Section I — Multiple Choice

Section II — Free Response

Section I — Multiple Choice

Section II — Free Response

AP Precalculus — Full Practice Exam

Time: 180 minutes (120 min MC + 60 min FRQ)

Section I: Multiple-Choice Questions (40 questions, 120 minutes)

Unit 1: Polynomial and Rational Functions

1. What is the end behavior of $f(x) = 2x^5 - 8x^3 + x - 4$? (A) As $x \to +\infty$, $f(x) \to +\infty$; as $x \to -\infty$, $f(x) \to -\infty$
(B) As $x \to +\infty$, $f(x) \to -\infty$; as $x \to -\infty$, $f(x) \to +\infty$
(C) As $x \to \pm\infty$, $f(x) \to +\infty$
(D) As $x \to \pm\infty$, $f(x) \to -\infty$

2. The polynomial $f(x) = (x-1)^2(x+3)(x-5)^3$ has how many distinct real zeros? (A) 2
(B) 3
(C) 6
(D) 7

3. What is the remainder when $f(x) = x^3 + 2x^2 - 5x + 1$ is divided by $(x + 2)$? (A) $-1$
(B) $1$
(C) $-17$
(D) $17$

4. Which function has a horizontal asymptote at $y = -2$? (A) $f(x) = \frac{x+1}{x-2}$
(B) $f(x) = \frac{2x-4}{x+1}$
(C) $f(x) = \frac{-2x^2+3}{x^2+1}$
(D) $f(x) = \frac{x^2-1}{2x+3}$

5. The function $f(x) = \frac{(x-2)(x+4)}{(x-2)(x+1)}$ has: (A) Vertical asymptotes at $x = 2$ and $x = -1$
(B) A hole at $x = 2$ and a vertical asymptote at $x = -1$
(C) A vertical asymptote at $x = 2$ and a hole at $x = -1$
(D) No vertical asymptotes

6. If $g(x) = 2f(x+3) - 1$ and $f(x) = x^2$, what is $g(0)$? (A) $7$
(B) $13$
(C) $17$
(D) $25$

7. A polynomial of degree 5 has at most how many turning points? (A) 3
(B) 4
(C) 5
(D) 6

8. The slant asymptote of $f(x) = \frac{x^3 - 2x + 5}{x^2 + 1}$ is: (A) $y = x$
(B) $y = x - 2$
(C) $y = x^2$
(D) There is no slant asymptote

9. Which of the following is equivalent to $(x - 1)$ being a factor of $f(x)$? (A) $f(0) = -1$
(B) $f(1) = 0$
(C) $f(-1) = 0$
(D) $f(0) = 1$

10. The graph of $f(x) = -\frac{1}{x+3}$ has a vertical asymptote at: (A) $x = 0$
(B) $x = -3$
(C) $x = 3$
(D) $y = 0$

Unit 2: Exponential and Logarithmic Functions

11. A substance decays according to $A(t) = A_0 \cdot e^{-0.12t}$. What is the half-life? (A) $\frac{\ln(0.5)}{-0.12}$
(B) $\frac{\ln(2)}{0.12}$
(C) Both (A) and (B)
(D) $\frac{0.12}{\ln(2)}$

12. $\log_5(1) =$ (A) $0$
(B) $1$
(C) $5$
(D) Undefined

13. Which expression equals $\log_2(48) - \log_2(3)$? (A) $\log_2(45)$
(B) $\log_2(16)$
(C) $\log_2(144)$
(D) $16$

14. The graph of $f(x) = \log_2(x)$ has a vertical asymptote at: (A) $x = 0$
(B) $x = 1$
(C) $y = 0$
(D) $y = 1$

15. Solve $5^x = 20$ for $x$. (A) $x = \frac{\ln(20)}{\ln(5)}$
(B) $x = \frac{\log(20)}{\log(5)}$
(C) $x = \log_5(20)$
(D) All of the above are equivalent

16. An account earns 6% interest compounded monthly. The effective annual yield is: (A) $6\%$
(B) $(1 + 0.06/12)^{12} - 1$
(C) $e^{0.06} - 1$
(D) $(1.06)^{12} - 1$

