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.
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 absolutevalue 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
When graphing functions, your challenges include the following:

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

Determining whether a parabola opens upward or downward and how steeply

Graphing radical functions with oddnumbered roots and recognizing the unlimited domain

Recognizing when polynomial functions don’t cross the xaxis at an intercept

Using asymptotes correctly as a guide in graphing

Reflecting functions vertically or horizontally, depending on the function transformation
Practice problems

Given the graph of a quadratic function, write its function equation in vertex form, y = a(x – h)^{2} + k.
Credit: Illustration by Thomson DigitalAnswer: y = –x^{2} + 9
The vertex is (0, 9), and the graph opens downward. Using the vertex form of a quadratic equation, y = a(x – h)^{2} + k, these characteristics are represented by y = a(x – 0)^{2} + 9, where a is a negative number.
The xintercepts are (‒3, 0) and (3, 0). Substitute (3, 0) into the equation and solve for a:
So the equation of the parabola is y = –1(x – 0)^{2} + 9.

Determine the intercepts of the graph of the polynomial. Then sketch the graph.
f(x) = (x – 3)^{2}(x + 2)^{2}
Answer: intercepts: (3,0), (‒2,0), (0, 36)
Find the xintercepts by letting y = 0 and solving for x. The xintercepts of y = (x – 3)^{2}(x + 2)^{2} are (3, 0) and (‒2, 0).
Find the yintercept by letting x = 0 and solving for y. The yintercept is (0, 36).
To sketch the graph, note that the exponents on the factors are even numbers, so the curve just touches the xaxis at each xintercept. The curve rises to the right as x approaches positive infinity, as determined when you test an x value greater than the rightmost intercept. Here’s the graph:
Credit: Illustration by Thomson Digital