write an equation for the function graphed below\nquestion help: video

Explanation:

Step1: Identify vertical asymptotes

The vertical asymptotes are $x=- 1$ and $x = 3$. So the denominator of a rational - function will have factors of $(x + 1)$ and $(x - 3)$, i.e., the denominator $q(x)=(x + 1)(x - 3)=x^{2}-2x - 3$.

Step2: Assume the form of the rational function

Let the rational function be $y=\frac{p(x)}{q(x)}=\frac{a(x - h)}{x^{2}-2x - 3}$. We can assume the numerator is a linear function $a(x - h)$ (since the graph has one horizontal asymptote and no holes).

Step3: Use a point on the graph

The graph passes through the point $(2,0)$. Substitute $x = 2$ and $y = 0$ into $y=\frac{a(x - h)}{x^{2}-2x - 3}$. When $x = 2$, $y = 0$, the numerator must be zero at $x = 2$. So the numerator is of the form $a(x - 2)$.

Step4: Use another point to find the value of $a$

Let's use the $y$ - intercept. When $x = 0$, $y=- \frac{2a}{-3}=\frac{2a}{3}$. From the graph, when $x = 0$, $y=-2$. So $\frac{2a}{3}=-2$, which gives $a=-3$.

Answer:

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