Practice Test 5 - AP Precalculus

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

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1. What is the period of the function $g(x)$?
2. What is the phase shift of the graph of the function $y = 3\cos(2x - \frac{\pi}{2})$?
3. A polynomial function $p(x)$ of degree $n \ge 2$ has a local maximum at $x=c$. Which of the following statements must be true about the behavior of $p(x)$ in the immediate vicinity of $x=c$?
4. Based on the problem described, what is the role of $t$ in $\vec r(t)=\langle 3\cos t,3\sin t\rangle$?
5. Find $\lim\limits_{x\to -2^+}\dfrac{5}{x+2}$; what is the behavior near the vertical asymptote?
6. Which of the following parametrizations does NOT describe the unit circle defined by $x^2 + y^2 = 1$?
7. The function $h(x) = a\tan(x)$ is a transformation of the parent tangent function. If $a = -2$, which of the following describes the transformation?
8. Using the polynomial provided, a quadratic $u(x)=3x^2-18x+31$ has $\Delta=324-372=-48$; calculate the complex zeros with the quadratic formula, rewriting $\sqrt{-48}=4\sqrt{3}\,i$ and simplifying.​
9. Two earthquakes have magnitudes $M_1=5.2$ and $M_2=6.0$ on the Richter scale, where $M=\log_{10}\!\left(\dfrac{I}{I_0}\right)$. To compare intensities, you use $M_2-M_1=\log_{10}\!\left(\dfrac{I_2}{I_1}\right)$. Based on the scenario above, solve for the intensity ratio $\dfrac{I_2}{I_1}$.​
10. In an economics model, the average cost (in dollars) per item is $A(x)=\dfrac{2000+50x}{x+10}$, where $x$ is the number of items produced and $x\ge0$ with $x\neq -10$. In the context of this model, how does $A(x)$ behave as production becomes very large? Use $\lim_{x\to\infty} A(x)$ and identify the horizontal asymptote.​
11. A town studies long-term population, but planners expect crowding to slow growth over time. Model 1: Exponential, $y=20000(1.04)^x$, where 1.04 is the annual growth factor. Model 2: Logarithmic, $y=20000+7000\log(x+1)$, where 7000 controls the added increase. Early measurements rise quickly and the exponential model matches the first few years well. After year 12, the yearly increase shrinks as housing becomes scarce. The exponential model keeps accelerating and predicts very large populations by year 50. The logarithmic model still rises but flattens, aligning with the crowding constraint. Based on the models described in the passage, which model best predicts long-term behavior in the context described?
12. Which of the following is a parametrization for a circle centered at the origin with a radius of 5, traced counterclockwise starting from the point (5, 0)?
13. Which rational function has a vertical asymptote at $x = -2$ but no vertical asymptote at $x = 3$?
14. The domain of the rational function $r(x)$ is all real numbers except $x=2$, and its range is all real numbers except $y=1$. Let $h(x) = r(x+4) - 5$. What are the domain and range of $h(x)$?
15. Consider the matrices $A = \begin{pmatrix} 3 & -2 \\ -4 & 3 \end{pmatrix}$ and $B = \begin{pmatrix} 3 & 2 \\ 4 & 3 \end{pmatrix}$. What is the product $AB$?
16. An investment is modeled by $A(t)=3000(1.06)^t$. Based on the scenario, how would you use logarithms to solve for $t$ when $A(t)\ge 6000$?​
17. A polynomial function of degree 4 has a negative leading coefficient. Which of the following must be true about the function?
18. A $\$12{,}000$ account increases 3.5% per year, modeled by $V(t)=12000\cdot(1.035)^t$. Which equation represents the scenario described?
19. The expression $\cos(x - \frac{\pi}{2})$ is equivalent to which of the following?

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