AP PreCalculus Flashcards: Rational Functions and End Behavior
Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026
Review key ideas with interactive flashcards. This set includes 16 cards to help you master important concepts.
What is the end behavior of a rational function if the polynomial in the denominator dominates the polynomial in the numerator?
If the polynomial in the denominator dominates the numerator for input values of large magnitude, the graph has a horizontal asymptote at y = 0.
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What is the end behavior of a rational function if the polynomial in the denominator dominates the polynomial in the numerator?
If the polynomial in the denominator dominates the numerator for input values of large magnitude, the graph has a horizontal asymptote at y = 0.
What is meant by 'end behavior' of a rational function?
End behavior describes how the function behaves for input values of large magnitude, as x approaches positive or negative infinity.
How are horizontal asymptotes and the concept of 'dominating' polynomials related?
Horizontal asymptotes exist when one polynomial does not dominate the other (non-zero asymptote) or when the denominator dominates the numerator (asymptote at y=0).
The leading term of a rational function's numerator is x⁴ and the denominator's is x². Does this function have a horizontal asymptote?
No, because the numerator dominates the denominator, the graph has the end behavior of the nonconstant polynomial quotient (x²).
The leading term in the numerator of a rational function is x², and in the denominator it is x³. What is the horizontal asymptote?
The polynomial in the denominator dominates the numerator, so the graph has a horizontal asymptote at y = 0.
If you are told that lim r(x) = 0 as x approaches infinity, what can you conclude about the polynomials in r(x)?
You can conclude that for large input values, the polynomial in the denominator dominates the polynomial in the numerator.
What does a rational function fundamentally measure?
It gives a measure of the relative size of the polynomial function in the numerator compared to the polynomial function in the denominator.
How is a horizontal asymptote at y = b for a function r(x) expressed using mathematical notation?
The notation is lim r(x) = b as x approaches positive infinity or lim r(x) = b as x approaches negative infinity.
What is the 'quotient of the leading terms'?
It is the term with the highest power in the numerator polynomial divided by the term with the highest power in the denominator polynomial.
A rational function's leading terms are 5x³ in the numerator and 2x³ in the denominator. What does this imply about its end behavior?
Since neither polynomial dominates, the quotient of the leading terms (5/2) is a constant, indicating a horizontal asymptote at y = 5/2.
What is a rational function?
A rational function is analytically represented as a quotient of two polynomial functions.
What determines which polynomial 'dominates' in a rational function for large input values?
The relative size of the leading terms of the polynomials in the numerator and denominator determines which one dominates.
Under what condition does a rational function's graph have a horizontal asymptote at a non-zero constant 'b'?
This occurs when neither the numerator nor denominator polynomial dominates the other, and the quotient of their leading terms is the constant 'b'.
What is the primary method for understanding the end behavior of a rational function?
The end behavior of a rational function can be understood by examining the corresponding quotient of the leading terms.
What is the end behavior of a rational function if the polynomial in the numerator dominates the polynomial in the denominator?
The graph has the end behavior of the nonconstant polynomial quotient, meaning it does not have a horizontal asymptote.
What does it mean for the end behavior if the quotient of the leading terms is a constant?
It means neither polynomial dominates the other for large input values, and that constant indicates the location of a horizontal asymptote.