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TOPICS:

1. increasing and decreasing functions

2. even and odd functions

3. vertical and horizontal shifts

4. reflecting graphs

1. increasing and decreasing functions

2. even and odd functions

3. vertical and horizontal shifts

4. reflecting graphs

1.
Pre-Calculus Honors

1.3: Graphs of Functions

HW: p.37 (8, 12, 14, 23-26 all,

38-42 even, 80-84 even)

Copyright © Cengage Learning. All rights reserved.

1.3: Graphs of Functions

HW: p.37 (8, 12, 14, 23-26 all,

38-42 even, 80-84 even)

Copyright © Cengage Learning. All rights reserved.

2.
Increasing and Decreasing Functions

Determine the intervals on which each function is

increasing, decreasing, or constant.

(a) (b) (c)

2

Determine the intervals on which each function is

increasing, decreasing, or constant.

(a) (b) (c)

2

3.
Increasing and Decreasing

The more you know about the graph of a function, the more

you know about the function itself. Consider the graph

shown in Figure 1.20. Moving from left to right, this graph

falls from x = –2 to x = 0, is constant from x = 0 to x = 2,

and rises from x = 2 to x = 4.

Figure 1.20

3

The more you know about the graph of a function, the more

you know about the function itself. Consider the graph

shown in Figure 1.20. Moving from left to right, this graph

falls from x = –2 to x = 0, is constant from x = 0 to x = 2,

and rises from x = 2 to x = 4.

Figure 1.20

3

4.
Even and Odd Functions

• A function whose graph is symmetric with respect to the

y -axis is an even function.

• A function whose graph is symmetric with respect to the

origin is an odd function.

• A graph has symmetry with respect to the y-axis if

whenever (x, y) is on the graph, then so is the point (–x, y).

• A graph has symmetry with respect to the origin if whenever

(x, y) is on the graph, then so is the point (–x, –y).

• A graph has symmetry with respect to the x-axis if

whenever (x, y) is on the graph, then so is the point (x, –y).

4

• A function whose graph is symmetric with respect to the

y -axis is an even function.

• A function whose graph is symmetric with respect to the

origin is an odd function.

• A graph has symmetry with respect to the y-axis if

whenever (x, y) is on the graph, then so is the point (–x, y).

• A graph has symmetry with respect to the origin if whenever

(x, y) is on the graph, then so is the point (–x, –y).

• A graph has symmetry with respect to the x-axis if

whenever (x, y) is on the graph, then so is the point (x, –y).

4

5.
Even and Odd Functions

A graph that is symmetric with respect to the x-axis is not

the graph of a function (except for the graph of y = 0).

Symmetric to y-axis. Symmetric to origin. Symmetric to x-axis.

Even function. Odd function. Not a function.

5

A graph that is symmetric with respect to the x-axis is not

the graph of a function (except for the graph of y = 0).

Symmetric to y-axis. Symmetric to origin. Symmetric to x-axis.

Even function. Odd function. Not a function.

5

6.
Even and Odd Functions

Algebraic Test for Even and Odd Functions:

•A function f is even when, for each x in the domain of f,

f(-x) = f(x).

•A function f is odd when, for each x in the domain of f,

f(-x) = -f(x).

6

Algebraic Test for Even and Odd Functions:

•A function f is even when, for each x in the domain of f,

f(-x) = f(x).

•A function f is odd when, for each x in the domain of f,

f(-x) = -f(x).

6

7.
Example 10 – Even and Odd Functions

Determine whether each function is even, odd, or neither.

a. g(x) = x3 – x

b. h(x) = x2 + 1

c. f (x) = x3 – 1

a. This function is odd because

g (–x) = (–x)3+ (–x)

= –x3 + x

= –(x3 – x)

= –g(x).

7

Determine whether each function is even, odd, or neither.

a. g(x) = x3 – x

b. h(x) = x2 + 1

c. f (x) = x3 – 1

a. This function is odd because

g (–x) = (–x)3+ (–x)

= –x3 + x

= –(x3 – x)

= –g(x).

7

8.
Example 10 – Solution

b. h(x) = x2 + 1

b. This function is even because

h (–x) = (–x)2 + 1

= x2 + 1

= h (x).

c. f (x) = x3 – 1

c. Substituting –x for x produces

f (–x) = (–x)3 – 1

= –x3 – 1.

So, the function is neither even nor odd.

8

b. h(x) = x2 + 1

b. This function is even because

h (–x) = (–x)2 + 1

= x2 + 1

= h (x).

c. f (x) = x3 – 1

c. Substituting –x for x produces

f (–x) = (–x)3 – 1

= –x3 – 1.

So, the function is neither even nor odd.

8

9.
Pre-Calculus Honors

1.4: Shifting, Reflecting, and

Stretching Graphs

Copyright © Cengage Learning. All rights reserved.

9

1.4: Shifting, Reflecting, and

Stretching Graphs

Copyright © Cengage Learning. All rights reserved.

9

10.
Library of Parent Functions: Commonly Used Functions

Label important characteristics of each parent function.