17. The function $f(x) = 3^{x+2} - 1$ has y-intercept: (A) $(0, 8)$
(B) $(0, 9)$
(C) $(0, 2)$
(D) $(0, -1)$

18. If $\ln(x) = 2.5$, then $x \approx$ (A) $7.39$
(B) $12.18$
(C) $148.41$
(D) $9.49$

19. What is the range of $f(x) = -2e^{x} + 5$? (A) $(-\infty, 5)$
(B) $(5, \infty)$
(C) $(-\infty, \infty)$
(D) $(-\infty, -5)$

20. Which data set is best modeled by an exponential function? (A) Consecutive differences are constant
(B) Consecutive ratios are constant
(C) Second differences are constant
(D) Neither

Unit 3: Trigonometric and Polar Functions

21. Convert $300°$ to radians. (A) $\frac{5\pi}{3}$
(B) $\frac{4\pi}{3}$
(C) $\frac{7\pi}{6}$
(D) $\frac{3\pi}{2}$

22. What is $\cos\left(\frac{5\pi}{4}\right)$? (A) $\frac{\sqrt{2}}{2}$
(B) $-\frac{\sqrt{2}}{2}$
(C) $\frac{1}{2}$
(D) $-\frac{1}{2}$

23. The period of $f(x) = 5\sin(4x)$ is: (A) $5\pi/2$
(B) $\pi/2$
(C) $2\pi$
(D) $8\pi$

24. If $\sin\theta = \frac{5}{13}$ and $\theta$ is in Quadrant II, what is $\cos\theta$? (A) $\frac{12}{13}$
(B) $-\frac{12}{13}$
(C) $\frac{5}{12}$
(D) $-\frac{5}{12}$

25. $\sin(2\theta) =$ (A) $2\sin\theta\cos\theta$
(B) $\sin^2\theta + \cos^2\theta$
(C) $2\cos^2\theta - 1$
(D) $\cos^2\theta - \sin^2\theta$

26. The function $\arctan(x)$ has range: (A) $[0, \pi]$
(B) $[-\pi/2, \pi/2]$
(C) $[-1, 1]$
(D) $(-\infty, \infty)$

27. In polar form, the rectangular point $(-1, 1)$ is: (A) $(\sqrt{2}, \pi/4)$
(B) $(\sqrt{2}, 3\pi/4)$
(C) $(\sqrt{2}, 5\pi/4)$
(D) $(-\sqrt{2}, \pi/4)$

28. The polar graph $r = 3\cos(2\theta)$ is a rose with how many petals? (A) 2
(B) 4
(C) 3
(D) 6

29. An arc of length $12\pi$ cm subtends an angle of $3\pi/2$ radians. The radius of the circle is: (A) $8$ cm
(B) $16$ cm
(C) $18$ cm
(D) $6$ cm

30. Which equation models a circle in polar coordinates? (A) $r = 2\sin\theta$
(B) $r = a\cos\theta$
(C) Both (A) and (B) are circles
(D) Neither is a circle

Unit 4: Parameters, Vectors, and Matrices

31. In the family $f(x) = m(x - 2) + 3$, the locus is: (A) $(2, 3)$
(B) $(0, 3)$
(C) $(-2, -3)$
(D) There is no single common point

32. The magnitude of $\mathbf{v} = \langle 5, -12 \rangle$ is: (A) $7$
(B) $13$
(C) $17$
(D) $\sqrt{169}$

33. If $\mathbf{u} \cdot \mathbf{v} = 0$ and neither vector is zero, then: (A) $\mathbf{u}$ and $\mathbf{v}$ point in the same direction
(B) $\mathbf{u}$ and $\mathbf{v}$ are orthogonal
(C) $\|\mathbf{u}\| = \|\mathbf{v}\|$
(D) $\mathbf{u} = -\mathbf{v}$

34. $\begin{bmatrix} 1 & 3 \\ 2 & 4 \end{bmatrix}\begin{bmatrix} -1 \\ 2 \end{bmatrix} =$ (A) $\begin{bmatrix} 5 \\ 6 \end{bmatrix}$
(B) $\begin{bmatrix} -7 \\ 6 \end{bmatrix}$
(C) $\begin{bmatrix} 5 \\ -6 \end{bmatrix}$
(D) $\begin{bmatrix} 1 \\ 8 \end{bmatrix}$

