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Prepare for the timed AP Precalculus test.

Your answers are held until submission. The final review identifies the units that need another pass.

AP Precalculus coverage and review map

Connect every question to the exact skill it rehearses. Work through the units in order, then return to any topic that still needs a hint or a second attempt.

01

Polynomial and Rational Functions

Zeros, rates of change, end behavior, transformations, and models

Goal: recognise a polynomial and rational functions problem from its wording, carry out the governing method, and check that the result is reasonable.

02

Exponential and Logarithmic Functions

Growth, decay, inverses, semi-log reasoning, and models

Goal: recognise a exponential and logarithmic functions problem from its wording, carry out the governing method, and check that the result is reasonable.

03

Trigonometric and Polar Functions

Periodic behavior, identities, equations, and polar representation

Goal: recognise a trigonometric and polar functions problem from its wording, carry out the governing method, and check that the result is reasonable.

04

Functions Involving Parameters, Vectors, and Matrices

Course extension; not assessed on the end-of-course exam

Goal: recognise a functions involving parameters, vectors, and matrices problem from its wording, carry out the governing method, and check that the result is reasonable.

AP Precalculus questions with worked answers

These 8 questions are printed in full on this page and are drawn across all 4 course units. Nothing here is generated on the fly. Cover the options, solve each one on paper, and only then open the worked answer to compare your method with the one shown.

Question 1: What is the average rate of change of f(x) = x^3 - 2x on the interval from x = 1 to x = 3?

AP Precalculus · Polynomial and Rational Functions

  1. 11
  2. 22
  3. 10
  4. 13
Show the worked answer
  1. Average rate of change is the change in output divided by the change in input.
  2. f(3) = 27 - 6 = 21 and f(1) = 1 - 2 = -1.
  3. The change in output is 21 - (-1) = 22, over an input change of 3 - 1 = 2.
  4. 22 / 2 = 11.

Answer: 11

Question 2: For f(x) = (3x^2 - 12) / (x^2 - x - 6), identify the hole, the vertical asymptote, and the horizontal asymptote.

AP Precalculus · Polynomial and Rational Functions

  1. hole at x = -2, asymptotes x = 3 and y = 3
  2. hole at x = 3, asymptotes x = -2 and y = 3
  3. hole at x = -2, asymptotes x = 3 and y = 1
  4. no hole, asymptotes x = 3, x = -2, and y = 3
Show the worked answer
  1. Factor the top: 3x^2 - 12 = 3(x - 2)(x + 2).
  2. Factor the bottom: x^2 - x - 6 = (x - 3)(x + 2).
  3. The shared factor (x + 2) cancels, leaving a hole at x = -2, and the leftover (x - 3) gives the asymptote x = 3.
  4. Top and bottom have the same degree, so the horizontal asymptote is 3 / 1 = 3.

Answer: hole at x = -2, asymptotes x = 3 and y = 3

Question 3: A population is modeled by P(t) = 240(1.15)^t with t in years. How long does it take to double, to the nearest tenth of a year?

AP Precalculus · Exponential and Logarithmic Functions

  1. 5.0 years
  2. 6.7 years
  3. 4.6 years
  4. 2.0 years
Show the worked answer
  1. Doubling means (1.15)^t = 2. The starting 240 cancels out.
  2. Take logs of both sides: t ln(1.15) = ln 2.
  3. ln 2 is about 0.6931 and ln 1.15 is about 0.1398.
  4. 0.6931 / 0.1398 = 4.959, which rounds to 5.0 years.

Answer: 5.0 years

Question 4: Solve 2 log base 5 of x = log base 5 of 36.

AP Precalculus · Exponential and Logarithmic Functions

  1. x = 6
  2. x = 18
  3. x = 36
  4. x = 6 or x = -6
Show the worked answer
  1. The coefficient 2 moves inside as a power: log base 5 of x^2 = log base 5 of 36.
  2. Same base and equal logs means equal inputs, so x^2 = 36.
  3. That gives x = 6 or x = -6.
  4. A logarithm needs a positive input, so x = -6 is rejected and x = 6 is the answer.

Answer: x = 6

Question 5: A sinusoid is modeled by y = 4 sin((pi/6)(t - 2)) + 10. What are its period, maximum, and minimum?

AP Precalculus · Trigonometric and Polar Functions

  1. period 12, maximum 14, minimum 6
  2. period 6, maximum 14, minimum 6
  3. period 12, maximum 4, minimum -4
  4. period 12, maximum 10, minimum 4
Show the worked answer
  1. The period is 2 pi divided by the coefficient of t, which is pi/6.
  2. 2 pi divided by pi/6 is 2 pi x 6 / pi = 12.
  3. The amplitude is 4 and the midline is 10.
  4. So the maximum is 10 + 4 = 14 and the minimum is 10 - 4 = 6.

Answer: period 12, maximum 14, minimum 6

Question 6: Convert the polar point (r, theta) = (4, 2 pi / 3) to rectangular coordinates.

AP Precalculus · Trigonometric and Polar Functions

  1. (-2, 2 times the square root of 3)
  2. (2, 2 times the square root of 3)
  3. (-2 times the square root of 3, 2)
  4. (-2, -2 times the square root of 3)
Show the worked answer
  1. Use x = r cos(theta) and y = r sin(theta).
  2. cos(2 pi / 3) = -1/2, so x = 4(-1/2) = -2.
  3. sin(2 pi / 3) = the square root of 3, over 2.
  4. So y = 4 times that, which is 2 times the square root of 3, about 3.46.

