AP Precalculus, AP exams, 30–40%

Unit 1: Polynomial and Rational Functions

Unit 1 establishes the analytical vocabulary of the course: how quantities change in tandem, average rates of change, and the behavior of polynomial and rational functions. End behavior, zeros, asymptotes, and holes are all analyzed through the structure of the expression. Transformations and function modeling close the unit and recur in Units 2 and 3.

Position: 1 of 4 units
Published weighting: 30–40% in the official framework
Topics covered: 6 topics in this guide

What is covered

Topics inside this unit.

These topics describe the scope students most often review for this part of AP Precalculus. Treat them as a study checklist rather than an exhaustive list of what a single exam form can sample.

Change in tandem and rates of change: Describing how one quantity changes as another does, and computing and interpreting average rates of change.
Polynomial functions and end behavior: Degree and leading coefficient as determinants of end behavior, and turning points and extrema.
Zeros, multiplicity, and complex zeros: How multiplicity shapes the graph at a zero, and what complex zeros imply about the real graph.
Rational functions: Vertical asymptotes, holes, and end behavior determined by comparing numerator and denominator degrees.
Equivalent representations: Rewriting polynomial and rational expressions to reveal a specific feature such as a zero or an asymptote.
Transformations and function modeling: Shifts, stretches, and reflections, and choosing an appropriate function type for a described situation.

Study tips

How to practice Polynomial and Rational.

Section-specific guidance that follows from how this material is usually tested.

Practice describing graph behavior in complete sentences, since the free-response section scores written reasoning.
Determine end behavior from structure rather than by graphing, which is faster and works on the no-calculator part.
Distinguish a hole from a vertical asymptote by examining the factored form, not the graph alone.

Where this fits

Unit 1: Polynomial and Rational Functions is 1 of 4 units in the AP Precalculus guide, and it carries a published weighting of 30–40% in the official framework. Use the full guide to balance it against everything else you have to review.

Back to the AP Precalculus guide Official exam information

Sample Polynomial and Rational questions

Practice questions, not yet expert-reviewed

Original practice questions written to the published framework for this section. No expert has reviewed them, and they are not taken from any official exam. Try each one before you open the answer.

Question 1Foundational
What is the average rate of change of f(x) = x^2 + 2x over the interval from x = 1 to x = 4?
1. A7
2. B3
3. C9
4. D21
Show answer
Answer: A. 7
The average rate of change is (f(4) - f(1))/(4 - 1). Since f(4) = 16 + 8 = 24 and f(1) = 1 + 2 = 3, the quotient is 21/3 = 7. Reporting 21 gives the change in the output without dividing by the change in the input.
Key insight: Average rate of change is the slope of the secant line: change in output over change in input.
Question 2Foundational
What is the end behavior of the polynomial function f(x) = -2x^3 + 5x?
1. AAs x → ∞, f(x) → -∞, and as x → -∞, f(x) → ∞
2. BAs x → ∞, f(x) → ∞, and as x → -∞, f(x) → -∞
3. Cf(x) → ∞ at both ends
4. Df(x) → -∞ at both ends
Show answer
Answer: A. As x → ∞, f(x) → -∞, and as x → -∞, f(x) → ∞
End behavior is governed by the leading term -2x^3. An odd degree sends the two ends in opposite directions, and the negative leading coefficient flips the standard cubic, so the function falls on the right and rises on the left. Matching ends would require an even degree.
Key insight: Odd degree means opposite ends; the sign of the leading coefficient decides which way each end goes.
Question 3Intermediate
At what x-value does the graph of f(x) = (x^2 - 9)/(x^2 - 2x - 3) have a hole?
1. Ax = 3
2. Bx = -1
3. Cx = -3
4. Dx = 1
Show answer
Answer: A. x = 3
Factoring gives (x - 3)(x + 3) over (x - 3)(x + 1). The common factor x - 3 cancels, so x = 3 is a removable discontinuity, a hole. The remaining factor x + 1 in the denominator gives a vertical asymptote at x = -1 instead.
Key insight: A factor that cancels creates a hole; a factor that remains in the denominator creates a vertical asymptote.
Question 4Intermediate
How does the graph of f(x) = (x - 2)^2 (x + 1) behave at x = 2?
1. AIt touches the x-axis and turns back without crossing
2. BIt crosses the x-axis
3. CIt has a vertical asymptote
4. DIt has a hole
Show answer
Answer: A. It touches the x-axis and turns back without crossing
The zero at x = 2 has multiplicity 2, an even multiplicity, so the graph is tangent to the x-axis there and returns to the same side. The zero at x = -1 has multiplicity 1, so the graph crosses there instead.
Key insight: Even multiplicity touches the axis; odd multiplicity crosses it.
Question 5Advanced
What is the equation of the slant asymptote of the rational function f(x) = (2x^2 + 3x - 1)/(x - 1)?
1. Ay = 2x + 5
2. By = 2x
3. Cy = 2x + 3
4. Dy = 2x - 5
Show answer
Answer: A. y = 2x + 5
Dividing 2x^2 + 3x - 1 by x - 1 gives a quotient of 2x + 5 with a remainder of 4, so f(x) = 2x + 5 + 4/(x - 1). As |x| grows the remainder term approaches 0, leaving the line y = 2x + 5. Keeping only the leading term gives the incomplete answer y = 2x.
Key insight: When the numerator degree is exactly one more than the denominator degree, the polynomial quotient is the slant asymptote.

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Unit 1: Polynomial and Rational Functions