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Vertical Stretching or Shrinking of the Graph of a Function

Suppose that a > 0. If a point (x, y) lies on the graph of y = (x), then the point (x, ay) lies on the graph of y = a(x).a.If a > 1, then the graph of y = a(x) is a vertical stretching of the graph of y = (x).b.If 0 < a <1, then the graph of y = a(x) is a vertical shrinking of the graph of y = (x)

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Horizontal Stretching or Shrinking of the Graph of a Function

Suppose a > 0. If a point (x, y) lies on the graph of y = (x), then the point ( , y) lies on the graph of y = (ax).a.If 0 < a < 1, then the graph of y = (ax) is horizontal stretching of the graph of y = (x).b.If a > 1, then the graph of y = (ax) is a horizontal shrinking of the graph of y = (x).

xa

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Reflecting

Forming a mirror image of a graph across a line is called reflecting the graph across the line.

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Example 2 REFLECTING A GRAPH ACROSS AN AXIS

Graph the function.a. ( )x xg

x (x) g(x)

0 0 0

1 1 – 1

4 2 – 2

x x

2 3

1

2

– 2

3

– 2

4

– 3

– 4

– 3– 4

4

x

y

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Example 2 REFLECTING A GRAPH ACROSS AN AXIS

Graph the function.b. ( )x x h

x (x) h(x)

– 4 undefined 2

– 1 undefined 1

0 0 0

1 1 undefined

4 2 undefined

x x

1 2 3

– 2

3

– 2

4

– 3

– 4

– 3– 4

4

x

y

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Reflecting Across an Axis

The graph of y = –(x) is the same as the graph of y = (x) reflected across the x-axis. (If a point (x, y) lies on the graph of y = (x), then (x, – y) lies on this reflection.The graph of y = (– x) is the same as the graph of y = (x) reflected across the y-axis. (If a point (x, y) lies on the graph of y = (x), then (– x, y) lies on this reflection.)

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Symmetry with Respect to An Axis

The graph of an equation is symmetric with respect to the y-axis if the replacement of x with – x results in an equivalent equation.The graph of an equation is symmetric with respect to the x-axis if the replacement of y with – y results in an equivalent equation.

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Symmetry with Respect to the OriginThe graph of an equation is symmetric with respect to the origin if the replacement of both x with – x and y with – y results in an equivalent equation.

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Important Concepts

1. A graph is symmetric with respect to both x- and y-axes is automatically symmetric with respect to the origin.

2. A graph symmetric with respect to the origin need not be symmetric with respect to either axis.

3. Of the three types of symmetry with respect to the x-axis, the y-axis, and the origin a graph possessing any two must also exhibit the third type.

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Symmetry with Respect to:

x-axis y-axis Origin

Equation is unchanged if:

y is replaced with – y

x is replaced with – x

x is replaced with – x and y is replaced with –

y Example

0x

y

0x

y

0x

y

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Even and Odd Functions

A function is called an even function if (– x) = (x) for all x in the domain of . (Its graph is symmetric with respect to the y-axis.)A function is called an odd function is (– x) = – (x) for all x in the domain of . (Its graph is symmetric with respect to the origin.)

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Example 5 DETERMINING WHETHER FUNCTIONS ARE EVEN, ODD, OR NEITHER

Decide whether each function defined is even, odd, or neither.

Solution Replacing x in (x) = 8x4 – 3x2 with – x gives:

a. 4 2( ) 8 3x x x

4 2 4 28( ) 3( ) 8 3( ) ( )x x x xx x

Since (– x) = (x) for each x in the domain of the function, is even.

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Example 5 DETERMINING WHETHER FUNCTIONS ARE EVEN, ODD, OR NEITHER

Decide whether each function defined is even, odd, or neither.

Solution

b. 3( ) 6 9x x x

3 9 )6 (x x x

The function is odd because (– x) = – (x).

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Example 5 DETERMINING WHETHER FUNCTIONS ARE EVEN, ODD, OR NEITHER

Decide whether each function defined is even, odd, or neither.

Solution

c. 2( ) 3 5x x x

2( ) 3 5x x

Since (– x) ≠ (x) and (– x) ≠ – (x), is neither even nor odd.

23 5x x x Replace x with – x

23 5x x

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Vertical Translations

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Horizontal Translations

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Horizontal Translations

If a function g is defined by g(x)= (x – c), where c is a real number, then for every point (x, y) on the graph of , there will be a corresponding point (x + c) on the graph of g. The graph of g will be the same as the graph of , but translated c units to the right if c is positive or c units to the left if c is negative. The graph is called a horizontal translation of the graph of .

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Caution Be careful when translating graphs horizontally. To determine the direction and magnitude of horizontal translations, find the value that would cause the expression in parentheses to equal 0. For example, the graph of y = (x – 5)2 would be translated 5 units to the right of y = x2, because x = + 5 would cause x – 5 to equal 0. On the other hand, the graph of y = (x + 5)2 would be translated 5 units to the left of y = x2, because x = – 5 would cause x + 5 to equal 0.

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Summary of Graphing TechniquesIn the descriptions that follow, assume that a > 0, h > 0, and k > 0. In comparison with the graph of y = (x):1.The graph of y = (x) + k is translated k units up.2.The graph of y = (x) – k is translated k units down.3.The graph of y = (x + h) is translated h units to the left.4.The graph of y = (x – h) is translated h units to the right. 5.The graph of y = a(x) is a vertical stretching of the graph of y = (x) if a > 1. It is a vertical shrinking if 0 < a < 1.6.The graph of y = a(x) is a horizontal stretching of the graph of y = (x) if 0 < a < 1. It is a horizontal shrinking if a > 1.7.The graph of y = – (x) is reflected across the x-axis.8.The graph of y = (– x) is reflected across the y-axis.