What is the average rate of change of f(x) = x^2 - 3x over the interval [1, 4]?
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This is a free 15-question sampler in the style of the AP Precalculus multiple-choice section, written by the StudyPDF team. It covers the three units that are tested on the exam, polynomial and rational functions, exponential and logarithmic functions, and trigonometric and polar functions, five questions each, and it mixes multiple choice, select all that apply and fill in the blank. Every question can be done by hand. Answer all 15, then check your score and read the worked explanation for every question.
All 15 questions at a glance
Prefer to read before you play? Every question in the sampler is listed below. Answers and explanations stay collapsed until you open them.
1.What is the average rate of change of f(x) = x^2 - 3x over the interval [1, 4]?
- A. 1
- B. 2
- C. 3
- D. 6
Answer: B. 2
The average rate of change is the slope of the secant line: (f(4) - f(1)) / (4 - 1). Here f(4) = 16 - 12 = 4 and f(1) = 1 - 3 = -2, so the change in output is 4 - (-2) = 6, and dividing by the change in input, 3, gives 2. The answer 6 forgets to divide, and 3 is the length of the interval.
2.Which describes the end behavior of f(x) = -2x^5 + 3x^2 - 1?
- A. f(x) approaches negative infinity as x approaches infinity, and f(x) approaches infinity as x approaches negative infinity
- B. f(x) approaches infinity as x approaches infinity, and f(x) approaches negative infinity as x approaches negative infinity
- C. f(x) approaches negative infinity as x approaches either infinity or negative infinity
- D. f(x) approaches infinity as x approaches either infinity or negative infinity
Answer: A. f(x) approaches negative infinity as x approaches infinity, and f(x) approaches infinity as x approaches negative infinity
For large inputs the leading term, -2x^5, dominates. The degree is odd, so the two ends of the graph go in opposite directions, and the leading coefficient is negative, so the right end goes down. That means f falls toward negative infinity on the right and rises toward infinity on the left. The other terms, 3x^2 and -1, have little influence once |x| is large.
3.Let r(x) = (x^2 - 4) / (x^2 - x - 6). Which statements are true?
Select all that apply.
- A. The graph has a hole at x = -2
- B. The graph has a vertical asymptote at x = 3
- C. The graph has a vertical asymptote at x = -2
- D. The graph has a horizontal asymptote at y = 1
- E. The graph has an x-intercept at x = -2
Answer: A. The graph has a hole at x = -2, B. The graph has a vertical asymptote at x = 3, D. The graph has a horizontal asymptote at y = 1
Factor both parts: r(x) = (x - 2)(x + 2) / ((x - 3)(x + 2)). The factor x + 2 cancels, so there is a hole at x = -2, not an asymptote, and not an x-intercept either because r is undefined there. The factor x - 3 is left in the denominator, so there is a vertical asymptote at x = 3. The numerator and denominator have the same degree, so the horizontal asymptote is the ratio of the leading coefficients, 1/1 = 1. The only x-intercept is x = 2.
4.Complete the sentence.
A cubic polynomial has a positive leading coefficient and zeros at x = -1 with multiplicity 2 and at x = 3 with multiplicity 1. At x = -1 the graph _____ the x-axis, and at x = 3 the graph _____ the x-axis.
Options for blank 1: crosses, touches but does not cross, stays above, stays below
Options for blank 2: crosses, touches but does not cross, stays above, stays below
Answer: touches but does not cross, crosses
The polynomial is p(x) = a(x + 1)^2 (x - 3) with a > 0. A zero of even multiplicity makes the graph touch the axis and turn back, because the factor (x + 1)^2 does not change sign. A zero of odd multiplicity makes the graph cross. So it touches at x = -1 and crosses at x = 3. The multiplicities add up to 3, which matches a cubic.
5.The point (1, 4) is on the graph of f. Which point must be on the graph of g(x) = 2f(x - 3) + 1?
