Free AP Precalculus Practice Exam (PDF)

This free AP Precalculus practice exam covers the three units the exam assesses: polynomial and rational functions, exponential and logarithmic functions, and trigonometric and polar functions. The multiple-choice section tests end behaviour, multiplicity, holes versus asymptotes, concavity from a table, semi-log plots, log properties, double-angle identities and the rate of change of a polar curve. The two free-response questions follow the real modelling format: fitting an exponential model to a described data set and building a sinusoidal model for a Ferris wheel. Students lose most marks by skipping the justification, so every solution states the reasoning the rubric rewards.

1. Question 12 marks · Polynomial functions
   Let $p(x) = (3 - x)(x + 1)^2(2x - 5)$. Which of the following describes the end behavior of $p$?
   • A.
     p(x) \to \infty$ as $x \to \infty$ and $p(x) \to \infty$ as $x \to -\infty

     LaTeX syntax issue: KaTeX parse error: Can't use function '$' in math mode at position 16: p(x) \to \infty$̲ as $x \to \inf…
   • B.
     p(x) \to -\infty$ as $x \to \infty$ and $p(x) \to \infty$ as $x \to -\infty

     LaTeX syntax issue: KaTeX parse error: Can't use function '$' in math mode at position 17: …(x) \to -\infty$̲ as $x \to \inf…
   • C.
     p(x) \to \infty$ as $x \to \infty$ and $p(x) \to -\infty$ as $x \to -\infty

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   • D.
     p(x) \to -\infty$ as $x \to \infty$ and $p(x) \to -\infty$ as $x \to -\infty

     LaTeX syntax issue: KaTeX parse error: Can't use function '$' in math mode at position 17: …(x) \to -\infty$̲ as $x \to \inf…
2. Question 22 marks · Polynomial functions
   Let $p(x) = (x + 2)^2 (x - 1)^3 (x - 4)$. Which statement about the graph of $p$ is true?
   • A.
     The graph crosses the $x$-axis at $x = -2$, $x = 1$ and $x = 4$.
   • B.
     The graph touches the $x$-axis without crossing at $x = -2$ and $x = 1$ and crosses at $x = 4$.
   • C.
     The graph touches the $x$-axis without crossing at $x = -2$ and crosses at $x = 1$ and $x = 4$.
   • D.
     The graph crosses the $x$-axis at $x = -2$ and touches without crossing at $x = 1$ and $x = 4$.
3. Question 32 marks · Rational functions
   Let $r(x) = \dfrac{2x^2 - 8}{x^2 - 5x + 6}$. Which of the following is true about the graph of $r$?
   • A.
     Vertical asymptotes at $x = 2$ and $x = 3$; horizontal asymptote $y = 2$
   • B.
     A hole at $x = 2$, a vertical asymptote at $x = 3$ and a horizontal asymptote $y = 0$
   • C.
     A hole at $x = 2$, a vertical asymptote at $x = 3$ and a horizontal asymptote $y = 2$
   • D.
     Vertical asymptotes at $x = 2$ and $x = 3$ and no horizontal asymptote
4. Question 42 marks · Rational functions
   As $x \to \infty$, the graph of $f(x) = \dfrac{x^2 + 3x - 1}{x - 1}$ approaches the line
   • A.
     $y = x$
   • B.
     $y = x + 2$
   • C.
     $y = x + 3$
   • D.
     $y = x + 4$
5. Question 52 marks · Rates of change
   A function $f$ has $f(1) = 3$, $f(2) = 7$, $f(3) = 10$ and $f(4) = 12$. Based on these values, which best describes $f$ on the interval $1 \leq x \leq 4$?
   • A.
     Increasing and concave up
   • B.
     Decreasing and concave down
   • C.
     Increasing and concave down
   • D.
     Linear with slope $3$
6. Question 62 marks · Exponential functions
   An exponential function $f(x) = a \cdot b^x$ satisfies $f(2) = 12$ and $f(5) = 96$. The value of $f(0)$ is
   • A.
     $1.5$
   • B.
     $3$
   • C.
     $4$
   • D.
     $6$
7. Question 72 marks · Logarithmic functions
   If $\log_b 2 = p$ and $\log_b 3 = q$, then $\log_b \dfrac{18}{b} =$
   • A.
     $p + 2q - 1$
   • B.
     $2p + q - 1$
   • C.
     $p + 2q$
   • D.
     $p + 2q - b$
8. Question 82 marks · Exponential modelling
   A set of data points $(x, y)$ is plotted on a semi-log graph with a logarithmic (base 10) scale on the vertical axis and a linear scale on the horizontal axis. The points lie on a straight line with slope $0.5$ and vertical intercept $2$. Which function models $y$ in terms of $x$?
   • A.
     $y = 2 + 0.5x$
   • B.
     $y = 10^{2x + 0.5}$
   • C.
     $y = 100 \cdot x^{0.5}$
   • D.
     $y = 100 \cdot 10^{0.5x}$
9. Question 92 marks · Exponential functions
   A sample of a radioactive substance decays exponentially. Its mass falls from 80 milligrams to 10 milligrams in 15 days. The half-life of the substance is
   • A.
     $3$ days
   • B.
     $5$ days
   • C.
     $7.5$ days
   • D.
     $\dfrac{15}{8}$ days
10. Question 102 marks · Logarithmic functions
    If $f(x) = 3 + \ln(x - 2)$ for $x > 2$, then the inverse function is $f^{-1}(x) =$
    • A.
      $e^{x - 3} + 2$
    • B.
      $e^{x - 2} + 3$
    • C.
      $e^{x + 2} - 3$
    • D.
      $\ln(x - 3) + 2$
11. Question 112 marks · Trigonometric identities & equations
    If $\sin\theta = \dfrac{3}{5}$ and $\dfrac{\pi}{2} < \theta < \pi$, then $\sin 2\theta =$
    • A.
      $\dfrac{24}{25}$
    • B.
      $-\dfrac{24}{25}$
    • C.
      $\dfrac{7}{25}$
    • D.
      $-\dfrac{7}{25}$
12. Question 122 marks · Sinusoidal models
    At a pier the water depth varies sinusoidally with time. The depth is at its minimum of 2 meters at 3 a.m. and reaches its next maximum of 8 meters at 9 a.m. Which function gives the depth $d(t)$ in meters, where $t$ is the number of hours after midnight?
    • A.
      $d(t) = 5 - 3\cos\left(\dfrac{\pi}{6}(t - 3)\right)$
    • B.
      $d(t) = 5 + 3\cos\left(\dfrac{\pi}{6}(t - 3)\right)$
    • C.
      $d(t) = 5 - 3\cos\left(\dfrac{\pi}{12}(t - 3)\right)$
    • D.
      $d(t) = 5 - 3\sin\left(\dfrac{\pi}{6}(t - 3)\right)$
13. Question 132 marks · Trigonometric identities & equations
    How many solutions does the equation $2\sin^2\theta - \sin\theta - 1 = 0$ have on the interval $0 \leq \theta < 2\pi$?
    • A.
      $1$
    • B.
      $2$
    • C.
      $3$
    • D.
      $4$
14. Question 142 marks · Polar functions
    The polar function $r = 3 + 3\cos\theta$ is graphed for $0 \leq \theta \leq 2\pi$. On which interval of $\theta$ is the distance between the point on the curve and the origin decreasing as $\theta$ increases?
    • A.
      $0 < \theta < \pi$
    • B.
      $\pi < \theta < 2\pi$
    • C.
      $\dfrac{\pi}{2} < \theta < \dfrac{3\pi}{2}$
    • D.
      0 < \theta < \dfrac{\pi}{2}$ and $\dfrac{3\pi}{2} < \theta < 2\pi

