write an equation for the function graphed below\n$y = $

Explanation:

Step1: Identify vertical asymptotes

The graph has vertical asymptotes at $x = -1$ and $x = 4$.

Step2: Identify the x-intercept

The graph crosses the x-axis at $x = 1$.

Step3: Identify the horizontal asymptote

As $x \to \pm\infty$, the graph approaches $y = 0$.

Step4: Determine the function's behavior

The graph changes sign at $x = -1$ and $x = 4$, suggesting odd multiplicity for denominators.

Step5: Formulate the rational function

$$y = \frac{a(x - 1)}{(x + 1)(x - 4)}$$

Step6: Solve for the constant $a$

Using the y-intercept $(0, 1)$: $1 = \frac{a(0 - 1)}{(0 + 1)(0 - 4)} \Rightarrow 1 = \frac{-a}{-4} \Rightarrow a = 4$.

Step7: Write the final equation

$$y = \frac{4(x - 1)}{(x + 1)(x - 4)}$$

Answer:

y = \frac{4(x - 1)}{(x + 1)(x - 4)}