b) x苸3⫺1, 2 4
c) x苸(⫺q, ⫺2 4 and x苸31, 2 4
d) x苸(⫺3, 0) and x苸(1, q )
e) x苸3⫺1, 3 4 and x苸34, q)
f) x苸(⫺q, 0) and x苸(2, 3)
2. a) x苸3⫺3, ⫺1 4 and x苸32, q )
b) x苸(⫺q, 1 4 and x苸32, q )
c) x苸(⫺q, ⫺4 4 and x苸31, q )
1
d) x苸(⫺q, ⫺5 4 and x苸 c ⫺1, d and
3
x苸31, q )
e) x苸(⫺q, ⫺1.5 4 and x苸3 1.5, 24 and x苸38, q )
f) x苸3⫺2, 0 4 and x苸37, q)
5. a) x苸(⫺5, ⫺1) and x苸(1, q )
b) {x苸R 冷 ⫺5 ⬍ x ⬍ ⫺1} and
{x苸R 冷x ⬎ 1}
c) no
d) x苸3⫺1, 1 4 and x苸(⫺q, ⫺5)
e) {x苸R 冷 ⫺1 ⱕ x ⱕ 1} and {x苸R 冷 x ⬍ ⫺5}
f) yes
6. a) x苸(⫺q, ⫺1) and x苸(⫺1, 2)
b) {x苸R 冷x ⬍ ⫺1} and
{x苸R 冷⫺1 ⬍ x ⬍ 2}
c) no
d) x苸(2, q )
e) {x苸R 冷x ⬎ 2}
f) yes
Copyright © 2009 by Nelson Education Ltd.
3. a) x ⫽ ⫺3, ⫺2, 1
b) x ⫽ ⫺2, 3
c) x苸(⫺2, ⫺1) and x苸(0, q )
d) x苸3⫺2, ⫺1 4 and x苸30, q )
e) x苸(⫺q, ⫺2) and x苸(⫺1, 0)
f) x苸(⫺q, ⫺2 4 and x苸3⫺1, 04
4. a) x苸(⫺q, 3) and x苸(3, q)
b) x苸(⫺7, 04 and x苸324, q )
c) x苸34, 54
d) x苸(7, q)
e) x苸33, q)
1
f) x苸(⫺q, ⫺34 and x苸 c , 1 d
2
10
Advanced Functions: Lesson 4.3 Extra Practice Answers
Lesson 4.4 Extra Practice
STUDENT BOOK PAGES 229–237
1. Determine the average rate of change from x ⫽ 3 to
x ⫽ 5 of the following functions.
a) f (x) ⫽ ⫺2x ⫺ 6
b) g(x) ⫽ x 4 ⫺ 1
c) h(x) ⫽ 6x 2 ⫺ 4x ⫹ 2
d) j(x) ⫽ x 3 ⫹ 2x 2 ⫺ 7x ⫹ 1
e) k(x) ⫽ 15
3
f) l(x) ⫽
x⫹7
Copyright © 2009 by Nelson Education Ltd.
2. Estimate the instantaneous rate of change at x ⫽ 4
and let h ⫽ 0.01 for the following functions.
a) f (x) ⫽ ⫺x ⫹ 5
b) g(x) ⫽ x 2 ⫺ 9x ⫹ 14
c) h(x) ⫽ 2x 3 ⫺ 3x 2 ⫺ 18x ⫺ 8
d) j(x) ⫽ x 4 ⫹ x 2 ⫺ 2
3
e) k(x) ⫽ x
f) l(x) ⫽ 兹x
3. Given the function
f (x) ⫽ 2x 4 ⫹ 4x 3 ⫺ 14x 2 ⫺ 16x ⫹ 24, use
graphing technology and state whether the
instantaneous rate of change is positive, negative,
or zero at the following values.
a) at x ⫽ ⫺4
b) at x ⫽ ⫺2.5
c) at x ⫽ ⫺1
d) at x ⫽ 1
e) at x ⫽ 0
f) at x ⫽ 1.5
5. Estimate the slope of the tangent line at x ⫽ 4 for
the following functions.
a) y ⫽ 6x ⫺ 2
b) y ⫽ 2x 2 ⫺ 4x ⫹ 1
c) y ⫽ x 3 ⫹ 7x ⫺ 1
x
d) y ⫽ ⫺ ⫺ 3
4
e) y ⫽ 2x 4 ⫺ x 3 ⫺ x 2 ⫹ 7
x5
f) y ⫽ ⫺
16
6. Consider the graph of the function f (x).
4
2
–4 –3 –2 –1 0
–2
f(x) –4
–6
–8
–10
y
x
1
a) Determine the average rate of change of f (x) on
the interval ⫺1 ⱕ x ⱕ 1.
b) Determine the equation of the secant line for
part a).
c) Determine the instantaneous rate of change at
x ⫽ ⫺3. Let h ⫽ 0.1.
d) Determine the instantaneous rate of change at
x ⫽ ⫺1. Let h ⫽ 0.01.
e) Determine the instantaneous rate of change at
x ⫽ 1. Let h ⫽ 0.001.
