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Chapter 3: Problem 113

Describe how to graph a rational function.

Short Answer

Expert verified
Graph a rational function by identifying the zeros, vertical and horizontal asymptotes, any holes and then approximate the overall shape of the function by placing each part around the asymptotes.

Step by step solution

01

Identify the Zeros of the Function

The zeros of the function are where the rational function crosses or touches the x-axis. They can be found by setting the numerator equal to zero, i.e., solving \(p(x) = 0\), and finding the real roots. The roots will be the x-coordinates where the function touches the x-axis. The point(s) will then be of the form (a, 0). Plot these points on the graph.
02

Identify the Poles or Vertical Asymptotes

The vertical asymptotes occur where the denominator of the rational function is zero, while the numerator is not. They can be found by solving the equation \(q(x) = 0\), whilst making sure the numerator doesn't zero at these points. The solution to this equation gives the vertical lines of asymptotes. Draw these as dashed vertical lines on the graph.
03

Identify the Horizontal Asymptotes

The horizontal asymptotes reveal how the function behaves for very large positive or negative x-values. If, for example, the degree of the polynomial in the numerator is less than the degree in the denominator, y=0 is the line of the horizontal asymptote. If the degrees are the same, the horizontal asymptote will be y = the ratio of the leading coefficients of the top and bottom. If the degree of the polynomial in the denominator is less than the degree in the numerator, there will be no horizontal asymptotes. Plot the identified horizontal asymptote as a dashed line on the graph.
04

Identify any Holes of the Function

Holes are points that have been removed from the function's domain. If a factor is the same in both the numerator and the denominator, then the function will have a hole at this x-value. This simultaneous factor can be found by simplifying the rational function. The hole is at the x-value that makes this factor zero. To find the y-coordinate of the hole, substitute the x-value of the hole into the simplified function.
05

Sketch the Function

Once the zeros, asymptotes, and possible holes have been identified and plotted, create a rough sketch of the rational function. The graph should approach the vertical asymptotes but never touch them. The graph will cross the x-axis at the zeros. If there are any holes at x=a, the graph will also trend towards this point but will leave an 'empty spot' right at the point (a,b).

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Most popular questions from this chapter

In Exercises 41鈥�64, a. Use the Leading Coefficient Test to determine the graph鈥檚 end behavior. b. Find the x-intercepts. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each intercept. c. Find the y-intercept. d. Determine whether the graph has y-axis symmetry, origin symmetry, or neither. e. If necessary, find a few additional points and graph the function. Use the maximum number of turning points to check whether it is drawn correctly. $$f(x)=x^{3}(x+2)^{2}(x+1)$$

Use everyday language to describe the behavior of a graph near its vertical asymptote if \(f(x) \rightarrow \infty\) as \(x \rightarrow-2^{-}\) and \(f(x) \rightarrow-\infty\) as \(x \rightarrow-2^{+}\).

In Exercises 41鈥�64, a. Use the Leading Coefficient Test to determine the graph鈥檚 end behavior. b. Find the x-intercepts. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each intercept. c. Find the y-intercept. d. Determine whether the graph has y-axis symmetry, origin symmetry, or neither. e. If necessary, find a few additional points and graph the function. Use the maximum number of turning points to check whether it is drawn correctly. $$f(x)=\frac{1}{2}-\frac{1}{2} x^{4}$$

Use a graphing utility to graph \(y=\frac{1}{x^{\prime}}, y=\frac{1}{x^{3}},\) and \(\frac{1}{x^{5}}\) in the same viewing rectangle. For odd values of \(n,\) how does changing \(n\) affect the graph of \(y=\frac{1}{x^{n}} ?\)

In Exercises 41鈥�64, a. Use the Leading Coefficient Test to determine the graph鈥檚 end behavior. b. Find the x-intercepts. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each intercept. c. Find the y-intercept. d. Determine whether the graph has y-axis symmetry, origin symmetry, or neither. e. If necessary, find a few additional points and graph the function. Use the maximum number of turning points to check whether it is drawn correctly. $$f(x)=-3(x-1)^{2}\left(x^{2}-4\right)$$
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