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Chapter 4

Graphing and Transforming Functions

You can graph functions fairly handily using a graphing calculator, but you’ll be frustrated using this technology if you don’t have a good idea of what you’ll find and where you’ll find it. You need to have a fairly good idea of how high or how low and how far left and right the graph extends. You get information on these aspects of a graph from the intercepts (where the curve crosses the axes), from any asymptotes (in rational functions), and, of course, from the domain and range of the function. A good working knowledge of the characteristics of different types of functions goes a long way toward making your graphing experience a success.

Another way of graphing functions is to recognize any transformations performed on basic function definitions. Just sliding a graph to the left or right or flipping the graph over a line is a lot easier than starting from scratch.

The Problems You’ll Work On

In this chapter, you’ll work with function graphs in the following ways:

Graphing both a function and its inverse

Determining the vertices of quadratic functions (parabolas)

Recognizing the limits of some radical functions when graphing

Pointing out the top or bottom point of an absolute-value function graph to establish its range

Solving polynomial equations for intercepts

Writing equations of the asymptotes of rational functions

Using function transformations to quickly graph variations on functions

What to Watch Out For

When graphing functions, your challenges include the following:

Taking advantage of alternate formats of function equations (slope-intercept form, factored polynomial or rational functions, and so on)

Determining whether a parabola opens upward or downward and how steeply

Graphing radical functions with odd-numbered roots and recognizing the unlimited domain

Recognizing when polynomial functions don’t cross the x-axis at an intercept

Using asymptotes correctly as a guide in graphing

Reflecting functions vertically or horizontally, depending on the function transformation

Functions and Their Inverses

181–190 Find the inverse of the function. Then graph both the function and its inverse.

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Sketching Quadratic Functions from Their Equations

191–195 Sketch the graph of the quadratic function.

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