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Chapter 3: Problem 81

Describe how to graph a rational function.

Short Answer

Expert verified
The graph of a rational function can be drawn by following these steps: 1) Identify the function. 2) Identify vertical asymptotes by setting the denominator equal to zero. 3) Identify horizontal asymptotes by taking the limit as x approaches infinity. 4) Find the x and y intercepts by setting y and x to zero respectively. 5) Plot the graph by marking the intercepts and asymptotes, and drawing the graph approaching but never crossing the asymptotes.

Step by step solution

01

Identify the function

The first step is to identify the function. Let's use a concrete example function \(f(x) = \frac{3x^2 - 2x + 1 }{x - 3}\).
02

Identify vertical asymptotes

The next step is to find the vertical asymptotes of the rational function. This is done by setting the denominator of the function equal to zero and solving for \(x\). In this case, \(x - 3 = 0\), so \(x = 3\) is the vertical asymptote.
03

Identify horizontal asymptotes

Next, identify horizontal asymptotes. This is done by considering the limit of the function as \(x\) approaches positive or negative infinity. As for our function, the degree of the numerator is equal to the degree of the denominator, so we use the ratio of the coefficients of the highest degree term in the numerator and denominator. That is, the horizontal asymptote is \(y = \frac{3}{1} = 3\).
04

Find the X and Y intercepts

To find the \(x\)-intercepts, set \(y\) to zero and solve for \(x\). To find the \(y\)-intercept, set \(x\) to zero, and solve for \(y\). In our function, \(f(x) = 0\) leads to \(x = 0.5, -0.6667\) and \(f(0) = -0.3333\), so these are our X and Y intercepts respectively.
05

Plotting the graph

The final step is to sketch the graph. Mark the asymptotes and intercepts on your graph. Then, draw the graph in regions divided by vertical asymptotes. Always remember that the graph should approach the asymptotes but never crosses them.

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Most popular questions from this chapter

Which one of the following is true? a. The graph of a rational function cannot have both a vertical and a horizontal asymptote. b. It is not possible to have a rational function whose graph has no \(y\) -intercept. c. The graph of a rational function can have three horizontal asymptotes. d. The graph of a rational function can never cross a vertical asymptote.

Use the four-step procedure for solving variation problems given on page 356 to solve. The volume of a gas varies directly as its temperature and inversely as its pressure. At a temperature of 100 Kelvin and a pressure of 15 kilograms per square meter, the gas occupies a volume of 20 cubic meters. Find the volume at a temperature of 150 Kelvin and a pressure of 30 kilograms per square meter.

Among all deaths from a particular disease, the percentage that are smoking related ( \(21-39\) cigarettes per day) is a function of the disease's incidence ratio. The incidence ratio describes the number of times more likely smokers are than nonsmokers to die from the disease. The following table shows the incidence ratios for heart disease and lung cancer for two age groups. Incidence Ratios $$\begin{array}{|l|cc|} \hline & \text { Heart Disease } & \text { Lung Cancer } \\ \hline \text { Ages } 55-64 & 1.9 & 10 \\ \text { Ages } 65-74 & 1.7 & 9 \\ \hline \end{array}$$ For example, the incidence ratio of 9 in the table means that smokers between the ages of 65 and 74 are 9 times more likely than nonsmokers in the same group to die from lung cancer. The rational function $$P(x)=\frac{100(x-1)}{x}$$ models the percentage of smoking-related deaths among all deaths from a disease, \(P(x),\) in terms of the disease's incidence ratio, \(x\). The graph of the rational function is shown. Use this function to solve Exercises . (graph can't copy) Find \(P(9) .\) Round to the nearest percent. Describe what this means in terms of the incidence ratio, 9 given in the table. Identify your solution as a point on the graph.

The width of a rectangular box is twice the height and the length is 7 inches more than the height. If the volume is 72 cubic inches, find the dimensions of the box. (GRAPH CANNOT COPY).

If you are given the equation of a rational function, explain how to find the horizontal asymptote, if any, of the functions graph.
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