Skip to main content
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 39
Chapter 4, Problem 39

Find the horizontal asymptote, if there is one, of the graph of each rational function. g(x)=12x2/(3x2+1)

Verified step by step guidance
1
Identify the degrees of the numerator and denominator polynomials in the rational function \(g(x) = \frac{12x^2}{3x^2 + 1}\). The degree of the numerator is 2, and the degree of the denominator is also 2.
Recall the rule for horizontal asymptotes of rational functions: If the degrees of numerator and denominator are equal, the horizontal asymptote is the ratio of the leading coefficients.
Determine the leading coefficient of the numerator, which is 12 (from \$12x^2\(), and the leading coefficient of the denominator, which is 3 (from \)3x^2$).
Write the horizontal asymptote as \(y = \frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}} = \frac{12}{3}\).
Simplify the fraction to express the horizontal asymptote in simplest form.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2m
Was this helpful?

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). Understanding the behavior of rational functions, especially their graphs, is essential for analyzing asymptotes and limits.
Recommended video:
6:04
Intro to Rational Functions

Horizontal Asymptotes

A horizontal asymptote describes the behavior of a function as x approaches infinity or negative infinity. It is a horizontal line y = L that the graph approaches but does not necessarily touch, indicating the end behavior of the function.
Recommended video:
4:48
Determining Horizontal Asymptotes

Degree of Polynomials in Rational Functions

The degrees of the numerator and denominator polynomials determine the horizontal asymptote. If degrees are equal, the asymptote is the ratio of leading coefficients; if numerator degree is less, asymptote is y=0; if greater, no horizontal asymptote exists.
Recommended video:
6:04
Intro to Rational Functions
Related Practice
Textbook Question

Find all zeros of the polynomial function or solve the given polynomial equation. Use the Rational Zero Theorem, Descartes's Rule of Signs, and possibly the graph of the polynomial function shown by a graphing utility as an aid in obtaining the first zero or the first root. f(x)=x3−4x2−7x+10

Textbook Question

In Exercises 33–40, use the Intermediate Value Theorem to show that each polynomial has a real zero between the given integers. f(x)=x5−x3−1; between 1 and 2

1
views
Textbook Question

Use synthetic division and the Remainder Theorem to find the indicated function value. f(x)=2x4−5x3−x2+3x+2; f(−1/2)

Textbook Question

An equation of a quadratic function is given. a) Determine, without graphing, whether the function has a minimum value or a maximum value. b) Find the minimum or maximum value and determine where it occurs. c) Identify the function's domain and its range. f(x)=3x2−12x−1

Textbook Question

Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. x3+x2+4x+4>0x^3+x^2+4x+4>0

4
views
Textbook Question

Use the Intermediate Value Theorem to show that each polynomial has a real zero between the given integers. f(x)=3x3−10x+9; between -3 and -2

2
views