Table of contents

0. Review of Algebra4h 18m
1. Equations & Inequalities3h 18m
2. Graphs of Equations43m
3. Functions2h 17m
4. Polynomial Functions1h 44m
5. Rational Functions1h 23m
6. Exponential & Logarithmic Functions2h 28m
7. Systems of Equations & Matrices4h 6m
8. Conic Sections2h 23m
9. Sequences, Series, & Induction1h 22m
10. Combinatorics & Probability1h 45m

5. Rational Functions

Graphing Rational Functions

College Algebra

5. Rational Functions

Graphing Rational Functions

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1:51 minutes

Problem 43

Textbook Question

In Exercises 37–44, find the horizontal asymptote, if there is one, of the graph of each rational function. f(x)=(−2x+1)/(3x+5)

Verified step by step guidance

Identify the rational function: $ f(x) = \frac{-2x + 1}{3x + 5} $.

Determine the degrees of the numerator and the denominator. Both are linear functions, so their degrees are 1.

Since the degrees of the numerator and the denominator are equal, the horizontal asymptote is found by dividing the leading coefficients.

The leading coefficient of the numerator is $-2$ and the leading coefficient of the denominator is $3$.

The horizontal asymptote is $ y = \frac{-2}{3} $.

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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 function that can be expressed as the ratio of two polynomials. The general form is f(x) = P(x)/Q(x), where P and Q are polynomials. Understanding the degrees of the polynomials in the numerator and denominator is crucial for analyzing the behavior of the function, especially as x approaches infinity.

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a function as x approaches positive 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 degree of the denominator, the horizontal asymptote is y=0; if they are equal, the asymptote is y = leading coefficient of P / leading coefficient of Q.

Leading Coefficients

The leading coefficient of a polynomial is the coefficient of the term with the highest degree. In the context of rational functions, the leading coefficients of the numerator and denominator are essential for determining the horizontal asymptote when the degrees of the polynomials are equal. This concept helps simplify the analysis of the function's end behavior.

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Related Practice

Textbook Question

Solve each problem. This rational function has two holes and one vertical asymptote. ƒ(x)=(x^3+7x^2-25x-175)/(x^3+3x^2-25x-75)

What are the x-values of the holes?

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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. g(x) = (2x - 4)/(x + 3)

Identify any vertical, horizontal, or oblique asymptotes in the graph of $y=f\left(x\right)$ . State the domain of $f$ .