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Chapter 3: Problem 113

Describe how to graph a rational function.

Short Answer

Expert verified
To graph a rational function, first understand that it's a quotient of two polynomials. Find the vertical and horizontal asymptotes, calculate the x and y intercepts, plot some additional points, and draw a curve that fits these points and asymptotes. The resulting graph provides a visual representation of the rational function's behavior.

Step by step solution

01

Define Rational Function

A rational function is a function that can be written as the quotient of two polynomial functions. If \(P(x)\) and \(Q(x)\) are polynomial functions then a rational function is given by \(R(x) = P(x) / Q(x)\). It's important that \(Q(x)\) doesn't equal to zero as we're not allowed to divide by zero.
02

Find the Asymptotes

We identify vertical and horizontal asymptotes. Vertical asymptotes are values of \(x\) where \(Q(x) = 0\), because the function will become undefined at these points. Horizontal asymptotes are the expected y-values when \(x\) approaches infinity or negative infinity, this is determined based on the degree of \(P(x)\) and \(Q(x)\). if the degree of \(P(x)\) is less than the degree of \(Q(x)\), then the x-axis (y = 0) is the horizontal asymptote.
03

Find the x and y intercepts

To find the x-intercepts set the numerator, \(P(x)\), equal to zero and solve for \(x\). The y-intercept is found by evaluating the function at \(x = 0\).
04

Plot the Points

Having found the intercepts and asymptotes, we choose a number of points for \(x\) in each of the intervals identified by the asymptotes and calculate the corresponding \(y\) values. Plot these points on the graph.
05

Draw the Graph

Finally, after the points and asymptotes are plotted, draw a smooth curve that fits the plotted points and asymptotes. The curve should approach the asymptotes but never cross them.

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Most popular questions from this chapter

Use the four-step procedure for solving variation problems given on page 424 to solve. \(y\) varies directly as \(x\) and inversely as the square of \(z . y=20\) when \(x=50\) and \(z=5 .\) Find \(y\) when \(x=3\) and \(z=6\).

Determine whether each statement makes sense or does not make sense, and explain your reasoning. My graph of \(y=\frac{x-1}{(x-1)(x-2)}\) has vertical asymptotes at \(x=1\) and \(x=2\)

The functions $$f(x)=0.0875 x^{2}-0.4 x+66.6$$ and $$g(x)=0.0875 x^{2}+1.9 x+11.6$$ model a car's stopping distance, \(f(x)\) or \(g(x),\) in feet, traveling at \(x\) miles per hour. Function \(f\) models stopping distance on dry pavement and function g models stopping distance on wet pavement. The graphs of these functions are shown for \(\\{x | x \geq 30\\} .\) Notice that the figure does not specify which graph is the model for dry roads and which is the model for wet roads. Use this information to solve. (GRAPH CANNOT COPY). a. Use the given functions to find the stopping distance on dry pavement and the stopping distance on wet pavement for a car traveling at 55 miles per hour. Round to the nearest foot. b. Based on your answers to part (a), which rectangular coordinate graph shows stopping distances on dry pavement and which shows stopping distances on wet pavement? c. How well do your answers to part (a) model the actual stopping distances shown in Figure 3.43 on page \(411 ?\) d. Determine speeds on wet pavement requiring stopping distances that exceed the length of one and one-half football fields, or 540 feet. Round to the nearest mile per hour. How is this shown on the appropriate graph of the models?

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Is every rational function a polynomial function? Why or why not? Does a true statement result if the two adjectives rational and polynomial are reversed? Explain.
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