Question

write an equation for the function graphed below.
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Explanation:

Step1: Identify vertical asymptotes

The vertical dashed lines are at $x=-3$ and $x=4$, so the denominator has factors $(x+3)$ and $(x-4)$. The denominator is $(x+3)(x-4) = x^2 - x - 12$.

Step2: Identify x-intercept

The graph crosses the x-axis at $x=3$, so the numerator has a factor $(x-3)$. Let the numerator be $a(x-3)$ where $a$ is a constant.

Step3: Solve for constant $a$

Use the y-intercept: when $x=0$, $y=1$. Substitute into $y=\frac{a(x-3)}{(x+3)(x-4)}$:
$1=\frac{a(0-3)}{(0+3)(0-4)}$
$1=\frac{-3a}{-12}$
$1=\frac{a}{4}$
$a=4$

Step4: Write final equation

Substitute $a=4$ into the function:
$y=\frac{4(x-3)}{(x+3)(x-4)}$ or expanded $y=\frac{4x-12}{x^2 - x - 12}$

Answer:

$y=\frac{4(x-3)}{(x+3)(x-4)}$