Question

Write an equation for a rational function with the given characteristics.Vertical asymptotes at x = -2 and x = 4, x-intercepts at (-3,0) and (1,0), horizontal asymptote at y = -2y =Additional MaterialseBookFind the Equation of a Rational FunctionExample Video

Answer 1

Given:

The characteristics of rational function:

Vertical asymptotes at x = -2 and x = 4

x-intercepts at (-3,0) and (1,0).

horizontal asymptote at y = -2

The foem of rational function is,

$\frac{f(x)}{g(x)}$

For the vertical asymtotes x = -2 and x = 4, That means denominator will have the terms,

$\begin{gathered} x=-2,x=4 \\ (x+2),(x-4) \end{gathered}$

For x intercept (-3,0) and (1,0) , the terms on the numerator is,

$\begin{gathered} (x+3)\text{ and (x-1)} \\ \text{Because this factors will given the values as x=-3 and x=1} \end{gathered}$

So, the rational function becomes,

$\frac{f(x)}{g(x)}=a\frac{(x+3)(x-1)}{(x+2)(x-4)}$

The horizontal asymtoes will describes the functions behaviour when x approaches to infinity.

So, a=-2.

$\frac{f(x)}{g(x)}=-2\frac{(x+3)(x-1)}{(x+2)(x-4)}$

Answer:

$\frac{-2(x+3)(x-1)}{(x+2)(x-4)}$