Answer: (-2, 2 times the square root of 3)

Question 7: What is the vector from the point (1, 5) to the point (7, -3), and what is its magnitude?

AP Precalculus · Functions Involving Parameters, Vectors, and Matrices

  1. vector (6, -8) with magnitude 10
  2. vector (-6, 8) with magnitude 10
  3. vector (6, -8) with magnitude 14
  4. vector (8, -2) with magnitude 8.25
Show the worked answer
  1. Subtract the starting point from the ending point.
  2. 7 - 1 = 6 and -3 - 5 = -8, so the vector is (6, -8).
  3. The magnitude is the square root of 6^2 + (-8)^2 = 36 + 64 = 100.
  4. The square root of 100 is 10.

Answer: vector (6, -8) with magnitude 10

Question 8: What is the inverse of the matrix [[2, 1], [3, 4]]?

AP Precalculus · Functions Involving Parameters, Vectors, and Matrices

  1. (1/5)[[4, -1], [-3, 2]]
  2. (1/5)[[2, -1], [-3, 4]]
  3. (1/11)[[4, -1], [-3, 2]]
  4. [[4, -1], [-3, 2]]
Show the worked answer
  1. The determinant is 2(4) - 1(3) = 8 - 3 = 5.
  2. For a 2 by 2 matrix, swap the two diagonal entries and change the sign of the other two.
  3. That gives [[4, -1], [-3, 2]].
  4. Divide by the determinant: the inverse is (1/5)[[4, -1], [-3, 2]].

Answer: (1/5)[[4, -1], [-3, 2]]

Which unit each printed question belongs to

Use this map after marking your work. If two misses share a unit, review that unit before starting a generated set.

The 8 printed AP Precalculus questions, the unit each one tests, and its answer.
QuestionUnitWhat it asksAnswer
1Polynomial and Rational FunctionsWhat is the average rate of change of f(x) = x^3 - 2x on the interval from x = 1 to x = 3?11
2Polynomial and Rational FunctionsFor f(x) = (3x^2 - 12) / (x^2 - x - 6), identify the hole, the vertical asymptote, and the horizontal asymptote.hole at x = -2, asymptotes x = 3 and y = 3
3Exponential and Logarithmic FunctionsA population is modeled by P(t) = 240(1.15)^t with t in years. How long does it take to double, to the nearest tenth of a year?5.0 years
4Exponential and Logarithmic FunctionsSolve 2 log base 5 of x = log base 5 of 36.x = 6
5Trigonometric and Polar FunctionsA sinusoid is modeled by y = 4 sin((pi/6)(t - 2)) + 10. What are its period, maximum, and minimum?period 12, maximum 14, minimum 6
6Trigonometric and Polar FunctionsConvert the polar point (r, theta) = (4, 2 pi / 3) to rectangular coordinates.(-2, 2 times the square root of 3)
7Functions Involving Parameters, Vectors, and MatricesWhat is the vector from the point (1, 5) to the point (7, -3), and what is its magnitude?vector (6, -8) with magnitude 10
8Functions Involving Parameters, Vectors, and MatricesWhat is the inverse of the matrix [[2, 1], [3, 4]]?(1/5)[[4, -1], [-3, 2]]

How the generated AP Precalculus sets are created

The 8 questions above are fixed and checked. The generator at the top of the page is different: it writes fresh questions with AI from the course and unit information shown here, then the application checks each one for a complete prompt, four choices, one keyed answer, and an explanation. Generated questions are original practice—not official or released exam questions—and AI can still make mathematical mistakes. Verify a disputed answer with the stated method, your course materials, or the MathGPT solver, and follow the site's academic-integrity guidance.

Questions about this AP Precalculus course test

What does this page cover?

It covers all 4 AP Precalculus units listed above.

When can I see correct answers and explanations?

Submit the full test first. The result screen then shows every response, the correct answer, and the explanation so you can review each miss.

How long is each section of the AP Precalculus exam?

Multiple choice: 1 hour 45 minutes, 42 questions. Free response: 1 hour 10 minutes, 4 questions.

Which AP Precalculus sections allow a graphing calculator?

Rules change from section to section. Multiple choice, Part A: no calculator. Multiple choice, Part B: graphing calculator required. Free response, Part A: graphing calculator required. Free response, Part B: no calculator. Practise each section under its own rule.

Which AP Precalculus units does this test cover?

All four: Polynomial and Rational Functions, Exponential and Logarithmic Functions, Trigonometric and Polar Functions, Functions Involving Parameters, Vectors, and Matrices. Study first: Polynomial and Rational Functions (30–40%) and Exponential and Logarithmic Functions (27–40%) tie for the largest published weighting.

What should I check first when I review a missed AP Precalculus question?

Rational functions: a hole and a vertical asymptote are different answers. Factor first, then decide which one the cancelled factor made.

What should I do with a missed question?

Classify the miss as a definition, setup, calculation, interpretation, or timing error. Re-solve it from a blank page, then use the MathGPT tutor for a hint or method check.