Question

0/1 pt 3 99 details
write an equation for the function graphed below
$y =$
question help: video read written example

Explanation:

Step1: Identify vertical asymptotes

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

Step2: Identify horizontal asymptote

The graph approaches $y=0$ as $x\to\pm\infty$, so the numerator degree < denominator degree. Assume numerator is a constant $k$ first, then use a point to solve for $k$.

Step3: Use intercept point $(2,0)$

Wait, correction: the graph crosses the x-axis at $x=2$, so numerator has a factor $(x-2)$. Now the function form is $y=\frac{k(x-2)}{(x+3)(x-4)}$.

Step4: Use y-intercept $(0,1)$

Substitute $x=0, y=1$:
$1=\frac{k(0-2)}{(0+3)(0-4)}$
$1=\frac{-2k}{-12}=\frac{k}{6}$
Solve for $k$: $k=6$

Step5: Final function

Substitute $k=6$ into the form:
$y=\frac{6(x-2)}{(x+3)(x-4)}$ or expanded: $y=\frac{6x-12}{x^2 - x -12}$

Answer:

$\boldsymbol{y=\frac{6(x-2)}{(x+3)(x-4)}}$ (or $\boldsymbol{y=\frac{6x-12}{x^2 - x -12}}$)