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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 [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?
  1. If [math]p(x)[/math] and [math]q(x)[/math] both have the same zeros.
  2. If [math]p(x)[/math] has more zeros than [math]q(x)[/math].
  3. If the degree of [math]p(x)[/math] is greater than the degree of [math]q(x)[/math].
  4. 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?
  1. [math]y=-7x+11[/math]
  2. [math]y=x+1[/math]
  3. [math]y=x - 8/3[/math]
  4. [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
  2. -1
  3. -7
  4. 4
B. 
What are the zeros of [math]q(x) ?[/math] Choose all correct answers.
  1. -3
  2. -1
  3. 1
  4. 5
C. 
Which of the following are vertical asymptotes of the function [math]f(x)?[/math] There may be more than one correct answer.
  1. [math]x=-3[/math]
  2. [math]x=-1[/math]
  3. [math]x=5[/math]
  4. This function has no vertical asymptotes.
D. 
Which of the following is correct, concerning the horizontal asymptote of the function [math]f(x) ?[/math]
  1. The horizontal asymptote is [math]x=1[/math].
  2. The horizontal asymptote is [math]x = 7/3[/math].
  3. The horizontal asymptote is [math]x = -1[/math].
  4. This function has no horizontal asymptote.
E. 
Graph the function [math]f(x)[/math]. Fully label the graph.
Graph 10x10



5. 
Graph the function [math]f(x) = (x^2 + x - 20)/(x^3 -9x)[/math]. Label the graph fully.
Graph 10x10



6. 
Graph the function [math]f(x) = (x^3 - x^2 - 16x + 16)/(x^2 + 8)[/math]. Label the graph fully.
Graph 10x10



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