Pre-Calculus: 1001 Practice Problems For Dummies (+ Free Online Practice)
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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 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

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

Practice problems

  1. Given the graph of a quadratic function, write its function equation in vertex form, y = a(xh)2 + k.

    [Credit: Illustration by Thomson Digital]
    Credit: Illustration by Thomson Digital

    Answer: y = –x2 + 9

    The vertex is (0, 9), and the graph opens downward. Using the vertex form of a quadratic equation, y = a(xh)2 + k, these characteristics are represented by y = a(x – 0)2 + 9, where a is a negative number.

    The x-intercepts 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.

  2. 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 x-intercepts by letting y = 0 and solving for x. The x-intercepts of y = (x – 3)2(x + 2)2 are (3, 0) and (‒2, 0).

    Find the y-intercept by letting x = 0 and solving for y. The y-intercept is (0, 36).

    To sketch the graph, note that the exponents on the factors are even numbers, so the curve just touches the x-axis at each x-intercept. The curve rises to the right as x approaches positive infinity, as determined when you test an x value greater than the right-most intercept. Here’s the graph:

    [Credit: Illustration by Thomson Digital]
    Credit: Illustration by Thomson Digital

About This Article

This article is from the book:

About the book author:

Mary Jane Sterling is the author of Algebra I For Dummies, Algebra Workbook For Dummies, and many other For Dummies books. She taught at Bradley University in Peoria, Illinois for more than 30 years, teaching algebra, business calculus, geometry, and finite mathematics.

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