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

y x Example 1 Stretching or Shrinking a Graph Graph each function x|x|

y x Example 1 Stretching or Shrinking a Graph Graph each function x|x|2|x|

y x Example 1 Stretching or Shrinking a Graph Graph each function x|x|

y x Example 1 Stretching or Shrinking a Graph Graph each function x|x|

y x Example 2 Reflecting a Graph Across an Axis Graph each function x

y x Example 2 Reflecting a Graph Across an Axis Graph each function x

y x Example 2 Reflecting a Graph Across an Axis Graph each function 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 x

In y = x 2 +4 replace x with -x

y x Example 3 Testing for Symmetry with Respect to an Axis Test for symmetry y

In x = y replace y with -y

y x Example 3 Testing for Symmetry with Respect to an Axis Test for symmetry

In x 2 +y 2 = 16 substitute –x for x and –y for y

y x Example 3 Testing for Symmetry with Respect to an Axis Test for symmetry x

In 2x + y = 4 substitute –x for x and –y for y

Another kind of symmetry occurs when a graph can be rotated 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.

Example 4 Testing for Symmetry with Respect to the Oigin Are the following graphs 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?

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) = 8x 4 - 3x 2

Example 5 Determining Whether Functions Are Even, Odd, or Neither Decide whether each function defined is even, odd, or neither. f(x) = 6x 3 - 9x

Example 5 Determining Whether Functions Are Even, Odd, or Neither Decide whether each function defined is even, odd, or neither. f(x) = 3x 2 + 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 Graph each function x|x||x|

y x Example 7 Translating a Graph Vertically Graph each function x|x||x|

y x Example 8 Using More Than One Trnasformation on Graphs Graph each function

y x Example 8 Using More Than One Trnasformation on Graphs Graph each function

y x Example 8 Using More Than One Trnasformation on Graphs Graph each function

Example 9 Graphing Trnaslations Given the Graph of y = f(x)