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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 81
Chapter 4, Problem 81

In Exercises 81–88, a. Find the slant asymptote of the graph of each rational function and b. Follow the seven-step strategy and use the slant asymptote to graph each rational function. f(x)=(x2−1)/x

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1
Identify the given rational function: \(f(x) = \frac{x^{2} - 1}{x}\).
To find the slant (oblique) asymptote, perform polynomial long division of the numerator \(x^{2} - 1\) by the denominator \(x\).
Divide \(x^{2}\) by \(x\) to get \(x\), then multiply \(x\) by \(x\) to get \(x^{2}\), subtract this from \(x^{2} - 1\) to find the remainder.
Next, divide the remainder by \(x\) to find the next term of the quotient, continue until the degree of the remainder is less than the degree of the divisor.
The quotient (without the remainder) will be the equation of the slant asymptote, which you can write as \(y = \) (quotient).

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Key Concepts

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

Rational Functions

A rational function is a ratio of two polynomials, expressed as f(x) = P(x)/Q(x), where Q(x) ≠ 0. Understanding the behavior of rational functions, including their domains and asymptotes, is essential for graphing and analyzing them.
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Intro to Rational Functions

Slant (Oblique) Asymptotes

Slant asymptotes occur when the degree of the numerator is exactly one more than the degree of the denominator. They represent the line that the graph approaches as x approaches infinity or negative infinity, found by performing polynomial division.
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Introduction to Asymptotes

Polynomial Division

Polynomial division is a method used to divide one polynomial by another, similar to long division with numbers. It helps find the quotient and remainder, which are used to determine slant asymptotes and simplify rational functions for graphing.
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Introduction to Polynomials
Related Practice
Textbook Question

In Exercises 81–88, a. Find the slant asymptote of the graph of each rational function and b. Follow the seven-step strategy and use the slant asymptote to graph each rational function. f(x)=(x2+1)/x

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Textbook Question

In Exercises 57–80, follow the seven steps to graph each rational function. f(x)=(3x2+x−4)/(2x2−5x)

Textbook Question

Solve the variation problems in Exercises 77–82. The pitch of a musical tone varies inversely as its wavelength. A tone has a pitch of 660 vibrations per second and a wavelength of 1.6 feet. What is the pitch of a tone that has a wavelength of 2.4 feet?

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Textbook Question

Find the inverse of f(x)=(x−10)/(x+10).

Textbook Question

Exercises 82–84 will help you prepare for the material covered in the next section. Solve: x2+4x+6=0

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Textbook Question

Exercises 82–84 will help you prepare for the material covered in the next section. Solve: x2+4x−1=0