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Chapter 3: Problem 113

Describe how to graph a rational function.

Short Answer

Expert verified
To graph a rational function, determine the horizontal and vertical asymptotes, find the x and y intercepts, identify any potential holes and then plot these onto the graph. After that, gradually sketch the function properly.

Step by step solution

01

Identifying the Horizontal Asymptote

The horizontal asymptote of a rational function can be found by examining the degrees of the polynomials in the numerator and the denominator. If the degree of the polynomial in the denominator is greater than the degree of the polynomial in the numerator, then the horizontal asymptote is y = 0. If the degrees are equal, you divide the leading coefficients of the numerator and denominator. If the degree of the polynomial in the numerator is greater than the denominator, there is no horizontal asymptote.
02

Identifying the Vertical Asymptotes

The vertical asymptotes occur at the x-values that make the denominator equal to zero. In other words, you find the vertical asymptotes by setting the denominator equal to zero and solving for x. Each solution will be a vertical asymptote. If the solution also makes the numerator zero, then it is not a vertical asymptote, but a hole.
03

Identifying the x and y-intercepts

The x-intercept(s) can be found by setting the entire function equal to zero, and solving for x. To find the y-intercept (there's only one), simply plug 0 in for x.
04

Plotting the Graph

Using the x and y-intercepts, asymptotes and possible holes identified in the earlier steps, sketch the graph carefully. Other points can be obtained by substituting x-values into the function. Ensure that your graph is asymptotic to the vertical and horizontal asymptotes you found in steps 1 and 2.

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Most popular questions from this chapter

Solve each inequality in Exercises \(86-91\) using a graphing utility. $$ x^{2}+3 x-10>0 $$

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 rain gutter is made from sheets of aluminum that are 12 inches wide by turning up the edges to form right angles. Determine the depth of the gutter that will maximize its cross-sectional area and allow the greatest amount of water to flow. What is the maximum cross-sectional area?

In Exercises \(61-64,\) find the domain of each function. $$ f(x)-\sqrt{\frac{2 x}{x+1}-1} $$

The rational function $$f(x)-\frac{27,725(x-14)}{x^{2}+9}-5 x$$ models the number of arrests, \(f(x)\), per \(100,000\) drivers, for driving under the influence of alcohol, as a function of a driver's age, \(x\). a. Graph the function in a \([0,70,5]\) by \([0,400,20]\) viewing rectangle. b. Describe the trend shown by the graph. c. Use the ZOOM and TRACE features or the the maximum function feature of your graphing utility to find the age that corresponds to the greatest number of arrests. How many arrests, per \(100,000\) drivers, are there for this age group?
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