- A. (-2, 9)
- B. (4, 8)
- C. (4, 7)
- D. (4, 9)
Answer: D. (4, 9)
To find the new point, work out which input of f the new graph uses. g(4) = 2f(4 - 3) + 1 = 2f(1) + 1 = 2(4) + 1 = 9, so (4, 9) is on the graph of g. The inside change x - 3 shifts the graph 3 units right, so the x-value goes from 1 to 4. The outside changes stretch the output by 2 and then shift it up 1, so 4 becomes 9. The answer (-2, 9) shifts the wrong way, and the others get the order of the output steps wrong.
6.Complete the sentence.
A geometric sequence with positive terms has g_3 = 24 and g_5 = 96. Its common ratio is _____ and its initial term g_0 is _____.
Options for blank 1: 2, 3, 4, 12
Options for blank 2: 6, 3, 12, 0.75
Answer: 2, 3
Going from the 3rd term to the 5th term multiplies by the ratio twice, so r^2 = 96/24 = 4. The terms are positive, so r = 2. The general term is g_n = g_0 times r^n, so g_3 = g_0 times 2^3 = 8 g_0 = 24, which gives g_0 = 3. A common slip is to treat 96/24 = 4 as the ratio itself instead of the ratio squared.
7.A town has a population of 8,000 that grows by 5 percent each year. Which function models the population P after t years?
- A. P(t) = 8000(0.05)^t
- B. P(t) = 8000 + 0.05t
- C. P(t) = 8000(5)^t
- D. P(t) = 8000(1.05)^t
Answer: D. P(t) = 8000(1.05)^t
Growth by 5 percent per year means each year the population is multiplied by 1 + 0.05 = 1.05, which is constant proportional change, so the model is exponential: P(t) = 8000(1.05)^t. The base 0.05 would shrink the population almost to nothing, the base 5 would multiply it fivefold every year, and adding 0.05t is linear growth of 0.05 people per year.
8.What is the solution of log_3(x + 1) = 2?
- A. 8
- B. 7
- C. 9
- D. 5
Answer: A. 8
Rewrite the logarithmic equation in exponential form: log_3(x + 1) = 2 means 3^2 = x + 1. So x + 1 = 9 and x = 8. Check the domain: x + 1 = 9 is positive, so the logarithm is defined. The value 9 is x + 1, not x, and 5 comes from multiplying 3 by 2 instead of raising 3 to the power 2.
9.Let f(x) = ln(x - 1) + 2. Which statements are true?
Select all that apply.
- A. The domain of f is x > 1
- B. The graph of f has a vertical asymptote at x = 1
- C. f(2) = 0
- D. The inverse function is f^(-1)(x) = e^(x - 2) + 1
- E. f is a decreasing function
Answer: A. The domain of f is x > 1, B. The graph of f has a vertical asymptote at x = 1, D. The inverse function is f^(-1)(x) = e^(x - 2) + 1
The logarithm is defined only when x - 1 > 0, so the domain is x > 1, and the graph approaches the vertical line x = 1 without touching it. To find the inverse, solve y = ln(x - 1) + 2 for x: subtract 2 and exponentiate to get x - 1 = e^(y - 2), so x = e^(y - 2) + 1. Then swap the roles of x and y, which gives f^(-1)(x) = e^(x - 2) + 1. For the other statements, f(2) = ln(1) + 2 = 2, not 0, and a natural logarithm with a positive coefficient is increasing, so f is not decreasing.
10.A student fits an exponential model to a data set, and the residual plot shows a clear curved pattern. What does this indicate?
- A. The model is appropriate, because the residuals are small
- B. The model is appropriate, because the residuals fall both above and below the axis
- C. The exponential model is not appropriate, because residuals from an appropriate model show no pattern
- D. The data must be periodic, so a sinusoidal model is required
Answer: C. The exponential model is not appropriate, because residuals from an appropriate model show no pattern
A residual is the actual value minus the value the model predicts. If a model fits well, its residuals scatter randomly around zero with no visible pattern. A curve, a trend or a fan shape means the model is missing some structure in the data, so it is not appropriate. Residuals landing on both sides of the axis happens even for bad models, and small residuals alone do not rule out a systematic pattern. A bad fit does not by itself mean the data are periodic.
11.Which gives all solutions of 2cos(theta) = 1 for 0 <= theta < 2pi?
