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Ch. 3 - Polynomial and Rational Functions
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 3 - Polynomial and Rational FunctionsProblem 58
Chapter 4, Problem 58

In Exercises 57–64, find the vertical asymptotes, if any, the horizontal asymptote, if one exists, and the slant asymptote, if there is one, of the graph of each rational function. Then graph the rational function. g(x) = (2x - 4)/(x + 3)

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Step 1: Identify the vertical asymptotes by setting the denominator equal to zero. Solve the equation x+3=0 to find the x-value(s) where the function is undefined.
Step 2: Determine the horizontal asymptote by comparing the degrees of the numerator and denominator. The numerator 2x-4 has degree 1, and the denominator x+3 also has degree 1. When the degrees are equal, the horizontal asymptote is given by the ratio of the leading coefficients.
Step 3: Check for a slant (oblique) asymptote. A slant asymptote exists if the degree of the numerator is exactly one more than the degree of the denominator. Since the degrees of the numerator and denominator are the same, there is no slant asymptote in this case.
Step 4: Summarize the asymptotes. The vertical asymptote is at the x-value found in Step 1, the horizontal asymptote is the ratio of the leading coefficients found in Step 2, and there is no slant asymptote as determined in Step 3.
Step 5: To graph the function, plot the asymptotes as dashed lines, and then evaluate the function at several x-values to determine the behavior of the graph near the asymptotes. Use these points to sketch the curve.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Vertical Asymptotes

Vertical asymptotes occur in rational functions where the denominator equals zero, leading to undefined values. To find vertical asymptotes, set the denominator of the function to zero and solve for x. These asymptotes indicate values that the function approaches but never reaches, often resulting in a division by zero.
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Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a function as x approaches infinity or negative infinity. For rational functions, the horizontal asymptote can be determined by comparing the degrees of the numerator and denominator. If the degree of the numerator is less than the denominator, the horizontal asymptote is y=0; if they are equal, it is the ratio of their leading coefficients.
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Slant Asymptotes

Slant (or oblique) asymptotes occur when the degree of the numerator is exactly one greater than the degree of the denominator. To find a slant asymptote, perform polynomial long division on the rational function. The quotient (ignoring the remainder) gives the equation of the slant asymptote, which describes the function's behavior as x approaches infinity.
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Related Practice
Textbook Question

Solve: x+x+5=5\(\sqrt{x}\) + \(\sqrt{x + 5}\) = 5

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Textbook Question

In Exercises 57–64, find the vertical asymptotes, if any, the horizontal asymptote, if one exists, and the slant asymptote, if there is one, of the graph of each rational function. Then graph the rational function. f(x) = 2x/(x^2 - 9)

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Textbook Question

Exercises 53–60 show incomplete graphs of given polynomial functions. a) Find all the zeros of each function. b) Without using a graphing utility, draw a complete graph of the function. f(x)=2x4−3x3−7x2−8x+6

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Textbook Question

In Exercises 57–64, find the vertical asymptotes, if any, the horizontal asymptote, if one exists, and the slant asymptote, if there is one, of the graph of each rational function. Then graph the rational function. h(x) = (x^2 - 3x - 4)/(x^2 - x -6)

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Textbook Question

Solve each rational inequality in Exercises 43–60 and graph the solution set on a real number line. Express each solution set in interval notation.1/(x - 3) < 1

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Textbook Question

Follow the seven steps to graph each rational function. f(x)=2x/(x2−4)

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