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Ch. 3 - Polynomial and Rational Functions
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063당신이 사용하는 게 아니라요?교과서 변경
4장, 문제 72

Graph each rational function. ƒ(x)=[(x+3)(x-5)]/[(x+1)(x-4)]

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Identify the rational function given: \(f(x) = \frac{(x+3)(x-5)}{(x+1)(x-4)}\). This function is a ratio of two polynomials.
Find the domain by determining the values of \(x\) that make the denominator zero. Solve \( (x+1)(x-4) = 0 \) to find the vertical asymptotes.
Find the zeros of the function by setting the numerator equal to zero: \( (x+3)(x-5) = 0 \). These values are the x-intercepts of the graph.
Determine the horizontal asymptote by comparing the degrees of the numerator and denominator. Since both are degree 2, the horizontal asymptote is the ratio of the leading coefficients.
Plot the vertical asymptotes, zeros, and horizontal asymptote on the coordinate plane. Then, choose test points in each interval to find the sign of \(f(x)\) and sketch the graph accordingly.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Rational Functions and Their Domains

A rational function is a ratio of two polynomials. Its domain includes all real numbers except where the denominator equals zero, causing vertical asymptotes or holes. Identifying these restrictions is essential before graphing.
추천 영상:
6:04
Intro to Rational Functions

Asymptotes of Rational Functions

Asymptotes are lines that the graph approaches but never touches. Vertical asymptotes occur at values that make the denominator zero, while horizontal or oblique asymptotes describe end behavior based on the degrees of numerator and denominator polynomials.
추천 영상:
6:24
Introduction to Asymptotes

Intercepts and Sign Analysis

Finding x-intercepts involves setting the numerator equal to zero, while y-intercepts are found by evaluating the function at x=0. Sign analysis between intercepts and asymptotes helps determine where the function is positive or negative, guiding accurate graphing.
추천 영상:
04:08
Graphing Intercepts