2

f ( x) x f ( x) x f ( x) x

3 1

f ( x) x f ( x) x f ( x)

x

10

Label important characteristics of each parent function.

2

f ( x) x f ( x) x f ( x) x

3 1

f ( x) x f ( x) x f ( x)

x

10

11.
Vertical Shift

Change each function so it shifts

up 2 units from the parent function.

2

f ( x) x f ( x) x f ( x) x

3 1

f ( x) x f ( x) x f ( x)

x

11

Change each function so it shifts

up 2 units from the parent function.

2

f ( x) x f ( x) x f ( x) x

3 1

f ( x) x f ( x) x f ( x)

x

11

12.
Horizontal Shift

Change each function so it shifts

right 3 units from the parent function.

3 1

f ( x) x f ( x) x f ( x)

x

12

Change each function so it shifts

right 3 units from the parent function.

3 1

f ( x) x f ( x) x f ( x)

x

12

13.
Vertical and Horizontal Shifts

13

13

14.
Example 1 – Shifts in the Graph of a Function

Compare the graph of each function with the graph of

f (x) = x3.

a. g (x) = x3 – 1 b. h (x) = (x – 1)3 c. k (x) = (x + 2)3 + 1

a. You obtain the graph of g by shifting the graph of f one

unit downward.

Vertical shift: one unit downward

Figure 1.37(a)

14

Compare the graph of each function with the graph of

f (x) = x3.

a. g (x) = x3 – 1 b. h (x) = (x – 1)3 c. k (x) = (x + 2)3 + 1

a. You obtain the graph of g by shifting the graph of f one

unit downward.

Vertical shift: one unit downward

Figure 1.37(a)

14

15.
Example 1 – Solution cont’d

Compare the graph of each function with the graph of f (x) = x3.

b. h (x) = (x – 1)3 : You obtain the graph of h by shifting the

graph of f one unit to the right.

Horizontal shift: one unit right

Figure 1.37 (b)

15

Compare the graph of each function with the graph of f (x) = x3.

b. h (x) = (x – 1)3 : You obtain the graph of h by shifting the

graph of f one unit to the right.

Horizontal shift: one unit right

Figure 1.37 (b)

15

16.
Example 1 – Solution cont’d

Compare the graph of each function with the graph of

f (x) = x3.

c. k (x) = (x + 2)3 + 1 : You obtain the graph of k by shifting the

graph of f two units to the left and then one unit upward.

Two units left and one unit upward

Figure 1.37 (c)

16

Compare the graph of each function with the graph of

f (x) = x3.

c. k (x) = (x + 2)3 + 1 : You obtain the graph of k by shifting the

graph of f two units to the left and then one unit upward.

Two units left and one unit upward

Figure 1.37 (c)

16

17.
Reflecting Graphs

17

17

18.
Example 5 – Nonrigid Transformations

Compare the graph of each function with the graph of

f (x) = | x |.

a. h (x) = 3| x |

b. g (x) = | x |

a. Relative to the graph of

f (x) = | x |, the graph of

h (x) = 3| x | = 3f (x)

is a vertical stretch (each

y-value is multiplied by 3)

of the graph of f (See Figure 1.45.)

Figure 1.45 18

Compare the graph of each function with the graph of

f (x) = | x |.

a. h (x) = 3| x |

b. g (x) = | x |

a. Relative to the graph of

f (x) = | x |, the graph of

h (x) = 3| x | = 3f (x)

is a vertical stretch (each

y-value is multiplied by 3)

of the graph of f (See Figure 1.45.)

Figure 1.45 18

19.
Example 5 – Solution cont’d

b. Similarly, the graph of g (x) = | x | = f (x) is a vertical

shrink (each y-value is multiplied by ) of the graph of f .

(See Figure 1.46.)

Figure 1.46

19

b. Similarly, the graph of g (x) = | x | = f (x) is a vertical

shrink (each y-value is multiplied by ) of the graph of f .

(See Figure 1.46.)

Figure 1.46

19

20.
Pre-Calculus Honors

1.3: Step Functions and

Piecewise-Defined Functions

HW: p.38 (56-62 even)

Copyright © Cengage Learning. All rights reserved.

20

1.3: Step Functions and

Piecewise-Defined Functions

HW: p.38 (56-62 even)

Copyright © Cengage Learning. All rights reserved.

20

21.
Example 8 – Sketching a Piecewise-Defined Function

Sketch the graph of

f (x) = 2x + 3, x ≤ 1

–x + 4, x > 1

by hand.

21

Sketch the graph of

f (x) = 2x + 3, x ≤ 1

–x + 4, x > 1

by hand.

21

22.
Sketch the piecewise function.

4 x , x 0

f ( x)

4 x , x 0

22

4 x , x 0

f ( x)

4 x , x 0

22

23.
Do Now: Sketch the piecewise function.

2 x 1, x 1

f ( x) 2

x 2, x 1

23

2 x 1, x 1

f ( x) 2

x 2, x 1

23