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Pre-Calculus Honors1.3: Graphs of Functions

HW: p.37 (8, 12, 14, 23-26 all, 38-42 even, 80-84 even)

2

Increasing and Decreasing Functions

Determine the intervals on which each function is increasing, decreasing, or constant.

(a) (b) (c)

3

Increasing and Decreasing FunctionsThe more you know about the graph of a function, the more you know about the function itself. Consider the graphshown in Figure 1.20. Moving from left to right, this graphfalls 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

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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 ifwhenever (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).

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Even and Odd FunctionsA graph that is symmetric with respect to the x-axis is notthe graph of a function (except for the graph of y = 0).

Symmetric to y-axis.Even function.

Symmetric to origin.Odd function.

Symmetric to x-axis.Not a function.

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Even and Odd FunctionsAlgebraic 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).

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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

Solution:

a. This function is odd because

g (–x) = (–x)3+ (–x)

= –x3 + x

= –(x3 – x)

= –g(x).

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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.

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Pre-Calculus Honors1.4: Shifting, Reflecting, and

Stretching Graphs

10

Library of Parent Functions: Commonly Used Functions

Label important characteristics of each parent function.

xxfxxfxxf

xxfxxfxxf

1)( )( )(

)( )( )(

3

2

11

Vertical Shift

Change each function so it shifts

up 2 units from the parent function.

xxfxxfxxf

xxfxxfxxf

1)( )( )(

)( )( )(

3

2

12

Horizontal Shift

Change each function so it shifts

right 3 units from the parent function.

xxfxxfxxf

1)( )( )( 3

13

Vertical and Horizontal Shifts

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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

Solution:

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

Vertical shift: one unit downwardFigure 1.37(a)

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Example 1 – SolutionCompare 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)

cont’d

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Example 1 – SolutionCompare 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)

cont’d

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Reflecting Graphs

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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 |

Solution:

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

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Example 5 – Solution

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.)

cont’d

Figure 1.46

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Pre-Calculus Honors1.3: Step Functions and

Piecewise-Defined FunctionsHW: p.38 (56-62 even)

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Example 8 – Sketching a Piecewise-Defined Function

Sketch the graph of

2x + 3, x ≤ 1

–x + 4, x > 1

by hand.

f (x) =

22

Sketch the piecewise function.

0,4

0,4)(

xx

xxxf

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Do Now: Sketch the piecewise function.

1,2

1,12)(

2 xx

xxxf