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This printable supports Common Core Mathematics Standard HSF-IF.C.7, HSF-IF.C.7d

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# Graphing Rational Functions (Grades 11-12)

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## Graphing Rational Functions

1.
Under which circumstances will the graph of the rational function $f(x) = (p(x))/(q(x))$, where $p(x)$ and $q(x)$ are both polynomial functions, have at least one vertical asymptote?
1. If $p(x)$ and $q(x)$ both have the same zeros.
2. If $p(x)$ has more zeros than $q(x)$.
3. If the degree of $p(x)$ is greater than the degree of $q(x)$.
4. If $q(x)$ has a zero which is not a zero of $p(x)$.
2.
Let $f$ be a rational function such that $f(x) = (p(x))/(q(x))$, where $p(x)$ is a polynomial of degree $m$ and leading coefficient $a$, while $q(x)$ is a polynomial of degree $n$ and leading coefficient $b$. For each choice about the relationship between the degrees of the polynomials $p(x)$ and $q(x)$, choose the correct fact concerning potential horizontal asymptotes of the graph of $f(x)$.
 $m>n$ $m=n$ $m < n$ No horizontal asymptote The horizontal asymptote is the x-axis The horizontal asymptote is the line $y =a/b$
3.
When graphed, the function $f(x) = (x^3 + x^2 -10x + 8)/(x^2-3)$ has a slant or oblique asymptote. What is the equation of this asymptote?
1. $y=-7x+11$
2. $y=x+1$
3. $y=x - 8/3$
4. $y = -10x + 8$
4.
For the following questions, let $f(x) = (p(x)) / (q(x)) = (x^3 + 4x^2 - 25x - 28) / (x^3 - x^2 - 17x - 15)$.
A.
What are the zeros of $p(x) ?$ Choose all correct answers.
1. 1
2. -1
3. -7
4. 4
B.
What are the zeros of $q(x) ?$ Choose all correct answers.
1. -3
2. -1
3. 1
4. 5
C.
Which of the following are vertical asymptotes of the function $f(x)?$ There may be more than one correct answer.
1. $x=-3$
2. $x=-1$
3. $x=5$
4. This function has no vertical asymptotes.
D.
Which of the following is correct, concerning the horizontal asymptote of the function $f(x) ?$
1. The horizontal asymptote is $x=1$.
2. The horizontal asymptote is $x = 7/3$.
3. The horizontal asymptote is $x = -1$.
4. This function has no horizontal asymptote.
E.
Graph the function $f(x)$. Fully label the graph.

5.
Graph the function $f(x) = (x^2 + x - 20)/(x^3 -9x)$. Label the graph fully.

6.
Graph the function $f(x) = (x^3 - x^2 - 16x + 16)/(x^2 + 8)$. Label the graph fully.

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