- A. pi/3 and 5pi/3
- B. pi/3 and 2pi/3
- C. pi/6 and 11pi/6
- D. pi/3 only
Answer: A. pi/3 and 5pi/3
Divide by 2 to get cos(theta) = 1/2. On the unit circle the x-coordinate is 1/2 at two angles in one turn: pi/3 in the first quadrant and 5pi/3 in the fourth. The answer 2pi/3 has cosine -1/2, and pi/6 and 11pi/6 have cosine sqrt(3)/2. Always look for the second solution, because the cosine takes most values twice per period.
12.Complete the sentence.
The function f(theta) = 3sin(2theta) + 1 has amplitude _____ and period _____.
Options for blank 1: 1, 2, 3, 6
Options for blank 2: pi/2, pi, 2pi, 4pi
Answer: 3, pi
In a form like a sin(b(theta - c)) + d, the amplitude is the absolute value of a, here 3, and the period is 2pi divided by the absolute value of b, here 2pi/2 = pi. The +1 moves the midline up to y = 1 but does not change the amplitude, so the function oscillates between -2 and 4. A larger b squeezes the graph horizontally, so doubling the input halves the period from 2pi to pi.
13.The height of a rider on a Ferris wheel is at most 40 feet and at least 4 feet, and one full turn takes 12 minutes. The rider is at the lowest point when t = 0 minutes. Which function models the height h(t) in feet?
- A. h(t) = 18cos(pi t / 6) + 22
- B. h(t) = -18cos(pi t / 6) + 22
- C. h(t) = -18cos(pi t / 12) + 22
- D. h(t) = -22cos(pi t / 6) + 18
Answer: B. h(t) = -18cos(pi t / 6) + 22
The midline is the average of the maximum and minimum, (40 + 4)/2 = 22, and the amplitude is half the range, (40 - 4)/2 = 18. The period is 12, so b = 2pi/12 = pi/6. A plain cosine starts at its maximum, but this rider starts at the minimum, so the cosine is reflected with a negative sign: h(0) = -18 + 22 = 4. The version with pi/12 would have a period of 24 minutes, and the last option swaps the amplitude and the midline.
14.Which polar coordinates name the same point as (3, pi/4)?
Select all that apply.
- A. (3, 9pi/4)
- B. (-3, 5pi/4)
- C. (3, 3pi/4)
- D. (-3, pi/4)
- E. (3, -7pi/4)
Answer: A. (3, 9pi/4), B. (-3, 5pi/4), E. (3, -7pi/4)
A point has many polar names. Adding or subtracting 2pi to the angle gives the same point, so (3, pi/4 + 2pi) = (3, 9pi/4) and (3, pi/4 - 2pi) = (3, -7pi/4) both work. A negative radius puts the point on the opposite side of the origin, so the same point can be named with the opposite direction, pi/4 + pi = 5pi/4, and radius -3. The point (3, 3pi/4) is a different location, in the second quadrant, and (-3, pi/4) is the reflection of the original point through the origin.
15.On the interval from theta = 0 to theta = pi/2, a polar function r = f(theta) is negative and increases from -4 to -1. As theta increases, what happens to the distance between the point (f(theta), theta) and the origin?
- A. It increases from 1 to 4
- B. It decreases from 4 to 1
- C. It increases from -4 to -1
- D. It stays constant
Answer: B. It decreases from 4 to 1
The distance from the origin is the absolute value of r, because a negative r only flips the direction of the point and not its distance. As r goes from -4 up to -1, |r| goes from 4 down to 1, so the point moves closer to the origin even though r itself is increasing. A distance cannot be negative, which rules out the third option. The same idea works for a positive r that decreases: the distance shrinks.
What the AP Precalculus exam looks like and which units are tested
The AP Precalculus exam is scored 1 to 5 and has two sections. According to the College Board, the format for the May 2027 exam is as follows. Section I is 42 multiple-choice questions in 1 hour 45 minutes, worth about 63 percent of the score: Part A has 29 questions in 65 minutes with no calculator, and Part B has 13 questions in 40 minutes with a graphing calculator required. Section II is 4 free-response questions in 1 hour 10 minutes, worth about 37 percent: Part A has 2 questions in 35 minutes with a graphing calculator (function concepts, then modeling a non-periodic context), and Part B has 2 questions in 35 minutes without one (modeling a periodic context, then symbolic manipulations). The exam is hybrid digital, so you answer multiple-choice questions in the Bluebook app and handwrite your free-response answers in a paper booklet. Sources written before the 2026-27 update list 40 multiple-choice questions and different timing, so trust the current College Board page.
