Question

write an equation for the function graphed below
y =
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Explanation:

Step1: Identify vertical asymptotes

The vertical dashed lines are at $x=-1$, $x=4$. So the denominator has factors $(x+1)(x-4)$. The function has the form $y=\frac{a(x-r)}{(x+1)(x-4)}$.

Step2: Identify x-intercept

The graph crosses the x-axis at $x=2$, so $r=2$. Now the function is $y=\frac{a(x-2)}{(x+1)(x-4)}$.

Step3: Solve for scaling factor $a$

Use a point on the graph, e.g., when $x=5$, $y\approx1$. Substitute into the equation:
$1=\frac{a(5-2)}{(5+1)(5-4)}$
$1=\frac{3a}{6}$
$3a=6$
$a=2$

Step4: Verify the function

Substitute $a=2$ to get the final function: $y=\frac{2(x-2)}{(x+1)(x-4)}$. We can expand the denominator: $(x+1)(x-4)=x^2-3x-4$, so $y=\frac{2x-4}{x^2-3x-4}$.

Answer:

$\boldsymbol{y=\frac{2(x-2)}{(x+1)(x-4)}}$ or $\boldsymbol{y=\frac{2x-4}{x^2-3x-4}}$