4. Consider the function, f (x) ⫽ x 2 ⫺ x ⫺ 20.
a) Determine the average rate of change in f (x) on
the interval 1 ⱕ x ⱕ 5.
b) Determine an equation for the secant line on the
same interval as part a).
c) Determine the instantaneous rate of change at
x ⫽ 1 and x ⫽ 2. Let h ⫽ 0.001.
d) Determine the instantaneous rate of change at
x ⫽ 3, x ⫽ 4, and x ⫽ 5. Let h ⫽ 0.001.
e) As x increases, what is the behaviour of the
instantaneous rate of change?
f) Let h ⫽ 0.0001. How does the instantaneous rate
of change at x ⫽ 1 compare to that in part c).
Lesson 4.4 Extra Practice
11
Lesson 4.4 Extra Practice Answers
1. a) ⫺2
b) 272
c) 44
d) 58
e) 0
f) ⫺0.025
2. a) ⫺1
b) ⫺1
c) 54
d) 264.97
e) ⫺0.1875
f) 0.25
5. a) 6
b) 12
c) 55
1
d) ⫺
4
e) 456
f) ⫺80
6. a) 5
b) y ⫽ 5x ⫺ 1
c) 7
d) ⫺1
e) 15
4. a) 5
b) y ⫽ 5x ⫺ 25
c) instantaneous rate of change ⫽ 1 at x ⫽ 1,
instantaneous rate of change ⫽ 3 at x ⫽ 2
d) instantaneous rate of change ⫽ 5 at x ⫽ 3,
instantaneous rate of change ⫽ 7 at x ⫽ 4, and
instantaneous rate of change ⫽ 9 at x ⫽ 5
e) As x increases by 1 the instantaneous rate increases
by 2.
f) The instantaneous rate of change is 1 and is the
same as in part c).
12
Advanced Functions: Lesson 4.4 Extra Practice Answers
Copyright © 2009 by Nelson Education Ltd.
3. a) negative
b) postive
c) positive
d) negative
e) negative
f) negative
Chapter 4 Review Extra Practice
STUDENT BOOK PAGES 238–241
1. Determine the zeros of the following functions
algebraically.
a) x 3 ⫽ 7x ⫹ 6
b) 16 ⫺ 12x ⫽ 8x 2 ⫺ 3x 3 ⫺ x 4
c) 2x 3 ⫺ 8x 2 ⫹ 2x ⫽ ⫺12
d) 2x 4 ⫽ ⫺6x 3 ⫹ 6x 2 ⫹ 14x ⫺ 12
e) ⫺20 ⫽ ⫺35x ⫹ 10x 2 ⫹ 5x 3
f) 7x 4 ⫺ 14x 2 ⫺ 56 ⫽ 21x 3 ⫺ 84x
Copyright © 2009 by Nelson Education Ltd.
2. Determine the zeros of the following functions with
graphing technology. Round to the nearest
hundredth, if necessary.
a) x 3 ⫹ 6x 2 ⫺ 5x ⫹ 12 ⫽ 0
b) x 4 ⫺ 5x 2 ⫹ 9x ⫽ 6
c) 3x 3 ⫹ x 2 ⫽ 12x
d) 2x 4 ⫽ x 3 ⫹ 1
e) 18 ⫽ ⫺7x 3 ⫹ x 2
1
7
f) ⫺ x 4 ⫹ x 3 ⫽ 9
2
2
3. Solve the following inequalities algebraically. Express
your solution in set notation.
a) 4x ⫺ 7 ⬎ 3x ⫺ 5
13
x ⬍⫺2 ⫹ 6x
b)
2
c) 3 ⫺ 4x ⱖ ⫺2 ⫺ 3x
5 ⫹ 3x
2x ⫹ 1
ⱖ
d)
2
4
7
x⫹6
⫹ 2x ⬍ ⫹ 2x
e)
3
3
6x
10 ⫹ 7x
ⱕ
f)
5
5
5. Solve the following inequalities.
a) (x ⫹ 2)(x ⫺ 4) ⬍ 0
b) (3x ⫺ 6)(x ⫺ 5) ⱖ 0
c) (2x ⫺ 2)(x ⫹ 6)(x ⫺ 1) ⬎ 0
d) ⫺2x(x ⫹ 4)(x ⫺ 6)(x ⫹ 1) ⱕ 0
e) (⫺3x ⫹ 9)(x ⫹ 1)(x ⫺ 4) ⬎ 0
f) x 2 (x ⫺ 4)(⫺5x ⫹ 10) ⱖ 0
6. Use graphing technology and state the intervals when
f (x) ⬍ 0. Express your solution in interval notation.
