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AP Precalculus -1.7 Rational Functions and End Behavior- MCQ Exam Style Questions - Effective Fall 2023

AP Precalculus -1.7 Rational Functions and End Behavior- MCQ Exam Style Questions – Effective Fall 2023

AP Precalculus -1.7 Rational Functions and End Behavior- MCQ Exam Style Questions – AP Precalculus- per latest AP Precalculus Syllabus.

AP Precalculus – MCQ Exam Style Questions- All Topics

Question 

The rational function \( r \) is given by \( r(x) = \frac{x^3 + 8x^2 + 17x + 10}{x^2 + 2x} = \frac{(x+1)(x+2)(x+5)}{x^2 + 2x} \). Which of the following gives equations for all the horizontal asymptotes, vertical asymptotes, and slant asymptotes of the graph of \( r \)?
(A) \( x = 0 \) and \( y = x + 6 \)
(B) \( x = 0 \) and \( x = -2 \)
(C) \( x = 0 \) and \( y = 5x + 10 \)
(D) \( x = 0 \) and \( y = x + 10 \)
▶️ Answer/Explanation
Detailed solution

Factor denominator: \( x^2 + 2x = x(x+2) \). Zeros of denominator: \( x = 0 \) and \( x = -2 \).
Numerator zeros: \( x = -1, -2, -5 \).
Cancel \( x+2 \) (common factor) → hole at \( x = -2 \), not a vertical asymptote.
Vertical asymptote only at \( x = 0 \).
Degree numerator (after cancel) = 2, degree denominator = 1 ⇒ slant asymptote.
Perform division: \( \frac{x^2 + 7x + 5}{x} = x + 7 + \frac{5}{x} \) or redo original before cancel: Divide \( x^3 + 8x^2 + 17x + 10 \) by \( x^2 + 2x \) → quotient \( x + 6 \), remainder \( 5x + 10 \).
Thus slant asymptote: \( y = x + 6 \).
✅ Answer: (A)

Question 

The function \( f \) is given by \( f(x) = -2x^7 + 5x^4 + 6x^2 – 3 \). Which of the following correctly describes the end behavior of \( f \) as the input values increase without bound?
(A) \( \lim_{x \to \infty} f(x) = \infty \)
(B) \( \lim_{x \to \infty} f(x) = -\infty \)
(C) \( \lim_{x \to -\infty} f(x) = \infty \)
(D) \( \lim_{x \to -\infty} f(x) = -\infty \)
▶️ Answer/Explanation
Detailed solution

Leading term: \( -2x^7 \), degree 7 (odd), coefficient negative.
For odd degree with negative leading coefficient:
As \( x \to \infty \), \( f(x) \to -\infty \).
As \( x \to -\infty \), \( f(x) \to \infty \).
The question asks for behavior as \( x \to \infty \).
✅ Answer: (B)

Question 

The function \( f \) is given by \( f(x) = 5x^6 – 2x^3 – 3 \). Which of the following describes the end behavior of \( f \)?
(A) \( \lim_{x \to -\infty} f(x) = -\infty \) and \( \lim_{x \to \infty} f(x) = -\infty \)
(B) \( \lim_{x \to -\infty} f(x) = \infty \) and \( \lim_{x \to \infty} f(x) = \infty \)
(C) \( \lim_{x \to -\infty} f(x) = -\infty \) and \( \lim_{x \to \infty} f(x) = \infty \)
(D) \( \lim_{x \to -\infty} f(x) = \infty \) and \( \lim_{x \to \infty} f(x) = -\infty \)
▶️ Answer/Explanation
Detailed solution

Leading term: \( 5x^6 \), degree 6 (even), coefficient positive.
For even degree with positive leading coefficient:
\( \lim_{x \to \pm\infty} f(x) = \infty \).
✅ Answer: (B)

Question 

The rational function \( f \) is given by \( f(x) = \frac{x^k(x-1)(x+3)}{x^5+2x-5} \), where \( k \) is a positive integer. For which of the following values of \( k \) will the graph of \( f \) have a horizontal asymptote at \( y = 0 \)?
(A) 2
(B) 3
(C) 4
(D) 5
▶️ Answer/Explanation
Detailed solution

The numerator is \( x^k(x-1)(x+3) \).
The degree of the numerator = \( k + 1 + 1 = k+2 \).
The denominator \( x^5 + 2x – 5 \) has degree \( 5 \).
Horizontal asymptote at \( y = 0 \) occurs when degree of denominator > degree of numerator.
So \( 5 > k+2 \) ⇒ \( k < 3 \).
Given \( k \) positive integer: possible \( k = 1 \) or \( 2 \).
Among choices, only \( k = 2 \) satisfies.
✅ Answer: (A) 2

Question 

The rational function \( r \) is given by \( r(x) = \frac{(2x-3)(x-4)(x-2)}{(3x-1)(2x+1)(x-1)} \) and is equivalent to \( r(x) = \frac{p(x)}{q(x)} \), where \( p \) and \( q \) are polynomial functions. Which of the following statements is true?
(A) The degree of \( p \) is less than the degree of \( q \), and \( \lim_{x \to \infty} r(x) = 0 \).
(B) The degree of \( p \) is greater than the degree of \( q \), and \( \lim_{x \to \infty} r(x) = \infty \).
(C) The degree of \( p \) is equal to the degree of \( q \), and \( \lim_{x \to \infty} r(x) = 0 \).
(D) The degree of \( p \) is equal to the degree of \( q \), and \( \lim_{x \to \infty} r(x) = \frac{1}{3} \).
▶️ Answer/Explanation
Detailed solution

