Question

1. write a rational function that has an x - intercept at x = - 3, a vertical asymptote at x = 1, and a horizontal asymptote of y = 2.

Explanation:

Step1: Determine the numerator for x - intercept

An x - intercept at $x=-3$ means $(x + 3)$ is a factor of the numerator. Let the numerator $N(x)=a(x + 3)$, where $a$ is a non - zero constant.

Step2: Determine the denominator for vertical asymptote

A vertical asymptote at $x = 1$ means $(x - 1)$ is a factor of the denominator. Let the denominator $D(x)=x - 1$.

Step3: Determine the value of the constant for horizontal asymptote

The general form of a rational function is $y=\frac{N(x)}{D(x)}=\frac{a(x + 3)}{x - 1}=\frac{ax+3a}{x - 1}$. For a rational function of the form $\frac{ax + b}{cx + d}$ ($c
eq0$), the horizontal asymptote is $y=\frac{a}{c}$. Since the horizontal asymptote is $y = 2$, and $c = 1$ in our case, then $a = 2$.

Answer:

$y=\frac{2(x + 3)}{x - 1}=\frac{2x+6}{x - 1}$