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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

A rectangular playground is to be fenced off and divided in two by another fence parallel to one side of the playground. Six hundred feet of fencing is used. Find the dimensions of the playground that maximize the total enclosed area. What is the maximum area?

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The graph of a rational function can never cross a vertical asymptote.

a. List all possible rational roots. b. Use synthetic division to test the possible rational roots and find an actual root. c. Use the quotient from part ( \(b\) ) to find the remaining roots and solve the equation. $$2 x^{3}-5 x^{2}-6 x+4=0$$

Why must every polynomial equation with real coefficients of degree 3 have at least one real root?

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 \(x \rightarrow-2^{+}\).
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