Practice Test 28 - Algebra 2

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

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1. On the complex plane, $z=-2+3i$ corresponds to the point $(-2,3)$. What is the distance from the origin to this point (i.e., the modulus $|z|$)?
2. How many terms are in the expression $2x^3 - 5x^2 + x - 8$? (Terms are separated by $+$ or $-$.)​
3. Solve and express solutions in $a \pm bi$ form: $x^2 + 4x + 8 = 0$.
4. Convert $4\,\text{cis}\,150^\circ$ to rectangular form $a+bi$. (Recall $r\,\text{cis}\,\theta=r(\cos\theta+i\sin\theta)$ and use $a=r\cos\theta$, $b=r\sin\theta$.)
5. What is the 4th term (counting from the first term as term 1) in the expansion of $(x+y)^6$?
6. A car worth $\$18{,}000$ depreciates by $14\%$ each year. Create an exponential model for its value $V(t)$ after $t$ years, then find how long it takes for the value to fall to $9000. Round to the nearest tenth of a year and interpret your result in context.
7. Graph the exponential function $f(x)=3\cdot 2^x-4$ showing the y-intercept, the horizontal asymptote, and the end behavior.
8. A lab uses the formula $T(h)=\dfrac{120}{h}$ to estimate the time $T$ (in minutes) it takes a chemical to cool when the heat setting is $h$. The heat setting must be a positive number, and the machine allows settings from 1 to 10 (including decimals). What is an appropriate realistic domain for $T(h)$?
9. Solve for $x$ and give both exact and approximate forms: $9\cdot 2^{x/5}=40.5$. (Express the exact answer using $\log_2$.)
10. For $f(x)=\frac{(x+2)(x-3)}{x(x-3)},$ identify any hole(s), vertical asymptote(s), zero(s), and the horizontal asymptote.​​​
11. Explain why $f(x)=x^2$ needs a domain restriction to have an inverse function. Which statement best connects this to the horizontal line test?
12. A system is written in matrix form as $AX=B$. Use technology to find $A^{-1}$ and then compute $X=A^{-1}B$. $A=\begin{bmatrix}1&1&1\\0&2&1\\2&-1&1\end{bmatrix},\quad B=\begin{bmatrix}6\\7\\3\end{bmatrix}$ What is the solution vector $X$?
13. Verify that the polynomial identity $a^3+b^3=(a+b)(a^2-ab+b^2)$ is true for all real numbers $a$ and $b$ by expanding the right-hand side.
14. Determine whether the coefficient matrix is invertible. If it is invertible, the system has a unique solution found by $X=A^{-1}B$. If it is not invertible (singular), $A^{-1}$ does not exist. $A=\begin{bmatrix}1&2\\2&4\end{bmatrix},\quad B=\begin{bmatrix}3\\6\end{bmatrix}$ Which statement is correct?
15. A quantity follows the model $Q(t)=Q_0\,(1.02)^{12t}$, where $t$ is in years. Which equivalent form reveals the annual growth factor?
16. Interpret the exponential expression by chunking: $y = A b^{ct}.$ Here $A$ is an initial amount, $b$ is a base, $c$ is a constant, and $t$ is time. Which statement correctly treats $b^{ct}$ as a single entity and identifies what does not depend on $A$?
17. A model for the value of a used laptop (in dollars) after $t$ years is $V(t)=900(0.8)^t.$ Describe the end behavior of $V(t)$ as $t \to \infty$ and interpret it in context.
18. Simplify using logarithm properties: $\log_4(64) - \log_4(4)$
19. A projectile's height might be modeled by a quadratic literal equation $h = ut - \dfrac{1}{2}gt^2$, where $h$ is height, $u$ is initial vertical velocity, $g$ is gravitational acceleration, and $t$ is time. Solve $h = ut - \dfrac{1}{2}gt^2$ for $t$.
20. Solve by completing the square: $x^2+6x+1=0.$

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