Practice Test 14 - Algebra 2

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

Start practice test

1. Solve by completing the square and express the solutions in complex form (if needed): $x^2-6x+13=0$
2. Use zeros to construct a rough graph of $h(x)=-(x-2)(x+1)(x+3)^2.$ Identify the zeros with multiplicities, indicate crossing vs touching, and state the end behavior.
3. Let $z=-6+2i$. Using the fact that taking the conjugate twice returns the original number, compute $\overline{\overline{z}}$.
4. Let $F(x)=5(x-2)^2+7(x-2)-3.$ If you chunk by letting $u=(x-2)$ and view $u$ as a single entity, which rewritten form correctly shows the structure in terms of $u$?
5. A gym membership costs $25$ per month plus a one-time sign-up fee of $60$. Let $m$ be the number of months. Write an equation for total cost $C$ (dollars) in terms of $m$ and choose appropriate axes labels and scale to graph for $0 \leq m \leq 12$. Which choice is correct?
6. A function is shown on the coordinate plane as a parabola opening upward with vertex at $(2,-3)$ and passing through $(0,1)$. Identify the $y$-intercept of the function.
7. Verify the inverse relationship between the temperature conversion formulas $C(F)=\dfrac{5}{9}(F-32)$ and $F(C)=\dfrac{9}{5}C+32$ by showing that $C(F(C))=C$ and $F(C(F))=F$.
8. What is the complex conjugate of $2-5i$?
9. Determine if $x=2$ is a zero of $p(x)=x^3-6x^2+11x-6$. (Hint: $x=2$ is a zero iff $p(2)=0$, which also means $(x-2)$ is a factor.)
10. A bacteria culture starts with 200 bacteria and doubles every 3 hours. Create an exponential model for the population after $t$ hours, then solve for when the population first reaches 10,000 bacteria. Round to the nearest tenth of an hour and interpret your result in context.
11. Use technology to solve the system by computing $X=A^{-1}B$. $\begin{cases} 3x+y+z=9\\ x+4y+z=9\\ x+y+5z=17 \end{cases}$ In matrix form, $AX=B$ with $A=\begin{bmatrix}3&1&1\\1&4&1\\1&1&5\end{bmatrix},\quad B=\begin{bmatrix}9\\9\\17\end{bmatrix}.$ What is $(x,y,z)$?
12. What is the LCD (least common denominator) for the rational expressions below? Finding the LCD is the key step for adding/subtracting rational expressions. Find the LCD for: $\frac{3}{x^2-4}\quad\text{and}\quad\frac{2}{x^2+4x+4}$​​
13. The function $h(x) = \sqrt{25 - x^2}$ represents the height of a semicircular arch, where $x$ is the horizontal distance from the center. Based on the graph shown, which statement about the domain is most accurate?
14. Convert $5\left(\cos 30^\circ+i\sin 30^\circ\right)$ to rectangular form $a+bi$ using $a=r\cos\theta$ and $b=r\sin\theta$.
15. Solve and evaluate: $2\cdot e^{0.5t}=20$. Write $t$ in exact logarithmic form and then approximate using technology.
16. Consider $f(x)=(x-3)^2+1$ with domain all real numbers. It fails the horizontal line test because some horizontal lines intersect the parabola twice. Which domain restriction makes $f$ invertible?
17. A student claims the system $$\begin{cases} 2x+3y=7\\ -4x+y=-1 \end{cases}$$ can be written as $$\begin{bmatrix}2&3\\-4&1\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}7\\-1\end{bmatrix}.$$ Which choice correctly states the connection between the system and the matrix equation?
18. A student is verifying whether $m(x) = \sqrt[3]{2x - 5}$ and $n(x) = \frac{x^3 + 5}{2}$ are inverse functions. After computing $m(n(x))$, the student gets $\sqrt[3]{2 \cdot \frac{x^3 + 5}{2} - 5} = \sqrt[3]{x^3 + 5 - 5} = \sqrt[3]{x^3} = x$. What should the student conclude?
19. To compare methods: long division works for any polynomial divisor, but synthetic division is faster when the divisor is $(x - c)$. Divide $P(x)=x^3+x^2-5x+2$ by $(x-1)$ using synthetic division (so $c=1$). Read the quotient coefficients and remainder from the bottom row (remainder $=P(1)$).
20. The complex number $z = -2 - 2\sqrt{3}i$ is written in polar form as $r(\cos\theta + i\sin\theta)$ where $\theta \in [0, 2\pi)$. What is the exact value of $r + \theta$?

Practice tests