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Slide 3 - 1 Copyright © 2009 Pearson Education, Inc. Active Learning Lecture Slides For use with Classroom Response Systems © 2009 Pearson Education, Inc. All rights reserved. Chapter 3 Functions and Their Graphs

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Page 1: Slide 3 - 1 Copyright © 2009 Pearson Education, Inc. Active Learning Lecture Slides For use with Classroom Response Systems © 2009 Pearson Education, Inc

Slide 3 - 1Copyright © 2009 Pearson Education, Inc.

Active Learning Lecture SlidesFor use with Classroom Response Systems

© 2009 Pearson Education, Inc.All rights reserved.

Chapter 3Functions and Their Graphs

Page 2: Slide 3 - 1 Copyright © 2009 Pearson Education, Inc. Active Learning Lecture Slides For use with Classroom Response Systems © 2009 Pearson Education, Inc

Slide 3 - 2Copyright © 2009 Pearson Education, Inc.

Determine whether the relation represents a function. If it is, state the domain and range.

{(–3, 13), (–2, 8), (0, 4), (2, 8), (4, 20)}

a. FunctionDomain: {–3, –2, 0, 2, 4}Range: {13, 8, 4, 20}

b. FunctionDomain: {–3, –2, 0, 2, 4}Range: {13, 8, 4, 20}

c. Not a function

Page 3: Slide 3 - 1 Copyright © 2009 Pearson Education, Inc. Active Learning Lecture Slides For use with Classroom Response Systems © 2009 Pearson Education, Inc

Slide 3 - 3Copyright © 2009 Pearson Education, Inc.

Determine whether the relation represents a function. If it is, state the domain and range.

{(–3, 13), (–2, 8), (0, 4), (2, 8), (4, 20)}

a. FunctionDomain: {–3, –2, 0, 2, 4}Range: {13, 8, 4, 20}

b. FunctionDomain: {–3, –2, 0, 2, 4}Range: {13, 8, 4, 20}

c. Not a function

Page 4: Slide 3 - 1 Copyright © 2009 Pearson Education, Inc. Active Learning Lecture Slides For use with Classroom Response Systems © 2009 Pearson Education, Inc

Slide 3 - 4Copyright © 2009 Pearson Education, Inc.

Determine whether the equation defines y as a function of x.

a. Function

b. Not a function

y

3x 1

x 2

Page 5: Slide 3 - 1 Copyright © 2009 Pearson Education, Inc. Active Learning Lecture Slides For use with Classroom Response Systems © 2009 Pearson Education, Inc

Slide 3 - 5Copyright © 2009 Pearson Education, Inc.

Determine whether the equation defines y as a function of x.

a. Function

b. Not a function

y

3x 1

x 2

Page 6: Slide 3 - 1 Copyright © 2009 Pearson Education, Inc. Active Learning Lecture Slides For use with Classroom Response Systems © 2009 Pearson Education, Inc

Slide 3 - 6Copyright © 2009 Pearson Education, Inc.

Find f (–x) when

a.

b.

c.

d.

f x 3x2 4x 3.

3x2 4x 3

3x2 4x 3

3x2 4x 3

3x2 4x 3

Page 7: Slide 3 - 1 Copyright © 2009 Pearson Education, Inc. Active Learning Lecture Slides For use with Classroom Response Systems © 2009 Pearson Education, Inc

Slide 3 - 7Copyright © 2009 Pearson Education, Inc.

Find f (–x) when

a.

b.

c.

d.

f x 3x2 4x 3.

3x2 4x 3

3x2 4x 3

3x2 4x 3

3x2 4x 3

Page 8: Slide 3 - 1 Copyright © 2009 Pearson Education, Inc. Active Learning Lecture Slides For use with Classroom Response Systems © 2009 Pearson Education, Inc

Slide 3 - 8Copyright © 2009 Pearson Education, Inc.

Find the domain of the function.

a. All real numbers

b.

c.

d.

h x x 4

x3 49x

x x 4

x x 0

x x 7,0,7

Page 9: Slide 3 - 1 Copyright © 2009 Pearson Education, Inc. Active Learning Lecture Slides For use with Classroom Response Systems © 2009 Pearson Education, Inc

Slide 3 - 9Copyright © 2009 Pearson Education, Inc.

