Question

Find a rational function ƒ having a graph with the given features.x-intercepts: (-1, 0) and (3, 0)y-intercept: (0, -3)vertical asymptote: x=1horizontal asymptote: y=1

Accepted Answer

Identify the zeros of the rational function from the x-intercepts. Since the function has x-intercepts at (-1, 0) and (3, 0), the numerator must have factors $$ (x + 1) $$ and $$ (x - 3) . So$$, the numerator can be written as $$ a(x + 1)(x - 3) $$, where $$ a  is a $$constant to be determined. Determine the denominator using the vertical asymptote. The vertical asymptote at $$ x = 1 $$ means the denominator must have a factor $$ (x - 1) . So$$, the denominator can be written as $$ (x - 1)  or $$possibly a higher degree polynomial with $$ (x - 1)  as a $$factor, but start with $$ (x - 1) $$ for simplicity. Use the horizontal asymptote to relate the degrees and leading coefficients. Since the horizontal asymptote is $$ y = 1 $$, the degrees of numerator and denominator must be the same, and the ratio of the leading coefficients must be 1. Currently, numerator is degree 2 and denominator is degree 1, so to have the same degree, multiply denominator by a linear factor $$ (x + b)  to $$make it degree 2: $$ (x - 1)(x + b) . $$Set up the function as $$ f(x) = \frac{a(x + 1)(x - 3)}{(x - 1)(x + b)} . $$Use the y-intercept $$ (0, -3)  to $$find constants $$ a $$ and $$ b  by $$substituting $$ x = 0 $$ and $$ f(0) = -3 . $$This gives the equation $$ -3 = \frac{a(0 + 1)(0 - 3)}{(0 - 1)(0 + b)} = \frac{a(1)(-3)}{(-1)(b)} = \frac{-3a}{-b} = \frac{3a}{b} . $$Solve the equation $$ -3 = \frac{3a}{b} $$ for one variable in terms of the other, then use the horizontal asymptote condition (leading coefficients ratio equals 1) to find the values of $$ a $$ and $$ b . $$This will give the explicit form of the rational function $$ f(x) .$$

Find a rational function ƒ having a graph with the given features.
x-intercepts: (-1, 0) and (3, 0)
y-intercept: (0, -3)
vertical asymptote: x=1
horizontal asymptote: y=1

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Rational Functions and Their Graphs

Intercepts of a Rational Function

Asymptotes of Rational Functions