AP PreCalculus Flashcards: Rational Functions and Vertical Asymptotes
Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026
Review key ideas with interactive flashcards. This set includes 10 cards to help you master important concepts.
If a value 'a' is a real zero of BOTH the numerator and the denominator, does the graph have a vertical asymptote at x = a?
No, according to the rule, a vertical asymptote does not exist at x = a because 'a' is also a real zero of the numerator.
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If a value 'a' is a real zero of BOTH the numerator and the denominator, does the graph have a vertical asymptote at x = a?
No, according to the rule, a vertical asymptote does not exist at x = a because 'a' is also a real zero of the numerator.
What is a 'real zero' of a polynomial function?
A real zero is a real number that, when substituted for the variable in the polynomial, results in the polynomial having a value of zero.
A rational function's denominator is zero at x = -1. The numerator is not zero at x = -1. What can you conclude?
You can conclude that the graph of the rational function has a vertical asymptote at x = -1.
If you find that x = 4 is a real zero of a rational function's denominator, what else must you check to confirm it's a vertical asymptote?
You must check if x = 4 is also a real zero of the polynomial function in the numerator. If it is not, then x = 4 is a vertical asymptote.
What is a vertical asymptote of a rational function's graph?
It is a vertical line, x = a, where the function's values increase or decrease without bound as x approaches a.
What does it mean for a function's values to 'increase or decrease without bound'?
It means the function's output values approach positive infinity (increase) or negative infinity (decrease) as the input approaches a certain value.
Describe the behavior of the values of a rational function, r, as the input x gets closer to a vertical asymptote.
Near a vertical asymptote, the values of the rational function r increase or decrease without bound.
Why is a vertical asymptote defined by a zero of the denominator?
A zero in the denominator makes the rational function undefined at that x-value, which is a necessary condition for the function's values to increase or decrease without bound.
What is the specific condition required for a rational function to have a vertical asymptote at x = a?
The value 'a' must be a real zero of the polynomial in the denominator and must not also be a real zero of the polynomial in the numerator.
What is the first step to determine the vertical asymptotes of a rational function?
The first step is to find all the real zeros of the polynomial function that is in the denominator.