a) f (x) ⫽ x 3 ⫹ x 2 ⫺ 44x ⫺ 84
b) f (x) ⫽ x 4 ⫹ 12x 3 ⫹ 52x 2 ⫹ 96x ⫹ 64
c) f (x) ⫽ 8x 2 ⫹ 8x ⫺ 48
d) f (x) ⫽ 8x 4 ⫺ 56x 2 ⫹ 48x
e) f (x) ⫽ 12x 4 ⫺ 84x 2 ⫹ 72x
f) f (x) ⫽ 14x 5 ⫹ 14x 4 ⫺ 56x 3 ⫺ 56x 2
7. Determine the average rate of change of the following
functions on the interval ⫺2 ⱕ x ⱕ 2.
a) f (x) ⫽ ⫺3x ⫺ 7
b) g(x) ⫽ x 2 ⫺ x ⫹ 9
c) h(x) ⫽ x 3 ⫺ x 2 ⫹ x ⫺ 1
d) j(x) ⫽ ⫺2x 3 ⫹ 5x 2 ⫺ 5
e) k(x) ⫽ 5
f) l(x) ⫽ ⫺6x 2 ⫹ 7
8. Estimate the instantaneous rate of change at x ⫽ 1
and let h ⫽ 0.001 for the following functions.
a) f (x) ⫽ ⫺3x ⫹ 2
b) g(x) ⫽ ⫺x 2 ⫺ 8
c) h(x) ⫽ 2 x ⫹ 1
d) j(x) ⫽ 兹x ⫹ 7 ⫺ 5
e) k(x) ⫽ 2 冷 x ⫹ 4 冷 ⫺ 1
f) l(x) ⫽ 3x 4 ⫺ x 2 ⫹ 8x ⫺ 1
4. Solve the following double inequalities algebraically
and determine if x ⫽ 2 is a solution. Express your
solution in interval notation.
a) 2x ⫹ 1 ⬍ 3x ⫹ 2 ⬍ 9 ⫹ 2x
x
1
3
b) ⫺ ⬍ ⱕ
2
4
2
c) ⫺2 ⫹ 5x ⬎ 4x ⬎⫺8 ⫹ 5x
d) 3 ⱕ 3x ⫺ 6 ⱕ 39
e) x ⫹ 5 ⬍ 2x ⫹ 4 ⬍ x ⫹ 6
f) 0 ⱕ (3x ⫹ 6) ⫹ (x ⫺ 13) ⱕ 3
Chapter 4 Review Extra Practice
13
Chapter 4 Review Extra Practice Answers
1. a) x ⫽ ⫺2, ⫺1, 3
b) x ⫽ ⫺4, ⫺2, 1, 2
c) x ⫽ ⫺1, 2, 3
d) x ⫽ ⫺3, ⫺2, 1
e) x ⫽ ⫺4, 1
f) x ⫽ ⫺2, 1, 2
5. a) x苸(⫺2, 4)
b) x苸(⫺q, 2 4 and x苸35, q )
c) x苸(⫺6, 1) and x苸(1, q )
d) x苸(⫺q, ⫺44 and x苸3⫺1, 0 4 and x苸36, q)
e) x苸(⫺q, ⫺1) and x苸(3, 4)
f) x苸32, 44 and x ⫽ 0
2. a) x ⫽ ⫺6.97
b) x ⫽ ⫺2.96, 1.28
c) x ⫽ ⫺2.17, 0, 1.84
d) x ⫽ ⫺0.74, 1
e) x ⫽ ⫺1.32
f) x ⫽ 1.48, 6.95
6. a) x苸(⫺q, ⫺6) and x苸(⫺2, 7)
b) no solution
c) x苸(⫺3, 2)
d) x苸(⫺3, 0) and x苸(1, 2)
e) x苸(⫺3, 0) and x苸(1, 2)
f) x苸(⫺q, ⫺2) and x苸(⫺1, 0) and x苸(0, 2)
{x苸R 冷 x ⬎ 2 }
{x苸R 冷 x ⬍⫺4 }
{x苸R 冷 x ⱕ 5 }
{x苸R 冷 x ⱖ 3 }
{x苸R 冷 x ⬍ 1 }
{x苸R 冷 x ⱖ ⫺10 }
4. a) x ⫽ 2 is a solution and x苸(⫺1, 7)
b) x ⫽ 2 is a solution and x苸(⫺6, 24
c) x ⫽ 2 is not a solution and x苸(2, 8)
d) x ⫽ 2 is not a solution and x苸33, 15 4
e) x ⫽ 2 is not a solution and x苸(1, 2)
7 5
f) x ⫽ 2 is a solution and x苸 c , d
4 2
14
7. a) ⫺3
b) ⫺1
c) 5
d) ⫺8
e) 0
f) 0
8. a) ⫺3
b) ⫺2
c) 1.39
d) 0.18
e) 2
f) 18
Advanced Functions: Chapter 4 Review Extra Practice Answers
Copyright © 2009 by Nelson Education Ltd.
3. a)
b)
c)
d)
e)
f)