AP PreCalculus Practice Quiz: Rational Functions and End Behavior
Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026
Test your understanding with short quizzes. This quiz has 14 questions to check your progress.
Question 1 of 14
All Questions (14)
A) As a product of two polynomial functions.
B) As a quotient of two polynomial functions.
C) As the leading term of a single polynomial.
D) As a constant value representing an asymptote.
Correct Answer: B
The content explicitly states that 'A rational function is analytically represented as a quotient of two polynomial functions'.
A) The y-intercept of the function.
B) The constant terms of the polynomials.
C) The quotient of the leading terms.
D) The entire polynomial in the denominator.
Correct Answer: C
The text states, 'The end behavior of a rational function can be understood by examining the corresponding quotient of the leading terms.'
A) The graph has a horizontal asymptote at y = 0.
B) The graph has a horizontal asymptote at y = 1.
C) The graph's end behavior is that of a nonconstant polynomial.
D) The graph has no predictable end behavior.
Correct Answer: A
The content specifies that 'If the polynomial in the denominator dominates the polynomial in the numerator for input values of large magnitude, the graph has a horizontal asymptote at y = 0.'
A) The graph has a vertical asymptote at x = b.
B) The graph has a y-intercept at y = b.
C) The graph has a horizontal asymptote at y = b.
D) The function's value is always equal to b.
Correct Answer: C
The text states, 'When the graph of a rational function r has a horizontal asymptote at y = b, the corresponding mathematical notation is lim r(x) = b or lim r(x) = b.'
A) When the polynomial in the denominator dominates the polynomial in the numerator.
B) When the polynomial in the numerator dominates the polynomial in the denominator.
C) When neither polynomial dominates the other.
D) When the quotient of the leading terms is a constant.
Correct Answer: B
This is stated in the text: 'If the polynomial in the numerator dominates the polynomial in the denominator for input values of large magnitude, the graph has the end behavior of the nonconstant polynomial quotient.'
A) The function has a horizontal asymptote at y = 0.
B) The function's end behavior is that of a nonconstant polynomial.
C) The function has a horizontal asymptote at the location indicated by that constant.
D) The function has no horizontal asymptote.
Correct Answer: C
The provided content says, 'If neither polynomial dominates the other... the quotient of the leading terms is a constant, and that constant indicates the location of a horizontal asymptote.'
A) The leading terms compared to the constant terms.
B) The function's domain compared to its range.
C) The polynomial in the numerator compared to the polynomial in the denominator.
D) The horizontal asymptote compared to the vertical asymptote.
Correct Answer: C
The definition in the text says a rational function 'gives a measure of the relative size of the polynomial function in the numerator compared to the polynomial function in the denominator'.
A) The numerator polynomial dominates the denominator polynomial.
B) The denominator polynomial dominates the numerator polynomial.
C) Neither polynomial dominates the other for input values of large magnitude.
D) The numerator and denominator are identical polynomials.
Correct Answer: C
The text explains that if a horizontal asymptote exists at a constant value (which would be non-zero in this case), it's because 'neither polynomial dominates the other for input values of large magnitude' and 'the quotient of the leading terms is a constant'.
A) Values close to zero.
B) Values where the denominator is zero.
C) Values of large magnitude.
D) All values in the function's domain.
Correct Answer: C
The text repeatedly qualifies the concept of dominance with the phrase 'for input values of large magnitude,' indicating this analysis applies to end behavior (as x approaches ±∞).
A) The quotient of the leading terms is a non-zero constant.
B) The polynomial in the numerator dominates the polynomial in the denominator.
C) The polynomial in the denominator dominates the polynomial in the numerator.
D) The quotient of the leading terms is a nonconstant polynomial.
Correct Answer: C
The text explicitly isolates one condition for a horizontal asymptote at y = 0: 'If the polynomial in the denominator dominates the polynomial in the numerator for input values of large magnitude, the graph has a horizontal asymptote at y = 0.' The other options describe different end behaviors.
A) cancel each other out, leaving only the constant terms.
B) are the primary determinants of the polynomials' relative sizes.
C) approach zero, simplifying the function.
D) create vertical asymptotes that define the function's limits.
Correct Answer: B
The text implies this by stating that end behavior is understood by the quotient of leading terms and then describing outcomes based on which polynomial 'dominates'. For large x, the leading term has the greatest impact on the value of a polynomial, thus determining its 'size' relative to another polynomial.
A) If the numerator is relatively larger, their quotient approaches zero, creating an asymptote at y=0.
B) If the numerator is relatively larger, their quotient grows without bound in the manner of a polynomial.
C) If the numerator is relatively larger, their relative size becomes a constant, creating a horizontal asymptote.
D) The 'relative size' is only relevant when the denominator dominates.
Correct Answer: B
The text connects these ideas. A rational function measures 'relative size'. If the numerator 'dominates' (is relatively larger for large inputs), the function's value will not be constant but will grow, and its graph will have the end behavior of the 'nonconstant polynomial quotient'.
A) In A, the denominator dominates; in B, the numerator dominates.
B) In A, the numerator dominates; in B, the denominator dominates.
C) In A, the numerator dominates; in B, neither polynomial dominates the other.
D) In A, neither polynomial dominates; in B, the denominator dominates.
Correct Answer: C
According to the text, end behavior like a nonconstant polynomial occurs when the numerator dominates. A horizontal asymptote at a constant (like y=5) occurs when neither polynomial dominates the other. This question requires synthesizing two separate rules from the text.
A) The location of the function's y-intercept.
B) The zeros of the polynomial in the denominator.
C) The relative growth rate of the numerator versus the denominator, as indicated by their leading terms.
D) The sum of the coefficients of the two polynomials.
Correct Answer: C
This question requires abstracting a single principle from all the provided rules. The text describes the function as a 'measure of the relative size' and explains all end behaviors in terms of which polynomial 'dominates' or if they are balanced. This 'dominance' is about their relative growth for large inputs, which is determined by the leading terms. Therefore, the relative growth rate is the unifying principle.