Graphing Rational Functions (Grades 11-12)
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Graphing Rational Functions
1.
Under which circumstances will the graph of the rational function [math]f(x) = (p(x))/(q(x))[/math], where [math]p(x)[/math] and [math]q(x)[/math] are both polynomial functions, have at least one vertical asymptote?
- If [math]p(x)[/math] and [math]q(x)[/math] both have the same zeros.
- If [math]p(x)[/math] has more zeros than [math]q(x)[/math].
- If the degree of [math]p(x)[/math] is greater than the degree of [math]q(x)[/math].
- If [math]q(x)[/math] has a zero which is not a zero of [math]p(x)[/math].
2.
Let [math]f[/math] be a rational function such that [math]f(x) = (p(x))/(q(x))[/math], where [math]p(x)[/math] is a polynomial of degree [math]m[/math] and leading coefficient [math]a[/math], while [math]q(x)[/math] is a polynomial of degree [math]n[/math] and leading coefficient [math]b[/math]. For each choice about the relationship between the degrees of the polynomials [math]p(x)[/math] and [math]q(x)[/math], choose the correct fact concerning potential horizontal asymptotes of the graph of [math]f(x)[/math].
[math]m>n[/math] | [math]m=n[/math] | [math]m < n[/math] | |
No horizontal asymptote | |||
The horizontal asymptote is the x-axis | |||
The horizontal asymptote is the line [math]y =a/b[/math] |
3.
When graphed, the function [math]f(x) = (x^3 + x^2 -10x + 8)/(x^2-3)[/math] has a slant or oblique asymptote. What is the equation of this asymptote?
- [math]y=-7x+11[/math]
- [math]y=x+1[/math]
- [math]y=x - 8/3[/math]
- [math]y = -10x + 8[/math]
4.
For the following questions, let [math]f(x) = (p(x)) / (q(x)) = (x^3 + 4x^2 - 25x - 28) / (x^3 - x^2 - 17x - 15)[/math].
A.
What are the zeros of [math]p(x) ?[/math] Choose all correct answers.
- 1
- -1
- -7
- 4
B.
What are the zeros of [math]q(x) ?[/math] Choose all correct answers.
- -3
- -1
- 1
- 5
C.
Which of the following are vertical asymptotes of the function [math]f(x)?[/math] There may be more than one correct answer.
- [math]x=-3[/math]
- [math]x=-1[/math]
- [math]x=5[/math]
- This function has no vertical asymptotes.
D.
Which of the following is correct, concerning the horizontal asymptote of the function [math]f(x) ?[/math]
- The horizontal asymptote is [math]x=1[/math].
- The horizontal asymptote is [math]x = 7/3[/math].
- The horizontal asymptote is [math]x = -1[/math].
- This function has no horizontal asymptote.
E.
Graph the function [math]f(x)[/math]. Fully label the graph.

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

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

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