Practice Test 1 - AP Precalculus

20 questions. Answer them in any order, then submit to see your score and every explanation.

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1. Data from an experiment shows a relationship where the output variable increases to a single maximum value and then decreases, appearing to be symmetric. Both a quadratic and a quartic ($4^{th}$ degree) polynomial model fit the data well. In the absence of a theoretical reason to prefer one over the other, why might a researcher choose the quadratic model?
2. Using the matrix provided, determine $\det(\mathbf{A})$ for $\mathbf{A}=\begin{pmatrix}1&2\\2&4\end{pmatrix}$ and interpret its significance.
3. What are the solutions for $\sqrt{3}\sin(x) = \cos(x)$ on the interval $[0, 2\pi]$?
4. An exponential function $f$ has the property that for any increase of 1 in the input variable $x$, the output $f(x)$ is multiplied by a factor of 4. Which of the following could be an expression for $f(x)$?
5. The tide level is modeled by $y=3\cos\left(\frac{\pi}{6}x\right)+5$, where $x$ is hours after midnight. The amplitude measures half the difference between high and low tide. The period gives the time for one full tidal cycle. Using the given function, how would the graph change if the amplitude is doubled?​
6. A population of deer is modeled by $f(t)=120\cdot 4^t$, where $t$ is measured in years. A wildlife manager needs the inverse to find the year when the herd reaches a given size. Here, $f(t)$ is the number of deer after $t$ years and 120 is the initial herd size. The base 4 indicates the herd quadruples each year in this model. What is the inverse of the function $f(t)$?
7. A Ferris wheel height is $h(t)=10\cos\!\left(\frac{\pi}{12}t\right)+12$, where $t$ is seconds. The amplitude is $10$, representing the radius, and the vertical shift is $12$. The period is $24$ seconds, so the motion repeats regularly. Doubling amplitude changes only the distance from the midline, not timing. Using the given function, how would the graph change if the amplitude is doubled?
8. A polynomial function $p(x)$ has distinct real zeros at $x=-3$, $x=1$, and $x=4$. Which of the following intervals must contain a local extremum of $p(x)$?
9. What is the determinant of the matrix $R = \begin{pmatrix} \sin\theta & \cos\theta \\ -\cos\theta & \sin\theta \end{pmatrix}$ for any real number $\theta$?
10. A fish population model uses $P(t)=-(t-3)^2(t+2)^3$; what does end behavior suggest as $t\to\infty$?
11. Stress along a beam: $S(x)=(x^2-9)(-2x+6)(x^2+1)$; identify degree and leading coefficient significance.
12. A robotics lab uses a rotating sensor to track a small rover whose planned path is modeled by $r(\theta)=5-2\sin(\theta)$, where $r$ (in meters) is the rover's distance from the sensor and $\theta$ (in radians) is the sensor's angle. The derivative $\dfrac{dr}{d\theta}$ represents the rate of change of the rover's distance per radian as the angle changes, found by differentiating the polar function with respect to $\theta$. Using $\dfrac{d}{d\theta}(\sin\theta)=\cos\theta$, the lab can determine whether the rover is moving inward or outward relative to the sensor as the scan angle increases. This helps tune the tracking algorithm so it can respond appropriately to rapid radial changes. Given $r(\theta)=5-2\sin(\theta)$, what is $\dfrac{dr}{d\theta}$ at $\theta=\pi/4$?
13. Average monthly temperatures in Denver, Colorado, follow a yearly cycle. Suppose average daily mean temperatures (°F) are: Jan 30, Feb 32, Mar 40, Apr 48, May 58, Jun 68, Jul 74, Aug 72, Sep 63, Oct 51, Nov 39, Dec 31. The maximum is 74 and the minimum is 30, so amplitude is 22 and vertical shift is 52. The period is 12 months, and the phase shift should place the peak near July. Based on the data presented, which sinusoidal function best models temperature $T(m)$, where $m$ is months after January (so $m=0$ is January)?
14. Which equation represents a phase shift of $\pi/2$ right for $y=\sin(x)$?
15. A curve is defined by the parametric equations $x(t) = 5 + 2\cos(t)$ and $y(t) = -1 + 6\sin(t)$. Which of the following is the equation of the curve in rectangular coordinates?
16. In astronomy, Kepler described planetary orbits as ellipses. Consider the standard form $\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1$ with $a>b>0$, where $(h,k)$ is the center. Based on the conic section described, how does increasing $a$ affect the ellipse's horizontal extent?
17. Consider the function $t(x)=\dfrac{x^2+3x}{x}$ for a production ratio; what is the location of the hole?
18. The height $h$, in meters, of a ball thrown upwards from a building is modeled by the function $h(t) = -4.9t^2 + 20t + 50$, where $t$ is the time in seconds after the ball is thrown. Which of the following describes a necessary restriction on the domain of the function for this model to be physically realistic?
19. A radioactive sample follows $A(t)=80\left(\tfrac12\right)^{t/6}$ (days). How long until $A(t)=10$ grams?
20. When multiplying a $2 \times 2$ transformation matrix $A$ by a $2 \times n$ matrix of input vectors, what is the result?

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