Practice Test 102 - Algebra 2

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

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1. A landscaper's total charge $T$ (dollars) depends on the number of labor hours $h$ and the number of shrubs planted $s$. The landscaper charges $30 per hour and $18 per shrub, plus a fixed trip fee of $25. Create an equation representing this relationship.
2. Find and verify the inverse of $f(x)=\dfrac{2x+3}{5}$ by using the idea that $f\big(f^{-1}(x)\big)=x$.
3. Which method is most efficient for solving $(2x + 1)^2 = 49$?
4. Write an expression for $f^{-1}(x)$ given $f(x)=x^3+5$.
5. Use polynomial division to write $\frac{x^4+1}{x^2+1}$ in the form $q(x)+\frac{r(x)}{x^2+1}$, where $\deg(r)<\deg(x^2+1)$.
6. What is the coefficient of $x^2y^4$ in the expansion of $(x+y)^6?$
7. Function $f$ is given algebraically by $f(x)=\dfrac{x+2}{x-1}$. Function $g$ is described verbally as: "$g$ is a rational function with a vertical asymptote at $x=-3$." Which function has the vertical asymptote closer to the origin?
8. On the complex plane, $z = 6 - 8i$ corresponds to the point $(6,-8)$. Using $|a+bi|=\sqrt{a^2+b^2}$, what is $|z|$?​
9. A town has a population of 48,000 and decreases by $2.5\%$ each year. Create an exponential model for the population $P(t)$ after $t$ years, then determine how many years it will take for the population to drop below 40,000. Round to the nearest tenth of a year and interpret your result in context.
10. A town's population is 48,000 and decreases by $1.6\%$ each year. Create an exponential model for the population $P(t)$ after $t$ years, then solve for when the population will fall to 40,000. Round to the nearest tenth of a year and interpret your result.
11. A student writes $AX=B$ for a system and claims the solution is found by multiplying both sides on the left by $A^{-1}$. Which expression correctly gives the solution vector (assuming $A$ is invertible and $AA^{-1}=I$)?
12. Consider the recursively defined sequence (a discrete function with domain $n\in\{1,2,3,\dots\}$): $a_1=2$ and $a_{n+1}=a_n^2-1$ for $n\ge 1$. What is $a_4$?
13. A company models the number of active users (in thousands) by $U(x)=x^3-4x$, where $x$ is years since launch. Does $U$ have symmetry? What type?
14. A car rental company charges a daily rate plus a fixed fee. The total cost (in dollars) for renting a car for $d$ days is $C=45d+30$. In this function, what does the slope $45$ represent in context?
15. Recognize $x^4-16$ as a difference of squares twice and factor completely using identities. Which answer is correct?​
16. Sketch $f(x)=- (x+3)^2(x-2)(x-1)$ showing all zeros with multiplicities, whether the graph crosses or touches at each, and the end behavior.
17. A rectangle is being designed for a poster. Let $L$ = length (in inches) and $W$ = width (in inches). The perimeter must be at most 40 inches, and the area must be at least 84 square inches. Also $L>0$ and $W>0$. Which system correctly represents these constraints?
18. Solve for $t$ and express the solution as a logarithm, then approximate using a calculator: $5\cdot 2^{3t}=70$.​
19. Use technology to solve the $3\times 3$ system by the inverse-matrix method: write $AX=B$, then compute $X=A^{-1}B$. $\begin{cases} 2x+y-z=1\\ x-y+2z=9\\ 3x+2y+z=0 \end{cases}$ What is $X=\begin{bmatrix}x\\y\\z\end{bmatrix}$?
20. A radioactive substance has a half-life of 8 years. If a sample contains 200 grams initially, which equation models the amount remaining after $t$ years, and what will be the amount after 12 years?

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