Write an equation for the function graphed below.

Steps to Solve

1. Identify the vertical asymptotes

The vertical asymptotes are $x = -1$ and $x = 4$. This means the denominator of the rational function will have factors of $(x+1)$ and $(x-4)$.

2. Identify the horizontal asymptote

The horizontal asymptote is $y = 0$. This tells us that the degree of the numerator is less than the degree of the denominator. Since we have two vertical asymptotes, we can assume the denominator will be a quadratic. Therefore with the horizontal asymptote at $y=0$, the numerator will be a constant.

3. Write a general form of the equation

Based on the vertical and horizontal asymptotes, the equation will look like: $$ y = \frac{A}{(x+1)(x-4)} $$ where $A$ is a constant.

4. Find a point on the graph and solve for A

The graph appears to pass through the point $(0, 0.5)$ or $(0, 1/2)$. Substitute this point into the equation and solve for $A$:

$$ \frac{1}{2} = \frac{A}{(0+1)(0-4)} $$ $$ \frac{1}{2} = \frac{A}{(1)(-4)} $$ $$ \frac{1}{2} = \frac{A}{-4} $$ $$ A = -2 $$

5. Write the final equation

Substitute the value of $A$ back into the general equation: $$ y = \frac{-2}{(x+1)(x-4)} $$