The course has four units, but only three are on the exam. The College Board gives these shares of the multiple-choice score: polynomial and rational functions 30 to 40 percent, exponential and logarithmic functions 25 to 40 percent, and trigonometric and polar functions 30 to 35 percent. Unit 4, functions involving parameters, vectors and matrices, is not assessed on the AP exam. A class may still teach it, but a practice test that spends questions on matrices or vectors is spending your time on something that will not appear.
A 5 is the top score, but what a score is worth is up to each college. The College Board recommends credit for a college precalculus course starting at a 3, and credit policies differ by school and by course. Look up the colleges you care about before you decide how hard to push.
How to use this sampler, and how to take it on paper
The 15 questions are split evenly, five per tested unit, to match how close the three units are in weight. They ask for the reasoning behind a result more than the button presses: why a factor that cancels leaves a hole instead of an asymptote, why a residual plot with a pattern means an exponential model is the wrong model, how to build a sinusoid from a maximum, a minimum and a period, why a negative radius still moves a point in a polar graph. That is the thinking the free-response section rewards too.
Take it in one sitting with scratch paper. Every question here can be done without a calculator, which is the Part A condition on the real exam, so the graphing-calculator skills of Part B are not practiced. Then read the explanation for every question, including the ones you got right. Treat a miss as a topic. If you missed the rational function question and the polynomial end behavior question, the gap is probably how the degree and the leading term control the graph. Math is written in plain text here, so x^2 means x squared and pi/3 means pi divided by 3.
There are no limits or derivatives in this course. If you are looking for those, try the AP Calculus AB sampler linked below. If you would rather work on paper, use the Download test as PDF button above the questions. The file has all 15 questions with an answer key on the last page and needs a free StudyPDF account. The questions are original and written to the published course outline. They are not released exam questions, and this sampler does not practice the free-response section, so pair it with the College Board's released free-response questions.
How StudyPDF builds AP Precalculus practice tests from your class notes
Fifteen questions can show you which unit is shaky. They cannot get you ready on their own, and a generic bank rarely matches the function families, notation and word problems your teacher actually uses.
StudyPDF works from your material instead. Upload your textbook chapters, your teacher's slides, a worked problem set or your review packet, press New material, and StudyPDF builds practice tests from those exact pages. Every question is grounded in your material and every explanation cites where the answer comes from, so you can go straight back to the page when a step does not make sense.
You do not need to upload anything to start. Name a topic instead, for example rational function asymptotes or modeling with sinusoidal functions, and Bo sets it up with its key concepts so you can build quizzes and flashcards on it. Your results track which concepts you keep missing, so the next practice test can target them. Starting is free.
Written by the StudyPDF team. Last updated August 19, 2026.
Good to know
Is this AP Precalculus practice test free?
Yes. All 15 questions, the score screen and every worked explanation are free, and the quiz runs in your browser. A free StudyPDF account is needed to download the test as a PDF with an answer key or to build more practice tests from your own notes.
Who wrote these questions?
The StudyPDF team wrote every question and explanation, following the published AP Precalculus course outline. They are original items, not real or released exam questions, and StudyPDF is not affiliated with the College Board, which does not endorse this page.
Why are there no questions on vectors, matrices or parametric functions?
Those topics are the course's fourth unit, which the College Board says is not assessed on the AP exam. This sampler covers the three units that are tested, with five questions each. If your class teaches the fourth unit, your teacher's own materials are the better source for it.
Can I get practice tests built from my own class materials?
Yes. Upload your textbook chapters, slides, problem sets or review packet to StudyPDF and you can build practice tests from those exact pages, with explanations tied to your sources. If you have nothing to upload, name a topic or a unit instead.