35. The determinant of $\begin{bmatrix} -3 & 7 \\ 2 & -5 \end{bmatrix}$ is: (A) $1$
(B) $-1$
(C) $29$
(D) $-29$

36. Which matrix does NOT have an inverse? (A) $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$
(B) $\begin{bmatrix} 2 & 4 \\ 1 & 2 \end{bmatrix}$
(C) $\begin{bmatrix} 3 & 1 \\ 0 & 2 \end{bmatrix}$
(D) $\begin{bmatrix} -1 & 5 \\ 0 & 4 \end{bmatrix}$

37. The unit vector in the direction of $\mathbf{v} = \langle 3, 4 \rangle$ is: (A) $\langle 3/4, 1 \rangle$
(B) $\langle 0.6, 0.8 \rangle$
(C) $\langle 1, 4/3 \rangle$
(D) $\langle 4/5, 3/5 \rangle$

38. A vector with magnitude 10 and direction $60°$ above the horizontal has components: (A) $\langle 5, 5\sqrt{3} \rangle$
(B) $\langle 5\sqrt{3}, 5 \rangle$
(C) $\langle 10\cos 60°, 10\sin 60° \rangle$ which equals $\langle 5, 5\sqrt{3} \rangle$
(D) Both (A) and (C) are correct

39. The matrix $\begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix}$ represents: (A) Rotation 90°
(B) Reflection across the y-axis
(C) Reflection across the x-axis
(D) Rotation 180°

40. To solve the system $3x + y = 7$ and $x - 2y = -4$ using matrices, write $\begin{bmatrix} x \\ y \end{bmatrix}$ as: (A) $A^{-1}\mathbf{b}$ where $A = \begin{bmatrix} 3 & 1 \\ 1 & -2 \end{bmatrix}$ and $\mathbf{b} = \begin{bmatrix} 7 \\ -4 \end{bmatrix}$
(B) $AB$ where $B = \begin{bmatrix} 7 \\ -4 \end{bmatrix}$
(C) $A + \mathbf{b}$
(D) $A\mathbf{b}$ directly

Section II: Free-Response Questions (4 questions, 60 minutes)

FRQ 1. (Unit 1 — Polynomial and Rational Functions)

Consider the function $f(x) = \frac{x^3 + 2x^2 - 9x - 18}{x^2 - 9}$.

(a) Factor the numerator and denominator completely. (b) Identify all holes, vertical asymptotes, and the horizontal asymptote (or slant asymptote if applicable). (c) Find all x- and y-intercepts. (d) Describe the end behavior of $f$.

FRQ 2. (Unit 2 — Exponential and Logarithmic Functions)

A hot object cools according to Newton's Law of Cooling. The temperature $T$ (in °C) after $t$ minutes is given by $T(t) = 22 + 78e^{-0.08t}$.

(a) What is the initial temperature of the object? (b) What temperature does the object approach as $t \to \infty$? (c) How long does it take for the object to cool to 50°C? Give your answer to the nearest minute. (d) What is the temperature after 30 minutes?

FRQ 3. (Unit 3 — Trigonometric and Polar Functions)

A mass on a spring oscillates with displacement $d(t) = 8\sin\left(\frac{\pi}{4}t + \frac{\pi}{6}\right)$ centimeters from equilibrium, where $t$ is in seconds.

(a) Find the amplitude, period, and phase shift. (b) What is the maximum displacement from equilibrium? (c) At what times during the first period is the mass at equilibrium ($d = 0$)? (d) Write a cosine function that produces the same graph.

FRQ 4. (Unit 4 — Parameters, Vectors, and Matrices)

(a) Two forces act on an object: $\mathbf{F}_1 = \langle 40, 30 \rangle$ newtons and $\mathbf{F}_2 = \langle -25, 45 \rangle$ newtons. Find the resultant force vector and its magnitude.

(b) Given the system of equations: $$3x - 2y = 8$$ $$5x + y = 13$$ Use the inverse matrix method to solve for $x$ and $y$. Show all work including finding the inverse of the coefficient matrix.

(c) In the function family $f(x) = a\sqrt{x - 2} + 1$, describe the effect of changing $a$ from $3$ to $-3$. What features remain the same? What changes?

END OF EXAM