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15. Question 158 marks · Exponential modelling
    Modelling a data set. A company launched a mobile app and recorded the number of active subscribers, in thousands, at the end of each of its first six months. At the end of month 1 it had 4.2 thousand subscribers; month 2, 7.2 thousand; month 3, 12.3 thousand; month 4, 20.9 thousand; month 5, 35.5 thousand; and month 6, 60.3 thousand. Let $S(t)$ be the number of subscribers, in thousands, at the end of month $t$.
    (a)

    Find the average rate of change of $S$ over the interval $1 \leq t \leq 3$ and over the interval $4 \leq t \leq 6$. Include units, and explain what the comparison tells you about the shape of the graph of $S$.

    [2]
    (b)

    Explain, using the data, why an exponential model is more appropriate than a linear model for $S$.

    [1]
    (c)

    Use the data for months 1 and 6 to write an exponential model of the form $S(t) = a \cdot b^t$. Round $a$ and $b$ to three decimal places.

    [2]
    (d)

    Use your model to predict the number of subscribers at the end of month 9. Then give one reason why this prediction may be unreliable.

    [2]
    (e)

    According to your model, during which month does the company first reach 150 thousand subscribers?

    [1]
16. Question 168 marks · Sinusoidal models
    Modelling a periodic context. A Ferris wheel has a diameter of 40 meters, and its lowest point is 2 meters above the ground. The wheel turns at a constant rate and completes one full rotation every 4 minutes. A rider boards at the lowest point at time $t = 0$ and the wheel starts turning immediately. Let $h(t)$ be the height of the rider above the ground, in meters, $t$ minutes after boarding.
    (a)

    State the midline, the amplitude and the period of $h$, and explain what each represents in the context of the Ferris wheel.

    [2]
    (b)

    Write a function $h(t)$ that models the height of the rider.

    [2]
    (c)

    Find the height of the rider $20$ seconds after boarding.

    [1]
    (d)

    Find the first two times after boarding at which the rider is exactly $32$ meters above the ground.

    [2]
    (e)

    On the interval $0 < t < 4$, find the interval of time during which the height of the rider is increasing at a decreasing rate. Explain your reasoning.

    [1]

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