Precalculus Rational Functions and Asymptotes Test with Point Values (PDF)
This rational functions test for Precalculus covers rational functions and asymptotes (Common Core A-APR.D.7, F-IF.C.7d, A-CED.A.1), with a free PDF to print.
Time and points
- Grade: Precalculus
- Chapter: Rational Functions and Asymptotes
- Standards: A-APR.D.7, F-IF.C.7d, A-CED.A.1
- Time: 45 minutes
- Total: 20 points
Rational functions test: questions
Question 1 – Domain and vertical asymptotes [2 pts]
For \(f(x)=\frac{x-4}{x^{2}-9}\), state the domain and the equations of the vertical asymptotes.
Question 2 – Horizontal asymptotes [2 pts]
Find the horizontal asymptote of (a) \(\frac{5x^{2}-3}{2x^{2}+x}\); (b) \(\frac{4x+1}{x^{2}+2}\).
Question 3 – Hole and asymptote [2 pts]
For \(f(x)=\frac{x^{2}-x-6}{x^{2}-2x-3}\), find the hole and the vertical asymptote.
Question 4 – Slant asymptote [2 pts]
Find the slant asymptote of \(f(x)=\frac{2x^{2}+3x-1}{x+1}\).
Question 5 – Read a rational graph [3 pts]
The graph of \(f\) is shown with its asymptotes dashed. (a) Write the equations of the asymptotes. (b) State the intercepts. (c) State the domain and range.
Question 6 – Solve a rational equation [2 pts]
Solve \(\frac{4}{x+1}=\frac{3}{x-2}\).
Question 7 – Extraneous solution [2 pts]
Solve \(\frac{x}{x-2}=\frac{2}{x-2}+3\).
Question 8 – Average cost [3 pts]
Making \(n\) bracelets costs \(\$800\) plus \(\$15\) each, so \(A(n)=\frac{800+15n}{n}\). (a) Find \(A(40)\). (b) Explain the horizontal asymptote. (c) Find \(n\) for an average cost of \(\$20\).
Question 9 – Rational inequality [2 pts]
Solve \(\frac{x+3}{x-1}< 0\).
Key facts behind this rational functions test
- Domain: all real numbers except zeros of the denominator.
- Vertical asymptote: a denominator zero that does not cancel. Hole: a cancelled factor.
- Degree of numerator < denominator: y = 0. Equal degrees: y = ratio of leading coefficients.
- Degree of numerator = degree of denominator + 1: slant asymptote from division.
- Always check rational equation answers for extraneous solutions.
How to use this rational functions test
Time yourself: the test is designed for an average student in 45 minutes. Answer the questions you are sure of first, then come back to the longer ones and check your units.
All Precalculus chapters: Precalculus math.
