COLLEGE ALGEBRA

2.7 - 1

10TH EDITION

LIAL

HORNSBY

SCHNEIDER

COLLEGE ALGEBRA

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2.7 Graphing TechniquesStretching and ShrinkingReflectingSymmetryEven and Odd FunctionsTranslations

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Example 1 STRETCHING OR SHRINKING A GRAPH

Graph the function

Solution Comparing the table of values for (x) = x and g(x) = 2x, we see that for corresponding x-values, the y-values of g are each twice those of . So the graph of g(x) is narrower than that of (x).

a. ( ) 2g x x

x (x)x

g(x)2x

– 2 2 4

– 1 1 2

0 0 0

1 1 2

2 2 4

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Graph the function

a. ( ) 2g x x

x (x)x

g(x)2x

– 2 2 4

– 1 1 2

0 0 0

1 1 2

2 2 4

1 2 3

– 2

3

– 2

4

– 3

– 4

– 3– 4

4

x

y

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Solution The graph of h(x) is also the same general shape as that of (x), but here the coefficient ½ causes the graph of h(x) to be wider than the graph of (x), as we see by comparing the tables of values.

b.1

( )2

h x x

x (x)|x|

h(x)½ |x|

– 2 2 1

– 1 1 ½

0 0 0

1 1 ½

2 2 1

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b.

x

y

1 2 3

– 2

3

– 2

4

– 3

– 4

– 3– 4

4

1( )

2h x x

x (x)|x|

h(x)½ |x|

– 2 2 1

– 1 1 ½

0 0 0

1 1 ½

2 2 1

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Solution Use Property 2 of absolute value

to rewrite

c. ( ) 2k x x

ab a b

2 .x

2 2 2k x x x x

Property 2

The graph of k(x) = 2 is the same as the graph of k(x) = 2xin part a.

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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.

Solution Every y-value of the graph is the negative of the corresponding y-value of

a. ( )x xg

( ) .x x

x (x) g(x)

0 0 0

1 1 – 1

2 2 – 2

x x

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Graph the function.a. ( )x xg

x (x) g(x)

0 0 0

1 1 – 1

2 2 – 2

x x

2 3

1

2

– 2

3

– 2

4

– 3

– 4

– 3– 4

4

x

y

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Graph the function.

Solution If we choose x-values for h(x) that are the negatives of those we use for (x), the corresponding y-values are the same. The graph h is a reflection of the graph across the y-axis.

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

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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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Example 3 TESTING FOR SYMMETRY WITH RESPECT TO AN AXIS

Test for symmetry with respect to the x-axis and the y-axis.

Solution Replace x with – x.

a. 2 4y x

2 4y x

24y x

2 4y x

Equivalent

2

– 2

4

x

y

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Solution The result is the same as the original equation and the graph is symmetric with respect to the y-axis.

a. 2 4y x

2 4y x

24y x

2 4y x

Equivalent

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Solution Replace y with – y.

b. 2 3x y

2 3x y

23x y

2 3y

Equivalent

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Solution The graph is symmetric with respect to the x-axis.

b. 2 3x y

2 3x y

23x y

2 3y

Equivalent 2

– 2

4

x

y

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Solution Substituting – x for x and – y in for y, we get (– x)2 + y2 = 16 and x2 + (– y)2 = 16. Both simplify to x2 + y2 = 16. The graph, a circle of radius 4 centered at the origin, is symmetric with respect to both axes.

c. 2 2 16x y

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Solution In 2x + y = 4, replace x with – x to get – 2x + y = 4. Replace y with – y to get 2x – y = 4. Neither case produces an equivalent equation, so this graph is not symmetric with respect to either axis.

d. 2 4x y

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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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Example 4 TESTING FOR SYMMETRY WITH RESPECT TO THE ORIGIN

Is this graph symmetric with respect to the origin?

Solution Replace x with – x and y with – y.

a. 2 2 16x y

2 2 16x y

2 216x y

2 2 16x y

Equivalent

Use parentheses around – x

and – y

The graph is symmetric with respect to the origin.

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Example 4 TESTING FOR SYMMETRY WITH RESPECT TO THE ORIGIN

Is this graph symmetric with respect to the origin?

Solution Replace x with – x and y with – y.

b. 3y x

3y x

3y x

3y x Equivalent

3y x

The graph is symmetric with respect to the origin.

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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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Solution

b. 3( ) 6 9x x x

3 9 )6 (x x x

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

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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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Example 6 TRANSLATING A GRAPH VERTICALLY

Solution

Graph g(x) = x– 4

x (x) g(x)

– 4 4 0

– 1 1 – 3

0 0 – 4

1 1 – 3

4 4 0

x 4x

4– 4

(0, – 4)

( ) 4x x g

( )x x

x

y

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Vertical Translations

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Example 7 TRANSLATING A GRAPH HORIZONTALLY

Graph g(x) = x – 4

Solution

x (x) g(x)

– 2 2 6

0 0 4

2 2 2

4 4 0

6 6 2

x 4x

4– 4

(0, – 4)

( ) 4x x g

( )x x

x

y

8

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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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Example 8 USING MORE THAN ONE TRNASFORMATION ON GRAPHS

Graph the function.

Solution To graph this, the lowest point on the graph of y = xis translated 3 units to the left and up one unit. The graph opens down because of the negative sign in front of the absolute value expression, making the lowest point now the highest point on the graph. The graph is symmetric with respect to the line x = – 3.

a. ( ) 3 1x x

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Graph the function.

a. ( ) 3 1x x

3

– 3– 2

– 6

( ) 3 1x x

x

y

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Graph the function.

Solution To determine the horizontal translation, factor out 2.

b. ( ) 2 4h x x

( ) 2 4h x x

22 x Factor out 2.

22 x ab a b

22 x 2 2

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Graph the function.

Solution Factor out 2.b. ( ) 2 4h x x

4

4

2

2

y x

( ) 2 4 2 2h x x x

x

y

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Graph the function.

Solution It will have the same shape as that of y = x2, but is wider (that is, shrunken vertically) and reflected across the x-axis because of the negative coefficient and then translated 4 units up.

c. 214

2x x g

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Graph the function.

Solution

c. 214

2x x g

4

3– 3

– 3

x

y

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Example 9 GRAPHING TRANSLATIONS GIVEN THE GRAPH OF y = (x)

Graph the function.

Solution The graph translates 3 units up.

a. ( ) ( ) 3x x g f( ) ( ) 3x x g f

– 4 – 2 3

3

0

x

y

( )y x f

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Graph the function.b. ( ) ( 3)x x h f

( ) ( 3)x x h f– 4 – 2 3

3

0

x

y

( )y x f

Solution The graph translates 3 units to the left.

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Graph the function.c. ( ) ( 2) 3x x k f

( ) ( 3)x x h f

– 4 – 2 5

3

0x

y

( )y x f

Solution The graph translates 2 units to the right and 3 units up.

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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 = (ax) 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.

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