Practice Test 69 - Algebra 2

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

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1. Convert the complex number $-3-3i$ from rectangular form $a+bi$ to polar form $r(\cos\theta+i\sin\theta)$. Use $r=\sqrt{a^2+b^2}$ and $\theta=\arctan\!\left(\frac{b}{a}\right)$ with quadrant adjustment. (Angle in degrees.)
2. A sinusoidal model for ocean tide height (in meters) at a pier is $T(t)=1.5\cos\left(\frac{\pi}{6}t\right)+2,$ where $t$ is time in hours. What is the period of $T(t)$, and what does it mean in context?
3. The graph shows a function $f$ drawn only on a restricted domain: it starts at a filled point at $x=-2$ and ends at an open point at $x=4$. From the graph, what is the domain of $f$? (Assume the curve exists for every $x$ between the endpoints.)
4. Find and verify the inverse of $f(x)=\dfrac{2x+3}{5}$ by choosing the correct $f^{-1}(x)$ (so that $f(f^{-1}(x))=x$).​​​
5. Use the change-of-base formula as needed and write in terms of natural logs: $\log_5(20).$ (Recall: $\log_b(x)=\frac{\log_c(x)}{\log_c(b)}$.)
6. A savings account balance is modeled by $B(t)=1200(1.03)^t$, where $B$ is the balance in dollars and $t$ is time in years. In the function $B(t)=1200(1.03)^t$, interpret the parameter 1200 in context.
7. A finite geometric series has first term $a=12$, common ratio $r=\tfrac{1}{3}$, and $n=7$ terms. What is $S_7$? (Note: If $|r|<1$, the infinite sum would converge, but here you must find the finite sum.)
8. A savings account has an annual growth factor of $1.15$. Which expression is equivalent to $(1.15)^t$ and makes the approximate monthly growth factor explicit? (You may use a calculator for the approximation.)
9. A savings account balance is tracked over several months. In month 1, the balance is $800. In month 3, the balance is $920. In month 5, the balance is $1040. In month 7, the balance is $1160. Which statement accurately describes the rate of change in the account balance?
10. Solve $4e^{0.5t}=50$ for $t$. Express the solution using a logarithm (exact form) and then approximate using technology.
11. Find the inverse function $f^{-1}(x)$ for $f(x)=2x^3$.​
12. Consider $h(x) = 2\sin(x) + 1$ on the interval $[0, 2\pi]$. To create an invertible function that achieves the maximum value of $h(x)$, which domain restriction should be used?
13. A gym charges a monthly membership fee plus a cost per class. The total cost (in dollars) after attending $c$ classes in a month is modeled by $C = 12c + 35$. In this function, what does the parameter $35$ represent in context?
14. A bacteria culture is measured every hour. The number of bacteria is shown below. Time (hours): 0, 1, 2, 3 Bacteria count: 500, 575, 661.25, 760.4375 Does this situation represent constant percent change? If yes, classify it and give the percent rate per hour.
15. In a finance model, one part of a calculation produces $(1.03)^{4t}(1.03)^{8t}$. Rewrite as a single exponential expression using the product of powers property.​​
16. The sequence $1,3,9,27,81,\dots$ can be treated as a function $g(n)$ with domain $n\in\{1,2,3,\dots\}$. Which recursive definition correctly generates this sequence?
17. A rental car company charges according to $C(m) = 45 + 0.25m$, where $C(m)$ is the total cost in dollars and $m$ is miles driven. If they change their pricing to $C(m) = 60 + 0.15m$, which statement best describes the effect of both parameter changes?
18. Solve $6\cdot 10^{0.2t}=48$ for $t$. Express your solution as a logarithm and then approximate using technology.
19. Let $f(x)=3^x$ (exponential) and $g(x)=x^4$ (polynomial). Use the table to identify the crossover point. At which listed $x$ does $f(x)$ first exceed $g(x)$?
20. Which set of matrices $A$, $X$, and $B$ correctly represents the system in the form $AX=B$ (use variable order $x, y$)? $$\begin{cases} 7x-3y=2\\ -5x+6y=9 \end{cases}$$

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