Practice Test 62 - Algebra 2

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

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1. The amount of a medication remaining in the bloodstream is modeled by $M(h)=250(0.88)^h$, where $M$ is in milligrams and $h$ is time in hours. In $M(h)=250(0.88)^h$, what does the 0.88 represent?
2. Solve the system using the inverse of the coefficient matrix. $$\begin{cases} 2x+y=5\\ 3x+4y=7 \end{cases}$$ Let $A=\begin{bmatrix}2&1\\3&4\end{bmatrix}$, $X=\begin{bmatrix}x\\y\end{bmatrix}$, and $B=\begin{bmatrix}5\\7\end{bmatrix}$. Find $A^{-1}$ and then compute $X=A^{-1}B$.
3. For the logarithmic function $q(x)=2\ln(x)+3$, identify the vertical asymptote and state the domain. (Connect to the parent function $y=\ln x$.)
4. Use polynomial division to write $\frac{x^4-3x^2+5}{x^2+1}$ in the form $q(x)+\frac{r(x)}{x^2+1}$, where $\deg(r)<\deg(x^2+1)$.
5. A linear function has $y$-intercept 6 and passes through the point $(4,14)$. Which function is correct?
6. Interpret the nested structure by chunking: $g(x)=2(3x+1)^3.$ View $(3x+1)$ as a single entity. Which statement correctly describes the structure and which part is independent of $x$?
7. Use the Binomial Theorem to compute $103^3$ by writing it as $(100+3)^3.$
8. All points $(x,y)$ that are 6 units from the origin form a circle. 1) Write an equation in two variables for this circle. 2) Choose the best graph setup (labels and scale) to draw it accurately. Which choice is correct?​
9. Demonstrate that the linear function $f(x)=3x+2$ has constant differences over unit intervals. Which statement correctly shows the constant difference for $\Delta x=1$?
10. Use technology (calculator or software) to solve the system by the matrix inverse method. System: \begin{align*} 2x+y-z&=1\\ x-y+2z&=8\\ 3x+2y+z&=7 \end{align*} Write the system as $AX=B$ and use $X=A^{-1}B$ (do not hand-calculate the $3\times 3$ inverse). What is $X=\begin{bmatrix}x\\y\\z\end{bmatrix}$?
11. Let $f(x)=\sqrt{x}$ (defined for $x\ge 0$). Consider $g(x)=f\left(\frac{x}{3}\right).$ How does the graph of $g$ compare to the graph of $f$?
12. If $w_1 = 1 + 2i$ and $w_2 = 3 + i$, what is the real part of $w_1 \overline{w_2} - \overline{w_1} w_2$?
13. Convert the system to matrix form $AX=B$ (variable order $x, y, z$): $\begin{cases} 6x-y+0z=3\\ 0x+2y-5z=-8\\ -x+0y+4z=1 \end{cases}$​
14. The table gives values of a one-to-one function $f$, so $f$ has an inverse. Use the table to find $f^{-1}(-4)$. (If $(a,b)$ is on $f$, then $(b,a)$ is on $f^{-1}$.)

    $x$	$f(x)$
    -3	5
    -1	1
    0	-2
    2	-4
    4	-7

15. Let $f(x)=x^3$ be the parent function. How does replacing $x$ with $x+2$ affect the graph, i.e., $g(x)=f(x+2)$?
16. Consider $f(x)=x^2-6x+8$ with domain all real numbers. Which domain restriction makes $f$ invertible by restricting to one side of its vertex (so it passes the horizontal line test)?
17. Consider the function $g(x)=x^4-3x^2.$ Does $g$ have symmetry? What type?
18. A water tank is being filled at a constant rate. The volume of water is modeled by $V = 8t + 120$, where $V$ is in liters and $t$ is time in minutes. In $V = 8t + 120$, what does it mean that the y-intercept is 120?
19. A laptop's value depreciates over time. The value (in dollars) is recorded each year: Year 0: 900 Year 1: 765 Year 2: 650.25 Year 3: 552.7125 Does this represent constant percent change? If so, is it growth or decay, and what is the percent rate per year?
20. A medication amount in the bloodstream is modeled by $M(t)=300(0.92)^t$, where $M$ is the amount in milligrams and $t$ is time in hours. What does it mean that the base is $0.92$ in this context?

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