AP PRECALCULUS Cheat Sheet 2026

The 30 highest-yield AP PRECALCULUS facts, distilled from real exam questions. Print it, save it as a PDF, or study it here — free, no sign-up.

1. Which set contains all solutions of 2 sin x = √3 on the interval [0, 2π)? → x = π/3 and x = 2π/3
2. Which is a polynomial of least degree with real coefficients that has 2 and 1 + i as zeros? → x³ − 4x² + 6x − 4
3. Which polar function has a graph that is symmetric about the y-axis (the line θ = π/2) but not about the x-axis? → r = 4 + 3 sin θ
4. For which values of x in [0, 2π) is cos x > 1/2? → 0 ≤ x < π/3 or 5π/3 < x < 2π
5. For g(x) = −2 cos(x) + 5, which pair gives the midline and the amplitude? → Midline y = 5, amplitude 2
6. The polar point (6, 2π/3) is converted to rectangular coordinates. Which is the result? → (-3, 3√3)
7. An exponential function f(x) = a·b^x with b > 0 satisfies f(2) = 45 and f(4) = 405. What is f(5)? → 1215
8. A student fits an exponential model to a data set and plots the residuals. Which observation about the residual plot best supports the model being appropriate? → The residuals are scattered randomly above and below zero with no pattern.
9. Which expression is equivalent to (2x² + 3x − 5)/(x − 1) for all x ≠ 1? → 2x + 5
10. What is the range of y = sec x? → (−∞, −1] ∪ [1, ∞)
11. Let f(x) = 2x + 3 and g(x) = x². What is the value of (f ∘ g)(2)? → 11
12. Which line is the horizontal asymptote of f(x) = (3x² − 5x + 1)/(x² + 4)? → y = 3
13. Consider f(x) = (x − 2)/((x − 2)(x + 3)). Which statement correctly describes the graph of f? → A vertical asymptote at x = −3 and a hole at x = 2
14. The rectangular point (-2, 2) is written in polar form (r, θ) with r > 0 and 0 ≤ θ < 2π. Which is correct? → (2√2, 3π/4)
15. What is the end behavior of f(x) = (x³ − 1)/(x² + 2)? → As x → ∞, f(x) → ∞, and as x → −∞, f(x) → −∞
16. What is the domain of f(x) = ln(5 − 2x)? → x < 5/2
17. Let g(x) = −3·2^x + 5. Which statement about g is true? → g has a horizontal asymptote at y = 5 and decreases without bound as x → ∞.
18. What is the exact value of log₄ 32? → 5/2
19. Which function is the inverse of h(x) = eˣ + 1? → h⁻¹(x) = ln(x − 1)
20. The point (4, 5) lies on the graph of y = f(x). Which point must lie on the graph of g(x) = −f(x − 3) + 2? → (7, −3)
21. On the interval (π/2, π), the graph of y = sin x is which of the following? → Decreasing and concave down
22. If cos θ = −3/5 and θ is in Quadrant II, what is cot θ? → −3/4
23. As x → ∞ and as x → −∞, the graph of f(x) = (x² + 1)/(x − 1) approaches which line? → y = x + 1
24. A geometric sequence begins 6, 18, 54, ... What is the sixth term? → 1458
25. Which function is equivalent to f(t) = 5·2^(3t) written in the form a·b^t? → 5·8^t
26. For the polar function r = 2 + 4 sin θ, which values of θ in [0, 2π) give r = 0, so the graph passes through the pole? → 7π/6 and 11π/6
27. An arithmetic sequence has a_3 = 11 and a_7 = 27. What is a_20? → 79
28. On a semi-log plot, log₁₀(y) is graphed against x, and the points form a straight line passing through (0, 1) and (2, 5). Which exponential model fits the data? → y = 10·100ˣ
29. The table gives values of a function: f(0) = 4, f(1) = 12, f(2) = 36, f(3) = 108. Which model best fits the data? → y = 4·3^x
30. What is the average rate of change of f(x) = x² − 4x + 1 over the interval [1, 5]? → 2