Find the domain of the function.

a. All real numbers

b.

c.

d.

h x x 4

x3 49x

x x 4

x x 0

x x 7,0,7

Page 10: Slide 3 - 1 Copyright © 2009 Pearson Education, Inc. Active Learning Lecture Slides For use with Classroom Response Systems © 2009 Pearson Education, Inc

Slide 3 - 10Copyright © 2009 Pearson Education, Inc.

Given

a.

b.

c.

d.

f x 6 x

f g x x2 18; x x 18

g x x 3,

find f • g and state its domain.

and

f g x 6 x x 3 ; x x 0

f g x 6 x x 3 ; x x 3, x 6

f g x 6 x x 3 ; x 3 x 6

Page 11: Slide 3 - 1 Copyright © 2009 Pearson Education, Inc. Active Learning Lecture Slides For use with Classroom Response Systems © 2009 Pearson Education, Inc

Slide 3 - 11Copyright © 2009 Pearson Education, Inc.

Given

a.

b.

c.

d.

f x 6 x

f g x x2 18; x x 18

g x x 3,

find f • g and state its domain.

and

f g x 6 x x 3 ; x x 0

f g x 6 x x 3 ; x x 3, x 6

f g x 6 x x 3 ; x 3 x 6

Page 12: Slide 3 - 1 Copyright © 2009 Pearson Education, Inc. Active Learning Lecture Slides For use with Classroom Response Systems © 2009 Pearson Education, Inc

Slide 3 - 12Copyright © 2009 Pearson Education, Inc.

Given

a.

b.

c.

d.

f x 9x 7

2x 9

f g x 11x 7

2x 9; x x

9

2, x

7

11

g x 2x

2x 9,

find f + g and state its domain.

and

f g x 7x 7

2x 9; x x

9

2

f g x 11x 7

2x 9; x x

9

2

f g x 11x 7

2x 9; x x 0

Page 13: Slide 3 - 1 Copyright © 2009 Pearson Education, Inc. Active Learning Lecture Slides For use with Classroom Response Systems © 2009 Pearson Education, Inc

Slide 3 - 13Copyright © 2009 Pearson Education, Inc.

Given

a.

b.

c.

d.

f x 9x 7

2x 9

f g x 11x 7

2x 9; x x

9

2, x

7

11

g x 2x

2x 9,

find f + g and state its domain.

and

f g x 7x 7

2x 9; x x

9

2

f g x 11x 7

2x 9; x x

9

2

f g x 11x 7

2x 9; x x 0

Page 14: Slide 3 - 1 Copyright © 2009 Pearson Education, Inc. Active Learning Lecture Slides For use with Classroom Response Systems © 2009 Pearson Education, Inc

Slide 3 - 14Copyright © 2009 Pearson Education, Inc.

Find and simplify the difference quotient of

a. 2x + h + 5

b.

c. 2x + h – 7

d. 1

f x x2 5x 7,

f x h f x h

, h 0.

2x2 2x 2xh h2 h 14

h

Page 15: Slide 3 - 1 Copyright © 2009 Pearson Education, Inc. Active Learning Lecture Slides For use with Classroom Response Systems © 2009 Pearson Education, Inc

Slide 3 - 15Copyright © 2009 Pearson Education, Inc.

Find and simplify the difference quotient of

a. 2x + h + 5

b.

c. 2x + h – 7

d. 1

f x x2 5x 7,

f x h f x h

, h 0.

2x2 2x 2xh h2 h 14

h

Page 16: Slide 3 - 1 Copyright © 2009 Pearson Education, Inc. Active Learning Lecture Slides For use with Classroom Response Systems © 2009 Pearson Education, Inc

Slide 3 - 16Copyright © 2009 Pearson Education, Inc.

Determine whether the graph is that of a function. If it is, state the domain, range, intercepts and symmetry.

a. Function; D:{x|x ≥ –2}R: {y|y ≤ 0}Int: (–2, 0), 0, –2), (2, 0)Sym: none

c. Function; D: all realsR: all realsInt: (–2, 0), 0, –2), (2, 0)Sym: none

b. Function; D:{x|x ≥ –2}R: {y|y ≤ 0}Int: (–2, 0), 0, –2), (2, 0)Sym: none

d. Not a function

Page 17: Slide 3 - 1 Copyright © 2009 Pearson Education, Inc. Active Learning Lecture Slides For use with Classroom Response Systems © 2009 Pearson Education, Inc

Slide 3 - 17Copyright © 2009 Pearson Education, Inc.