Numerator \( p(x) = (2x-3)(x-4)(x-2) \):
Leading term: \( 2x \cdot x \cdot x = 2x^3 \), degree 3.
Denominator \( q(x) = (3x-1)(2x+1)(x-1) \):
Leading term: \( 3x \cdot 2x \cdot x = 6x^3 \), degree 3.
Thus degree of \( p \) = degree of \( q \) = 3.
For large \( x \), \( r(x) \approx \frac{2x^3}{6x^3} = \frac{1}{3} \).
So \( \lim_{x \to \infty} r(x) = \frac{1}{3} \).
✅ Answer: (D)

Question 

The rational function \( h \) is expressed as the quotient of two polynomial functions \( f \) and \( g \) by \( h(x) = \frac{f(x)}{g(x)} \). The function \( f \) is given by \( f(x) = 6x^3 – x^2 + 60x – 25 \). If the graph of \( h \) has a slant asymptote of \( y = 2x – 1 \), which of the following describes \( g \)?
(A) \( g \) has degree 2 with leading coefficient 3.
(B) \( g \) has degree 2 with leading coefficient 12.
(C) \( g \) has degree 3 with leading coefficient 3.
(D) \( g \) has degree 4 with leading coefficient 12.
▶️ Answer/Explanation
Detailed solution

A rational function has a slant (oblique) asymptote when the degree of the numerator is exactly one more than the degree of the denominator.
Given \( f \) is degree 3, so \( g \) must be degree 2.
Slant asymptote \( y = 2x – 1 \) is the quotient from polynomial long division.
Leading term of \( f \) is \( 6x^3 \), leading term of \( g \) is \( ax^2 \).
\( \frac{6x^3}{ax^2} = \frac{6}{a}x \), which equals \( 2x \) (from \( 2x – 1 \)).
So \( \frac{6}{a} = 2 \) ⇒ \( a = 3 \).
Thus \( g \) has degree 2, leading coefficient 3.
✅ Answer: (A)

Question 

The rational function \( h \) is given by \( h(x) = \frac{2x^5 + 5x^3 – 2x^2 – 13}{3x^2 – 2x + 7} \). Which of the following describes the end behavior of \( h \)?
(A) As \( x \) increases without bound, \( h(x) \) increases without bound, and as \( x \) decreases without bound, \( h(x) \) decreases without bound.
(B) As \( x \) increases without bound, \( h(x) \) increases without bound, and as \( x \) decreases without bound, \( h(x) \) increases without bound.
(C) As \( x \) increases without bound, \( h(x) \) decreases without bound, and as \( x \) decreases without bound, \( h(x) \) increases without bound.
(D) As \( x \) increases without bound, \( h(x) \) decreases without bound, and as \( x \) decreases without bound, \( h(x) \) decreases without bound.
▶️ Answer/Explanation
Detailed solution

Degree of numerator: 5, degree of denominator: 2.
Since numerator degree > denominator degree, no horizontal asymptote; end behavior ≈ \( \frac{2x^5}{3x^2} = \frac{2}{3}x^3 \).
Cubic \( \frac{2}{3}x^3 \) has positive leading coefficient.
For large positive \( x \), \( x^3 \) → +∞ ⇒ \( h(x) \) → +∞.
For large negative \( x \), \( x^3 \) → −∞ ⇒ \( h(x) \) → −∞.
Thus: as \( x \to +\infty \), \( h(x) \to +\infty \); as \( x \to -\infty \), \( h(x) \to -\infty \).
✅ Answer: (A)

Question 

A polynomial function \( p \) has three distinct zeros each with multiplicity 1, and its leading coefficient is positive. The polynomial function \( q \) has exactly one zero with multiplicity 3, and its leading coefficient is negative. The rational function \( h \) can be written as the quotient of \( p \) and \( q \) by \( h(x) = \frac{p(x)}{q(x)} \). Which of the following statements about \( h \) must be true?
(A) The graph of \( h \) has a horizontal asymptote at \( y = 0 \).
(B) The graph of \( h \) has a horizontal asymptote at \( y = a \), where \( a > 0 \).
(C) The graph of \( h \) has a horizontal asymptote at \( y = a \), where \( a < 0 \).
(D) The graph of \( h \) has no horizontal asymptote.
▶️ Answer/Explanation
Detailed solution

\( p \) has three distinct zeros, each multiplicity 1 ⇒ degree \( p \) = 3, leading coefficient positive.
\( q \) has exactly one zero with multiplicity 3 ⇒ degree \( q \) = 3, leading coefficient negative.
Thus degree of numerator = degree of denominator = 3.
Horizontal asymptote exists at \( y = \frac{\text{leading coefficient of } p}{\text{leading coefficient of } q} \).
Let leading coefficient of \( p \) be \( A > 0 \), of \( q \) be \( B < 0 \).
Then \( a = \frac{A}{B} < 0 \).
So horizontal asymptote is \( y = a \) where \( a < 0 \).
✅ Answer: (C)

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