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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 you have difficulty obtaining the functions to be maximized. Read Example 2 in Section \(1.10 .\) The annual yield per cherry tree is fairly constant at 50 pounds per tree when the number of trees per acre is 30 or fewer. For each additional tree over \(30,\) the annual yield per tree for all trees on the acre decreases by 1 pound due to overcrowding. How many cherry trees should be planted per acre to maximize the annual yield for the acre? What is the maximum number of pounds of cherries per acre?

Determine whether each statement makes sense or does not make sense, and explain your reasoning. When all is said and done, it seems to me that direct variation equations are special kinds of linear functions and inverse variation equations are special kinds of rational functions.

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?

An equation of a quadratic function is given. a. Determine, without graphing, whether the function has a minimum value or a maximum value. b. Find the minimum or maximum value and determine where it occurs. c. Identify the function鈥檚 domain and its range. $$f(x)=-4 x^{2}+8 x-3$$

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.
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