004 Sample Final A, Problem 9

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Graph the function. Give equations of any asymptotes, and list any intercepts.     ${\displaystyle y={\frac {6}{x^{2}-x-2}}}$

Foundations
What are the vertical asymptotes of ${\displaystyle f(x)}$?
Answer:
The vertical asymptotes are ${\displaystyle x=a}$ where ${\displaystyle f(a)}$ is undefined.

Solution:

Step 1:
Since ${\displaystyle y={\frac {6}{(x-2)(x+1)}}}$, the vertical asymptotes are ${\displaystyle x=2}$ and ${\displaystyle x=-1}$.
Step 2:
The horizontal asymptote is ${\displaystyle y=0}$.
Step 3:
There is no ${\displaystyle x}$ intercept since ${\displaystyle y\neq 0}$ for any ${\displaystyle x}$.
Plugging in ${\displaystyle x=0}$, we get ${\displaystyle y=-3}$. So, the ${\displaystyle y}$ intercept is (0,3)
Final Answer:
The vertical asymptotes are ${\displaystyle x=2}$ and ${\displaystyle x=-1}$. The horizontal asymptote is ${\displaystyle y=0}$. There is no ${\displaystyle x}$ intercept and the ${\displaystyle y}$ intercept is (0,3).