Determine whether the graph is that of a function. If it is, state the domain, range, intercepts and symmetry.

a. Function; D:{x|x ≥ –2}R: {y|y ≤ 0}Int: (–2, 0), 0, –2), (2, 0)Sym: none

c. Function; D: all realsR: all realsInt: (–2, 0), 0, –2), (2, 0)Sym: none

b. Function; D:{x|x ≥ –2}R: {y|y ≤ 0}Int: (–2, 0), 0, –2), (2, 0)Sym: none

d. Not a function

Page 18: Slide 3 - 1 Copyright © 2009 Pearson Education, Inc. Active Learning Lecture Slides For use with Classroom Response Systems © 2009 Pearson Education, Inc

Slide 3 - 18Copyright © 2009 Pearson Education, Inc.

Here is the graph of f. For what numbers x is f (x) > 0?

a. [–10, –6), (7, 10)

b. (–6, 7)

c. (–6, ∞)

d. (–∞, –6)

Page 19: Slide 3 - 1 Copyright © 2009 Pearson Education, Inc. Active Learning Lecture Slides For use with Classroom Response Systems © 2009 Pearson Education, Inc

Slide 3 - 19Copyright © 2009 Pearson Education, Inc.

Here is the graph of f. For what numbers x is f (x) > 0?

a. [–10, –6), (7, 10)

b. (–6, 7)

c. (–6, ∞)

d. (–∞, –6)

Page 20: Slide 3 - 1 Copyright © 2009 Pearson Education, Inc. Active Learning Lecture Slides For use with Classroom Response Systems © 2009 Pearson Education, Inc

Slide 3 - 20Copyright © 2009 Pearson Education, Inc.

The graph of a function is given. Decide if it is even, odd, or neither.

a. even

b. odd

c. neither

Page 21: Slide 3 - 1 Copyright © 2009 Pearson Education, Inc. Active Learning Lecture Slides For use with Classroom Response Systems © 2009 Pearson Education, Inc

Slide 3 - 21Copyright © 2009 Pearson Education, Inc.

The graph of a function is given. Decide if it is even, odd, or neither.

a. even

b. odd

c. neither

Page 22: Slide 3 - 1 Copyright © 2009 Pearson Education, Inc. Active Learning Lecture Slides For use with Classroom Response Systems © 2009 Pearson Education, Inc

Slide 3 - 22Copyright © 2009 Pearson Education, Inc.

Determine algebraically whether the function is even, odd, or neither.

a. even

b. odd

c. neither

f x 2x

x

Page 23: Slide 3 - 1 Copyright © 2009 Pearson Education, Inc. Active Learning Lecture Slides For use with Classroom Response Systems © 2009 Pearson Education, Inc

Slide 3 - 23Copyright © 2009 Pearson Education, Inc.

Determine algebraically whether the function is even, odd, or neither.

a. even

b. odd

c. neither

f x 2x

x

Page 24: Slide 3 - 1 Copyright © 2009 Pearson Education, Inc. Active Learning Lecture Slides For use with Classroom Response Systems © 2009 Pearson Education, Inc

Slide 3 - 24Copyright © 2009 Pearson Education, Inc.

Determine whether the function is increasing, decreasing, or constant on the interval (–∞, –8).

a. increasing

b. decreasing

c. constant

Page 25: Slide 3 - 1 Copyright © 2009 Pearson Education, Inc. Active Learning Lecture Slides For use with Classroom Response Systems © 2009 Pearson Education, Inc

Slide 3 - 25Copyright © 2009 Pearson Education, Inc.

Determine whether the function is increasing, decreasing, or constant on the interval (–∞, –8).

a. increasing

b. decreasing

c. constant

Page 26: Slide 3 - 1 Copyright © 2009 Pearson Education, Inc. Active Learning Lecture Slides For use with Classroom Response Systems © 2009 Pearson Education, Inc

Slide 3 - 26Copyright © 2009 Pearson Education, Inc.

Find the values for which f has local maxima and give the local maxima value.

a. at x = 5 loc. max. is –8; at x = 3.9, it is 2

b.at x = –8 loc. max. is 5; at x = 2.2, it is 3.9

c. at x = –3.3 loc. max. is –2.5; at x = –2.5, it is 5

d.at x = –2.5 loc. max. is –3.3; at x = 5, it is –2.5

Page 27: Slide 3 - 1 Copyright © 2009 Pearson Education, Inc. Active Learning Lecture Slides For use with Classroom Response Systems © 2009 Pearson Education, Inc

Slide 3 - 27Copyright © 2009 Pearson Education, Inc.

