Practice Test 93 - Algebra 2

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

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1. Factor $h(x)=4x^2-9$ to find its zeros and x-intercepts.
2. In physics, kinetic energy is $KE=\tfrac{1}{2}mv^2$, where $m$ is mass and $v$ is speed. Solve $KE=\tfrac{1}{2}mv^2$ for $v$ (include $\pm$ where appropriate).
3. Two gym membership plans track total cost after $m$ months: Plan P: $25, 50, 75, 100$ (for $m=1,2,3,4$) Plan Q: $25, 27.50, 30.25, 33.275$ (for $m=1,2,3,4$) Which plan shows constant percent change (exponential), and which shows constant additive change (linear)?
4. Apply the Rational Zeros Theorem to help factor completely: $P(x)=2x^3-3x^2-8x+12.$
5. Check if $f(x)=\dfrac{x+3}{2}$ and $g(x)=2x+3$ are inverse functions by computing both $f(g(x))$ and $g(f(x))$ and comparing each to $x$.​
6. Let $G(x)=5(x-2)^2+7(x-2)-3.$ If you treat $u=(x-2)$ as a single unit, which rewrite best highlights the structure of $G$ as a polynomial in $u$?
7. Consider the piecewise function $f(x) = \begin{cases} x^2 & \text{if } x \leq 0 \\ 2x + 3 & \text{if } x > 0 \end{cases}$. To make this function invertible by further restricting the domain, which analysis is correct?
8. Use the geometric series sum formula to find the sum of the first 6 terms of the series $3+6+12+24+48+96.$
9. On a coordinate plane, a straight line passes through the points $(-1,1)$ and $(2,10)$. Construct the linear function $f(x)=mx+b$ that matches the line.
10. Sketch $q(x)=(x+1)(x-2)(x-5)^2$ showing all zeros, the behavior at each zero (cross or touch), and the end behavior.
11. If $z = a + bi$ where $a$ and $b$ are real numbers, and $z^2 = -5 + 12i$, what is the value of $a^2 + b^2$?
12. If $g(x) = f(kx)$ where $k > 0$, and the period of $f(x)$ is 8, what value of $k$ would make the period of $g(x)$ equal to 2?
13. A plumber can drain a tank in 5 hours. A leak refills the tank and would fill it in 20 hours if the drain were closed. If the drain is open and the leak continues, set up a rational equation and solve for the net time to empty the tank. Let $t$ = time in hours to empty the tank with both processes happening.
14. A car rental company charges a flat fee of $30 plus $0.20 per mile. You have $80 to spend. Create an inequality for the total cost and find the maximum number of miles you can drive. Let $m$ = number of miles driven.
15. A square garden has area $A(s)=s^2$, where $s$ is the side length in meters. Side length can be measured to any precision, but it cannot be negative. Compare the mathematical domain of $A(s)=s^2$ with the realistic domain in this context.
16. Function $f$ is shown on the coordinate plane as a line passing through the points $(-2,-1)$ and $(2,7)$. Function $g$ is given algebraically by $g(x)=3x+1$. Which function has the greater rate of change (slope)?
17. Use log properties to expand the expression: $\log_2\left(\frac{(xy^2)^3}{\sqrt{z}}\right).$
18. A population model is $N(t)=A\,b^{ct}.$ If you treat $b^{ct}$ as one entity, which interpretation correctly identifies what depends on $t$ and what does not?
19. Every quadratic equation $ax^2+bx+c=0$ has exactly 2 complex solutions counting multiplicity (2 distinct real if $b^2-4ac>0$, 1 repeated real if $b^2-4ac=0$, or 2 complex conjugates if $b^2-4ac<0$). For the equation $(x-3)^2=0$, how many solutions are there in the complex numbers counting multiplicity?
20. Why do the $x$-coordinates of intersection points of the graphs $y=f(x)$ and $y=g(x)$ solve the equation $f(x)=g(x)$?​

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