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Chapter 2.6
Graphing Techniques
One of the main objectives of this course is to recognize and learn to graph various functions. Graphing techniques presented in this section show how to graph functions that are defined by altering a basic function.
Stretching and Shrinking
We begin by considering how the graph of
.xf y ofgraph the
tocompares or
axfyxfay
y
x
Example 1 Stretching or Shrinking a GraphGraph each function
xxf
x |x|
-2-1012
y
x
Example 1 Stretching or Shrinking a Graph
xxf
Graph each function
xxg 2
x |x| 2|x|
-2-1012
21012
y
x
Example 1 Stretching or Shrinking a GraphGraph each function
xxh 21
xxf
x |x| x21
-2-1012
21012
y
x
Example 1 Stretching or Shrinking a GraphGraph each function
xxk 2
xxf
x |x| x2
-2-1012
21012
2x rewrite to
)baab(
R2)(Section 19 pageon 3property Use
(a).part in
x2xg ofgraph theas same theis
2xxk ofgraph theTherefore,
y
x
Example 2 Reflecting a Graph Across an AxisGraph each function
xxf
x
0
1
4
9
x
y
x
Example 2 Reflecting a Graph Across an AxisGraph each function
xxf
x
0
1
4
9
x x0
1
2
3
y
x
Example 2 Reflecting a Graph Across an AxisGraph each function
xxf
x
0
-1
-4
-9
x x
Symmetry
The graph of f shown in Figure 75(a) is cut in half by the y-axis with each half the mirror image of the other half.
A graph with this property is said to be symmetric with respect to the y-axis. As this graph suggests, a graph is symmetric with respect to the y-axis if the point (-x, y) is on the graph whenever the point (x, y) is on the graph.
Similarly, if the graph of g in Figure 75(b) were folded in half along the x-axis, the portion at the top would exactly match the portion at the bottom.
Such a graph is symmetric with respect to the x-axis: the point (x, -y) is on the graph whenever the point (x, y) is on the graph.
y
x
Example 3 Testing for Symmetry with Respect to an Axis
Test for symmetry
42 xy
x
012
-1
2x 42 x
-2
In y = x2 +4 replace x with -x
y
x
Example 3 Testing for Symmetry with Respect to an Axis
Test for symmetry
32 yx
y
012
-1
2y 32 y
-2
In x = y2 - 3 replace y with -y
y
x
Example 3 Testing for Symmetry with Respect to an Axis
Test for symmetry
1622 yx
In x2 +y2 = 16
substitute –x for x and –y for y
y
x
Example 3 Testing for Symmetry with Respect to an Axis
Test for symmetry
42 yx
x
012
-1
42 x
-2
In 2x + y = 4
substitute –x for x and –y for y
Another kind of symmetry occurs when a graph can be rotated 1800 around the origin, with the result coinciding exactly with the original graph. Symmetry of this type is called symmetry with respect to the origin. A graph is symmetric with respect to the origin if the point (-x, -y) is on the graph whenever the point (x, y) is on the graph.
Figure 78 shows two graphs that are symmetric with respect to the origin.
Figure 78 shows two graphs that are symmetric with respect to the origin.
Example 4 Testing for Symmetry with Respect to the Oigin
Are the following graphs symmetric with respect to the origin?
1622 yx
Example 4 Testing for Symmetry with Respect to the Oigin
Are the following graphs symmetric with respect to the origin?
3xy
A graph symmetric with respect to both the x- and y-axes is automatically symmetric with respect to the origin. However, a graph symmetric with respect to the origin need not be symmetric with respect to either axis.
See figure 80.
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.
Even and Odd Functions
The concepts of symmetry with respect to the y-axis and symmetry with respect to the origin are closely associated with the concepts of even and off functions.
Example 5 Determining Whether Functions Are Even, Odd, or Neither
Decide whether each function defined is even, odd, or neither.
f(x) = 8x4 - 3x2
Example 5 Determining Whether Functions Are Even, Odd, or Neither
Decide whether each function defined is even, odd, or neither.
f(x) = 6x3 - 9x
Example 5 Determining Whether Functions Are Even, Odd, or Neither
Decide whether each function defined is even, odd, or neither.
f(x) = 3x2 + 5x
Translations
The next examples show the results of horizontal and vertical shifts, called translations, of the graph f(x) = |x|
y
x
Example 6 Translating a Graph Vertically
xxf
Graph each function
4 xxg
x |x| |x|-4
-2-1012
21012
y
x
Example 7 Translating a Graph Vertically
xxf
Graph each function
4 xxg
x |x| |x|-4
-2-1012
21012
y
x
Example 8 Using More Than One Trnasformation on Graphs
xxf
Graph each function
13 xxf
y
x
Example 8 Using More Than One Trnasformation on Graphs
xxf
Graph each function
42 xxh
y
x
Example 8 Using More Than One Trnasformation on Graphs Graph each
function
42
1 2 xxg
Example 9 Graphing Trnaslations Given the Graph of y = f(x)
Example 9 Graphing Trnaslations Given the Graph of y = f(x)
3 xfxg
Example 9 Graphing Trnaslations Given the Graph of y = f(x)
3 xfxg
Example 9 Graphing Trnaslations Given the Graph of y = f(x)
3 xfxg
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