Copyright © 2013, 2009, 2005 Pearson Education, Inc. 1 2 Graphs and Functions Copyright © 2013,...

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 1

2Graphs and Functions

Copyright © 2013, 2009, 2005 Pearson Education, Inc.


2.7 Graphing Techniques• Stretching and Shrinking• Reflecting• Symmetry• Even and Odd Functions• Translations


Example 1 STRETCHING OR SHRINKING A GRAPH

Graph each 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 xx (x) = x g(x) = 2x

– 2 2 4

– 1 1 2

0 0 0

1 1 2

2 2 4



Graph each function. (a) ( ) 2g x x

1 2 3

– 2

3

– 2

4

– 3

– 4

– 3– 4

4

x

y

x (x) = x g(x) = 2x

– 2 2 4

– 1 1 2

0 0 0

1 1 2

2 2 4



Solution The graph of h(x) is also the same general shape as that of (x), but here the coefficient 1/2 causes a vertical shrink. The graph of h(x) is wider than the graph of (x), as we see by comparing the tables of values.

(b)1

( )2

h x x

x (x) = h(x) =

– 2 2 1

– 1 1 1/2

0 0 0

1 1 1/2

2 2 1

x 12 x



(b)

x

y

1 2 3

– 2

3

– 2

4

– 3

– 4

– 3– 4

4

1( )

2h x x x (x) = h(x) =

– 2 2 1

– 1 1 1/2

0 0 0

1 1 1/2

2 2 1

x 12 x



Solution Use Property 2 of absolute value

to rewrite

(c) ( ) 2k x x

ab a b 2 .x

2 2 2k xx x x

Property 2

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


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).


Horizontal 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 ( , 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


Reflecting

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


Example 2 REFLECTING A GRAPH ACROSS AN AXIS


Solution Every y-value of the graph of g(x) is the negative of the corresponding y-value of

(a) ( )x xg

( ) .x x

x (x) = g(x) =

0 0 0

1 1 – 1

4 2 – 2

x x



Graph each function.(a) ( )x xg

x (x) = g(x) =

0 0 0

1 1 – 1

4 2 – 2

x x




Solution Choosing x-values for h(x) that are the negatives of those used for (x), we see that the corresponding y-values are the same. The graph of 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



Graph each function.( )x x h

x (x) = h(x) =

– 4 undefined 2

– 1 undefined 1

0 0 0

1 1 undefined

4 2 undefined

x x

(b)


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.)


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.


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




Solution The result is the same as the original equation, so the graph is symmetric with respect to the y-axis. Substituting –y for y does not result in an equivalent equation, and thus the graph is not symmetric with respect to the x-axis.

(a) 2 4y x




Solution Replace y with – y.

(b) 2 3x y

2 3x y

23x y

2 3y

Equivalent




Solution The graph is symmetric with respect to the x-axis. It is not symmetric with respect to the y-axis.

(b) 2 3x y

2

– 2

4

x

y




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

(c) 2 2 16x y




Solution

(c) 2 2 16x y




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




Solution(d) 2 4x y


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 at the same time results in an equivalent equation.


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.


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.


Important Concepts of Symmetry

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

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

• Of the three types of symmetrywith respect to the x-axis, the y-axis, and the origina graph possessing any two types must also exhibit the third type of symmetry.


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


Even and Odd Functions

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


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

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

Solution Replacing x with – x gives:

(a) 4 2( ) 8 3x x x

4 2 4 2( ) 8( ) 3

( (

3

)

(

)

) 8x x

x x

x xx

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

4 2( ) 8 3x x x




Solution(b) 3( ) 6 9x x x

3

( ) (

) 9

)

( 6x x x

x x

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

3

3

( ) 6 9

( ) 6( ) 9( )

x x x

x x x




Solution

(c) 2( ) 3 5x x x

2( ) 3 5xx x

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

23 5x x x Replace x with – x.

23 5x x x


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


Vertical Translations

If a function g is defined by g(x) = f(x) + c, where c is a real number, then for every point (x,y) on the graph of f, there will be a corresponding point (x, y + c) on the graph of g.

The graph of g will be the same as the graph of f, but translated c units up if c is positive or │c│ units down if c is negative. The graph of g is called a vertical translation of the graph of f.


Vertical Translations


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

(4, 0)

( ) 4x x g

( )x x

x

y

8


Horizontal Translations

If a function g is defined by g(x) = f(x – c), where c is a real number, then for every point (x,y) on the graph of f, there will be a corresponding point (x + c,y) on the graph of g.

The graph of g will be the same as the graph of f, but translated c units to the right if c is positive or │c │units to the left is c is negative. The graph of g is called a horizontal translationof the graph of f.


Horizontal Translations


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.


Example 8 USING MORE THAN ONE TRNASFORMATION


Solution To graph this function, the lowest point on the graph of y = xis translated 3 units to the left and 1 unit up. 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




(a) ( ) 3 1x x

3

– 3– 2

– 6

( ) 3 1x x

x

y




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




Solution(b) ( ) 2 4h x x

4

4

2

2

y x

( ) 2 4 2 2h x x x

x

y




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

(c) 214

2x x g




Solution

(c) 214

2x x g


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

A graph of a function defined by y = f(x) is shown in the figure. Use this graph to sketch each of the following graphs.



Graph the function.

Solution The graph is the same, translated 3 units up.

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



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

( ) ( 3)x x h f

Solution The graph must be translated 3 units to the left.



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

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


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.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 1 2 Graphs and Functions Copyright © 2013,...

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Transcript of Copyright © 2013, 2009, 2005 Pearson Education, Inc. 1 2 Graphs and Functions Copyright © 2013,...