AP PreCalculus Flashcards: Rational Functions and Holes
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.
What must be true about a specific input value for a hole to even be a possibility in a rational function?
The input value must be a real zero of both the numerator and the denominator of the rational function.
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What must be true about a specific input value for a hole to even be a possibility in a rational function?
The input value must be a real zero of both the numerator and the denominator of the rational function.
How is the precise location of a hole at x = c determined for a rational function?
The location is found by examining the output values (y-values) that correspond to input values (x-values) sufficiently close to c.
Explain the role of multiplicity in determining whether a shared zero between the numerator and denominator results in a hole.
A hole is created only when the multiplicity of the shared zero is greater in the numerator, or equal in both, allowing the discontinuity to be 'removable'.
A real zero has equal multiplicity in both the numerator and denominator. Does the graph have a hole at this input value?
Yes, the graph has a hole because the condition states that the numerator's multiplicity must be greater than or equal to the denominator's multiplicity.
What is a hole in the graph of a rational function?
A hole is a point of discontinuity that occurs at an input value where a real zero in the numerator has a multiplicity greater than or equal to its multiplicity in the denominator.
Why is it necessary to examine output values 'sufficiently close' to c to find a hole's location, rather than at c itself?
The function is undefined at x = c due to division by zero, so we must examine the function's behavior as it approaches c to find the y-value of the hole.
If a rational function has a hole at x = c, what can you infer about the factors of the numerator and denominator?
Both the numerator and denominator share a factor of (x - c), and the exponent of that factor in the numerator is greater than or equal to its exponent in the denominator.
What is the general process for determining holes in the graphs of rational functions?
The process involves identifying shared real zeros in the numerator and denominator and comparing their multiplicities to see if the condition for a hole is met.
A real zero has a multiplicity of 2 in the numerator and 1 in the denominator. What graphical feature exists at this input value?
The graph of the rational function has a hole at the corresponding input value because the numerator's multiplicity is greater than the denominator's.
What condition related to the multiplicities of a real zero causes a hole in the graph of a rational function?
A hole occurs if the multiplicity of a real zero in the numerator is greater than or equal to its multiplicity in the denominator.