AP Precalculus – Rational Functions: Asymptotes & Holes – Drill 1

About This Drill

AP Precalculus – Rational Functions: Asymptotes & Holes – Drill 1 is a Math practice drill covering Rational Functions: Asymptotes & Holes. It contains 5 original questions created by Brian Stewart, a Barron's test prep author with over 20 years of tutoring experience.

This AP® Precalculus drill focuses on vertical asymptotes and holes in rational functions (Topics 1.9–1.10). Practice identifying when a common factor creates a removable discontinuity (hole) versus when a denominator factor produces a vertical asymptote. Master this distinction — it's a high-frequency trap on the AP® exam.

Questions in This Drill

1. Which of the following rational functions has a vertical asymptote at x = −3 and a hole at x = 4?
2. Let \( h(x) = \dfrac{x^2 - 9}{x^2 + 2x - 15} \). Which of the following correctly identifies all holes and vertical asymptotes of h?
3. The table below shows values of a rational function r(x) for inputs near x = 2 and x = 5.
   
   

   x	1.9	1.99	2.01	2.1	4.9	4.99	5.01	5.1
   r(x)	0.31	0.330	0.336	0.36	−98	−998	1002	102

   
   Which statement is best supported by the table?
4. Let \( p(x) = \dfrac{x^3 - 4x}{x^2 - x - 2} \). Which of the following correctly identifies the x-intercepts, holes, and vertical asymptotes of p?
5. A student analyzes \( r(x) = \dfrac{x^2 - 5x + 6}{x^2 - 4} \) and claims: "Since the denominator equals zero at x = 2, the function has a vertical asymptote at x = 2." Which of the following best evaluates this claim?