Find the values for which f has local maxima and give the local maxima value.

a. at x = 5 loc. max. is –8; at x = 3.9, it is 2

b.at x = –8 loc. max. is 5; at x = 2.2, it is 3.9

c. at x = –3.3 loc. max. is –2.5; at x = –2.5, it is 5

d.at x = –2.5 loc. max. is –3.3; at x = 5, it is –2.5

Page 28: Slide 3 - 1 Copyright © 2009 Pearson Education, Inc. Active Learning Lecture Slides For use with Classroom Response Systems © 2009 Pearson Education, Inc

Slide 3 - 28Copyright © 2009 Pearson Education, Inc.

Use a graphing utility to approximate any local maxima and local minima and where the graph is increasing and decreasing on the given interval.

a. loc max at (2, –1) loc min at (0, 3) incr on (–1, 0) & (2, 3)decr on (0, 2)

c. loc max at (0, 3)loc min at (2, –1)incr on (0, 2) decr on (–1, 0) & (2, 3)

f x x3 3x2 3; 1,3

b. loc max at (0, 3)loc min at (2, –1)incr on (–1, 0) & (2, 3)decr on (0, 2)

d. loc max at (2, –1) loc min at (0, 3)incr on (–1, 0)decr on (0, 2)

Page 29: Slide 3 - 1 Copyright © 2009 Pearson Education, Inc. Active Learning Lecture Slides For use with Classroom Response Systems © 2009 Pearson Education, Inc

Slide 3 - 29Copyright © 2009 Pearson Education, Inc.

Use a graphing utility to approximate any local maxima and local minima and where the graph is increasing and decreasing on the given interval.

a. loc max at (2, –1) loc min at (0, 3) incr on (–1, 0) & (2, 3)decr on (0, 2)

c. loc max at (0, 3)loc min at (2, –1)incr on (0, 2) decr on (–1, 0) & (2, 3)

f x x3 3x2 3; 1,3

b. loc max at (0, 3)loc min at (2, –1)incr on (–1, 0) & (2, 3)decr on (0, 2)

d. loc max at (2, –1) loc min at (0, 3)incr on (–1, 0)decr on (0, 2)

Page 30: Slide 3 - 1 Copyright © 2009 Pearson Education, Inc. Active Learning Lecture Slides For use with Classroom Response Systems © 2009 Pearson Education, Inc

Slide 3 - 30Copyright © 2009 Pearson Education, Inc.

Find the average rate of change for the function

a. –2

b.

c. –28

d.

from 1 to 5. f x 2x 1

1

2

1

6

Page 31: Slide 3 - 1 Copyright © 2009 Pearson Education, Inc. Active Learning Lecture Slides For use with Classroom Response Systems © 2009 Pearson Education, Inc

Slide 3 - 31Copyright © 2009 Pearson Education, Inc.

Find the average rate of change for the function

a. –2

b.

c. –28

d.

from 1 to 5. f x 2x 1

1

2

1

6

Page 32: Slide 3 - 1 Copyright © 2009 Pearson Education, Inc. Active Learning Lecture Slides For use with Classroom Response Systems © 2009 Pearson Education, Inc

Slide 3 - 32Copyright © 2009 Pearson Education, Inc.

Graph

a. b.

c. d.

f x x 5 if x 1

2 if x 1

Page 33: Slide 3 - 1 Copyright © 2009 Pearson Education, Inc. Active Learning Lecture Slides For use with Classroom Response Systems © 2009 Pearson Education, Inc

Slide 3 - 33Copyright © 2009 Pearson Education, Inc.

Graph

a. b.

c. d.

f x x 5 if x 1

2 if x 1

Page 34: Slide 3 - 1 Copyright © 2009 Pearson Education, Inc. Active Learning Lecture Slides For use with Classroom Response Systems © 2009 Pearson Education, Inc

Slide 3 - 34Copyright © 2009 Pearson Education, Inc.

Match the correct function to the graph.

a.

b.

c.

d.

y x 1

y x

y x 1

y x 1

Page 35: Slide 3 - 1 Copyright © 2009 Pearson Education, Inc. Active Learning Lecture Slides For use with Classroom Response Systems © 2009 Pearson Education, Inc

Slide 3 - 35Copyright © 2009 Pearson Education, Inc.

