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Chapter 2: Problem 117

Describe how to graph a rational function.

Short Answer

Expert verified
To graph a rational function, identify the function, find the y-intercept by setting x=0, the x-intercept by setting y=0, identify horizontal and vertical asymptotes and plot on the graph. The graph should approach asymptotes and curve around them.

Step by step solution

01

Identify the Rational Function

The first step is to identify the function that one is dealing with. The rational function is any function that can be written as the ratio of two polynomial functions. Both these polynomials are real numbers.
02

Find the Y-Intercept

To find the y-intercept of a rational function, simply substitute x = 0 into the function and solve for y.
03

Find the X-Intercept

To find the x-intercept of a rational function, set y = 0 and solve for x.
04

Identify the Horizontal Asymptote

A line is a horizontal asymptote if the function approaches that line as x goes to either positive or negative infinity. If the degree of the numerator is equal to degree of denominator then the horizontal asymptote will be the ratio of leading coefficients. If the degree of numerator is less than degree of denominator the x-axis (y=0) is the horizontal asymptote. If the degree of numerator is more than degree of denominator there will be no horizontal asymptote.
05

Identify the Vertical Asymptote(s)

A line is a vertical asymptote if the function approaches plus or minus infinity as x approaches the value of the asymptote. Vertical asymptote is a value 鈥榓鈥� such that either (lim_{x \to a^-}f(x) = 卤鈭�) or (lim_{x \to a^+}f(x) = 卤鈭�). The denominator of function should equal zero for such 鈥榓鈥� and this value should not make numerator zero. Set denominator to zero and solve for x to get vertical asymptote(s).
06

Draw the Graph

After identifying all the important points and asymptotes, plot these points on the graph. Usually, start by drawing the asymptotes as they provide a neat framework to sketch the graph. Then plot x and y intercepts and few more points on both sides of vertical asymptote(s). Note that the curve should approach the asymptotes while not touching or crossing them. Make sure that graph on left and right end approaches horizontal asymptotes.

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Most popular questions from this chapter

If \(f\) is a polynomial or rational function, explain how the graph of \(f\) can be used to visualize the solution set of the inequality \(f(x)<0\).

Find the horizontal asymptote, if there is one, of the graph of rational function. $$h(x)=\frac{12 x^{3}}{3 x^{2}+1}$$

The equation for \(f\) is given by the simplified expression that results after performing the indicated operation. Write the equation for \(f\) and then graph the function. $$\frac{2}{x^{2}+3 x+2}-\frac{4}{x^{2}+4 x+3}$$

Will help you prepare for the material covered in the next section. Solve: \(x^{3}+x^{2}=4 x+4\)

a. Find the slant asymptote of the graph of each rational function and \(\mathbf{b} .\) Follow the seven-step strategy and use the slant asymptote to graph each rational function. $$f(x)=\frac{x^{3}-1}{x^{2}-9}$$
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