Write an equation for a rational function with: Vertical asymptotes at

enfurezca3x

enfurezca3x

Answered question

2021-12-03

Write an equation for a rational function with:
Vertical asymptotes at x=2 and x=4
x intercepts at x=1 and x=3
Horizontal asymptote at y=8

Answer & Explanation

Prioned

Prioned

Beginner2021-12-04Added 11 answers

Step1
A function of the form f(x)n(x) where f(x) and n(x) are polynomials is called a rational function.
The graphs of rational functions can be recognized by the fact that they often break into two or more parts.
These parts go out of the coordinate system along an imaginary straight line called an asymptote.
Step2
As we have vertical asymptote at x=2 and x=4.
We have in denominator, n(x)=(x+2)(x+4)
n(x)=2+6x+8
f(x) which is in the numerator must be of the same degree as the denominator.
As we have x intercepts at x=1 and x=3
So, f(x)=(x1)(x3)
Rational function,R(x)=kf(x)n(x)=k(x1)(x3)x2+6x+8
As we have the horizontal asymptote at y=8
So at x=0,R(0)=8
8=k(01)(03)(02+6x0+8)8=3k8k=643
Hence, rational function,
R(x)=(643)(x1)(x3)(x2+6x+8)R(x)=64(x24x+3)3(x2+6x+8)

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