Practice Test 16 - Algebra 2

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

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1. A laptop loses value each year due to depreciation. Its value is $1200 after 0 years, $1020 after 1 year, $867 after 2 years, and $736.95 after 3 years. From the data, what is the percent rate of change per year, and is it growth or decay?
2. Order these functions by eventual growth rate (fastest to slowest) as $x\to\infty$: - $f(x)=0.1\cdot 4^x$ (exponential) - $g(x)=x^7$ (polynomial) - $h(x)=500x^2$ (quadratic) - $p(x)=12x$ (linear)
3. A gym membership fee starts at $25 for month 1 and increases by $4 each month. Treat the cost as a function $f(n)$ with domain the positive integers, where $f(n)$ is the cost in month $n$. Model this situation with both a recursive and an explicit formula for $f(n)$.
4. Let $z = 2 + 5i$. What is the product of $z$ and its conjugate $\overline{z}$? (Recall $(a+bi)(a-bi)=a^2+b^2$.)
5. Solve by completing the square: $x^2 + 6x + 2 = 0$.​
6. A student claims that $f(x)=x^2$ (with domain restricted to $x\ge 0$) and $g(x)=\sqrt{x}$ are inverse functions. Verify or disprove by computing $f(g(x))$ and $g(f(x))$ using the stated domain restriction.​
7. Which example correctly represents the case of a quadratic having exactly 2 solutions in the complex numbers but only 1 distinct solution value because it has multiplicity 2? (Recall: for $ax^2+bx+c=0$, the discriminant $b^2-4ac$ determines whether there are 2 distinct real solutions, 1 repeated real solution counted twice, or 2 complex conjugates.)
8. Find the expansion of $(x+2)^5$ using the Binomial Theorem (or Pascal's Triangle).
9. Complete the square by using structure. Rewrite $x^2+8x+20$ in vertex form. Which rewrite is correct?
10. Use the inverse relationship to simplify: $2^{\log_2(17)}$.
11. A parking garage has two types of spaces: regular spaces that are $8$ feet wide and compact spaces that are $7$ feet wide. The garage has $180$ feet of width available for one row of parking spaces. City regulations require that at least $60\%$ of spaces must be regular spaces. The garage wants to maximize the total number of parking spaces in this row. If $r$ represents regular spaces and $c$ represents compact spaces, which constraint correctly represents the city regulation requirement?
12. A phone battery starts at 100% charge and drains by 12% each hour (so 88% remains each hour). Write and solve an exponential equation to find how many hours until the battery is at 30%. Let $t$ = time in hours.
13. Subtract and simplify to lowest terms. Subtract: $\dfrac{3}{x^2-4}-\dfrac{1}{x-2}$ (assume denominators are nonzero).
14. Two phone plans track the number of users over months. Plan A adds 250 users each month. Plan B increases the number of users by 5% each month. Which statement is correct about the type of change for each plan?
15. Solve the system by graphing. (If you read the intersection from the graph, use the exact ordered pair.) $\begin{cases} y = x + 2 \\ y = -2x + 8 \end{cases}$
16. Divide and express in standard form $a+bi$ by multiplying by the conjugate of the denominator: $\frac{3+2i}{1-i}.$
17. A quadratic with real coefficients has the form $ax^2+bx+c=0$ and always has exactly 2 complex solutions counting multiplicity (2 distinct real, 1 repeated real, or 2 complex conjugates). Suppose a real-coefficient quadratic has one solution $2+3i$. What must the other solution be, and why?
18. A water tank contains 200 liters at time $t=0$ and drains at a constant rate. The volume (in liters) after $t$ minutes is $V(t)=200-5t$. The tank is used only until it reaches empty. What domain makes sense for $t$ in this context?
19. Solve the system. The solution(s) are the intersection point(s) of the line and the parabola. $\begin{cases} y = -x + 2 \\ y = x^2 \end{cases}$
20. Use the definition $\log_b(x)=y \Leftrightarrow b^y=x$ to evaluate $\ln(e^6)$.​

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