Practice Test 3 - AP Precalculus

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

Average monthly temperatures in Phoenix, Arizona, show a smooth seasonal rise and fall. Using typical normals (°F), January averages about 57°F and July about 95°F, giving a mean near 76°F. A sinusoidal model uses amplitude to represent half the seasonal swing, period of 12 months, phase shift to place the peak in July, and vertical shift to match the mean. Using the information given in the scenario, explain the impact of vertical shift on the sinusoidal function in this scenario.
On the unit circle, let $\theta=\arccos\left(\frac{\sqrt{3}}{2}\right)$ with $0\le\theta\le\pi$ . Use inverse cosine's principal range to determine $\theta$ exactly in radians.
Matrices can represent geometric transformations of points. A rotation or reflection can be applied by multiplying a transformation matrix by a coordinate vector. For example, reflecting points across the $y$ -axis uses $\mathbf{R}=\begin{bmatrix}-1&0\\0&1\end{bmatrix}.$ Addition and subtraction compare positions before and after a move, while multiplication applies the transformation. Using the matrix model described, what operation would you use to reflect point $\begin{bmatrix}x\\y\end{bmatrix}$ across the $y$ -axis?
A medicine decays as $M(t)=200(0.92)^t$ (hours). When will $M(t)=50$ milligrams?
A surveyor uses $\tan\theta=\dfrac{\text{opposite}}{\text{adjacent}}$ to model an angle of elevation. If $\tan\theta=\dfrac{3}{4}$ , which statement is correct?
Based on the passage, a quadratic polynomial $f(x)=2x^2-4x+13$ has degree $2$ , so it has two zeros counting multiplicity. Complex zeros satisfy $f(x)=0$ even when the graph on the real coordinate plane never crosses the $x$ -axis. Using the quadratic formula, set $2x^2-4x+13=0$ so $x=\dfrac{4\pm\sqrt{(-4)^2-4(2)(13)}}{2\cdot 2}=\dfrac{4\pm\sqrt{16-104}}{4}=\dfrac{4\pm\sqrt{-88}}{4}$ . Rewrite $\sqrt{-88}=\sqrt{88}\,i=2\sqrt{22}\,i$ , giving $x=\dfrac{4\pm 2\sqrt{22}\,i}{4}=1\pm\dfrac{\sqrt{22}}{2}i$ . On the complex plane, these zeros plot at $(1,\pm\tfrac{\sqrt{22}}{2})$ using real and imaginary axes. Calculate the complex zeros of the polynomial using the quadratic formula.
A road's grade uses slope, and tangent matches slope by $\tan\theta=\dfrac{\text{rise}}{\text{run}}$ . Based on this, how does tangent relate to slope?
A set of data relates an independent variable $x$ to a dependent variable $y$ . When analyzing the differences in $y$ -values for uniform increases in $x$ -values, it is found that the first differences are not constant, but the second differences are constant and non-zero. Which type of function is the most appropriate to model this data?
Bacteria doubles hourly in a geometric sequence with $a_1=200$ and $r=2$ : 200; 400; 800. Using the given terms, what is the $n$ th term of the sequence?
A rotating arm follows $\vec{r}(t)=\langle 3\cos(\omega t),\,3\sin(\omega t)\rangle$ ; what does increasing $\omega$ change?
A sinusoidal function $f(x) = a\sin(bx)+d$ has a maximum value of 10 and a minimum value of -2. What is the amplitude of the function?
A radioactive isotope has half-life $8$ days and initial mass $50$ mg, modeled by $A(t)=50\left(\tfrac12\right)^{t/8}$ . Determine the time when $A(t)=12.5$ mg.
In an input-output model, $A=\begin{pmatrix}0.2&0.1\\0.3&0.2\end{pmatrix}$ and demand $\vec d=\begin{pmatrix}50\\40\end{pmatrix}$ ; what is $A\vec d$ ?
The zeros of the function $f(x) = \tan(x)$ occur at which values of $x$ ?
Which of the following values is the greatest?
A sound sensor models intensity by $g(x)=10^x$ and converts to decibels with $f(I)=10\log_{10}(I)$ . Based on the scenario, what is $f(g(3))$ ?
A startup's user base grows according to $f(t)=150\cdot 4^t$ , where $t$ is weeks. The product team uses the inverse to find when users reach a benchmark. Here, $f(t)$ is the number of users after $t$ weeks and 150 is the initial number. Assume the weekly growth factor remains 4. Using the inverse, solve for $t$ when $f(t)=U$ .
A rational function is a quotient of polynomials, $f(x)=\frac{p(x)}{q(x)}$ . Zeros are the $x$ -values where $f(x)=0$ , so $p(x)=0$ and $q(x)\neq 0$ . Vertical asymptotes occur at real solutions of $q(x)=0$ that do not cancel with the numerator. Consider the rational function $f(x)=\frac{x^2+1}{x^2-1}.$ Here, $x^2-1=(x-1)(x+1)$ , so the function is undefined at $x=\pm 1$ , producing vertical asymptotes at $x=1$ and $x=-1$ because nothing cancels. The numerator $x^2+1=0$ has solutions $x=\pm i$ , which are complex zeros; this means there are no real $x$ -intercepts. Using the function provided, determine the vertical asymptotes of $f(x)=\frac{x^2+1}{x^2-1}$ .
A curve is parameterized by $x(t) = 2\sin(t) + 3$ and $y(t) = \cos^2(t)$ for all real $t$ . What is the range of possible y-values for this curve?
As an angle $\theta$ increases from $\pi$ to $\frac{3\pi}{2}$ , what is the behavior of $\sin\theta$ and $\cos\theta$ ?

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