PreCalculus Honors Unit Test #3

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For all questions on this test, let $f\left(x\right)=\frac{{x}^{3}-{x}^{2}-2x}{{x}^{2}-4}$.

 1 The graph of $f\left(x\right)$ has one hole. Give its coordinates. (3)

First, the factorization: $\frac{\left(x\right)\left(x-2\right)\left(x+1\right)}{\left(x+2\right)\left(x-2\right)}$. The common factor of $\left(x-2\right)$ indicates a hole at $x=2$; its y-coordinate is at $\frac{\left(2\right)\left(2+1\right)}{\left(2+2\right)}=\frac{3}{2}$. For all other questions, the rational function is really $\frac{\left(x\right)\left(x+1\right)}{x+2}=\frac{{x}^{2}+x}{x+2}$.

 2 The graph of $f\left(x\right)$ has one vertical asymptote. Give its equation. (3)

It is at $x=-2$.

 3 The graph of $f\left(x\right)$ has two axis intercepts. Give their coordinates. (4)

They are at $\left(0,0\right)$ and $\left(-1,0\right)$.

 4 Determine the equation of the end behavior asymptote of $f\left(x\right)$. (4)

Division time!

$\begin{array}{rrrr}\hfill -2& \hfill 1& \hfill 1& \hfill 0\\ \hfill & \hfill & \hfill -2& \hfill 2\\ \hfill & \hfill 1& \hfill -1& \hfill 2\end{array}$

The end behavior asymptote is at $y=x-1$.

 5 Give a labeled sketch of the graph of $f\left(x\right)$ in the space below. (10)

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