Practice Test 42 - Algebra 2

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

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1. Consider $f(x) = \frac{x^2 + 5x + 6}{2x^2 + 7x + 3}$. If this function is graphed, which description best characterizes its end behavior and critical features?
2. Treat $z=a+bi$ as the point $(a,b)$ on the complex plane. The distance between $z_1=a+bi$ and $z_2=c+di$ is $|z_1-z_2|=\sqrt{(a-c)^2+(b-d)^2}$, and the midpoint is $\dfrac{z_1+z_2}{2}=\dfrac{a+c}{2}+\dfrac{b+d}{2}i$. For $z_1=-4+0i$ and $z_2=2+6i$, what are the distance and midpoint?
3. Identify the amplitude, period, and midline of the trigonometric function $f(x)= -3\sin(2x)+1$. (Relate to the parent function $y=\sin x$.)
4. Evaluate $\log_{10}(1)$ using the definition $\log_b(x)=y \Leftrightarrow b^y=x$.
5. Determine whether the system has one solution, no solution, or infinitely many solutions. If it has a solution, find it. $\begin{cases} 2x - y = 1 \\ 4x - 2y = 5 \end{cases}$
6. A company models total cost as $C(q) = 1200 + 35q,$ where $q$ is the number of items produced. Which interpretation correctly identifies independent parts by chunking the expression into a fixed part and a part that depends on $q$?
7. Two printers work at constant rates. Printer X can print a batch of flyers in 5 hours, and Printer Y can print the same batch in 10 hours. Write and solve an equation to find how long it takes to print one batch if both printers work together. Let $t$ = time in hours to print one batch together.
8. If $z_1$ and $z_2$ are the roots of $3x^2 - 12x + 15 = 0$, what is the value of $|z_1|^2 + |z_2|^2$?
9. Consider the finite geometric series $\sum_{k=0}^{9} 4\left(\frac{1}{3}\right)^k.$ What is its exact value? Note: Since $\left|\frac{1}{3}\right|<1$, the infinite geometric series would converge, but this question asks for the finite sum with $n=10$ terms.
10. Write the system of linear equations as a single matrix equation $AX=B$ (use variable order $x, y$): $$\begin{cases} 3x+2y=8\\ x-5y=-3 \end{cases}$$
11. Use the identity $(a+b)^2=a^2+2ab+b^2$ to compute $101^2$ mentally by rewriting $101$ as $100+1$. Which result is correct?​
12. Apply the sum of cubes identity to factor the polynomial $x^3+8$ completely over the integers.
13. The table shows some values of function $p(x)$. If $p$ has an inverse, which of the following could be the value of $p(6)$?
14. Factor completely by recognizing repeated difference of squares structure: $x^6 - 64$
15. A population model uses the annual growth expression $(1.08)^t$, where $t$ is in years. Which equivalent expression best reveals the quarterly growth factor (4 quarters per year)?
16. A sequence is a function whose domain is a subset of the integers, so it is defined only at discrete inputs. Which set is the most appropriate domain for a sequence written as $a_1, a_2, a_3, \dots$?
17. Given the explicit formula $a_n = 18\left(\tfrac{2}{3}\right)^{n-1}$ for $n\ge 1$, write an equivalent recursive definition for the sequence.
18. A complex number $z=a+bi$ represents the point $(a,b)$ on the complex plane. Using the coordinate-geometry distance formula, the distance between $z_1=a+bi$ and $z_2=c+di$ is $|z_1-z_2|=\sqrt{(a-c)^2+(b-d)^2}$, and the midpoint is $\dfrac{z_1+z_2}{2}$. For $z_1=-2-i$ and $z_2=4+3i$, what are the distance and the midpoint?
19. A manufacturing company plans to increase production by 8% each quarter for the next 5 quarters, starting with an initial production of 2,400 units in Quarter 1. What is the total number of units produced over all 5 quarters?
20. Solve and evaluate using technology: $12\cdot 10^{t/4}=500$. Give $t$ in exact logarithmic form and as a decimal approximation.

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