Match the correct function to the graph.

a.

b.

c.

d.

y x 1

y x

y x 1

y x 1

Page 36: Slide 3 - 1 Copyright © 2009 Pearson Education, Inc. Active Learning Lecture Slides For use with Classroom Response Systems © 2009 Pearson Education, Inc

Slide 3 - 36Copyright © 2009 Pearson Education, Inc.

Write an equation that results in the absolute value function, shifted 6 units upward.

a.

b.

c.

d.

y x 6

y x 6

y x 6

y x 6

Page 37: Slide 3 - 1 Copyright © 2009 Pearson Education, Inc. Active Learning Lecture Slides For use with Classroom Response Systems © 2009 Pearson Education, Inc

Slide 3 - 37Copyright © 2009 Pearson Education, Inc.

Write an equation that results in the absolute value function, shifted 6 units upward.

a.

b.

c.

d.

y x 6

y x 6

y x 6

y x 6

Page 38: Slide 3 - 1 Copyright © 2009 Pearson Education, Inc. Active Learning Lecture Slides For use with Classroom Response Systems © 2009 Pearson Education, Inc

Slide 3 - 38Copyright © 2009 Pearson Education, Inc.

Graph

a. b.

c. d.

y 1

4x .

Page 39: Slide 3 - 1 Copyright © 2009 Pearson Education, Inc. Active Learning Lecture Slides For use with Classroom Response Systems © 2009 Pearson Education, Inc

Slide 3 - 39Copyright © 2009 Pearson Education, Inc.

Graph

a. b.

c. d.

y 1

4x .

Page 40: Slide 3 - 1 Copyright © 2009 Pearson Education, Inc. Active Learning Lecture Slides For use with Classroom Response Systems © 2009 Pearson Education, Inc

Slide 3 - 40Copyright © 2009 Pearson Education, Inc.

Graph

a. b.

c. d.

f x x

Page 41: Slide 3 - 1 Copyright © 2009 Pearson Education, Inc. Active Learning Lecture Slides For use with Classroom Response Systems © 2009 Pearson Education, Inc

Slide 3 - 41Copyright © 2009 Pearson Education, Inc.

Graph

a. b.

c. d.

f x x

Page 42: Slide 3 - 1 Copyright © 2009 Pearson Education, Inc. Active Learning Lecture Slides For use with Classroom Response Systems © 2009 Pearson Education, Inc

Slide 3 - 42Copyright © 2009 Pearson Education, Inc.

The figure shows a rectangle inscribed in an isosceles right triangle whose hypotenuse is 6 units long. Express the area A of the rectangle in terms of x.

a. b.

c. d.

A x 2x 3 x A x x 3 x

A x 2x2A x 2x x 3

Page 43: Slide 3 - 1 Copyright © 2009 Pearson Education, Inc. Active Learning Lecture Slides For use with Classroom Response Systems © 2009 Pearson Education, Inc

Slide 3 - 43Copyright © 2009 Pearson Education, Inc.

The figure shows a rectangle inscribed in an isosceles right triangle whose hypotenuse is 6 units long. Express the area A of the rectangle in terms of x.

a. b.

c. d.

A x 2x 3 x A x x 3 x

A x 2x2A x 2x x 3

Page 44: Slide 3 - 1 Copyright © 2009 Pearson Education, Inc. Active Learning Lecture Slides For use with Classroom Response Systems © 2009 Pearson Education, Inc

Slide 3 - 44Copyright © 2009 Pearson Education, Inc.

A box with an open top is to be constructed from a rectangular piece of cardboard with dimensions 15 inches by 30 inches by cutting out equal squares of sides x at each corner and then folding up the sides.

a.

b.

c.

d.

V x 15 2x 30 2x V x x 15 x 30 x

V x x 15 2x 30 2x

Express the volume V of the box as a function of x.

V x 15 x 30 x

Page 45: Slide 3 - 1 Copyright © 2009 Pearson Education, Inc. Active Learning Lecture Slides For use with Classroom Response Systems © 2009 Pearson Education, Inc

Slide 3 - 45Copyright © 2009 Pearson Education, Inc.

A box with an open top is to be constructed from a rectangular piece of cardboard with dimensions 15 inches by 30 inches by cutting out equal squares of sides x at each corner and then folding up the sides.

a.

b.

c.

d.

V x 15 2x 30 2x V x x 15 x 30 x

V x x 15 2x 30 2x

Express the volume V of the box as a function of x.

V x 15 x 30 x