Practice Test 1 - Algebra 2

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

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1. A projectile's vertical displacement can be modeled by the kinematic equation $s=ut+\tfrac{1}{2}at^2$, where $s$ is displacement, $u$ is initial velocity, $a$ is acceleration, and $t$ is time. Solve $s=ut+\tfrac{1}{2}at^2$ for $t$ (a quadratic in $t$).
2. Factor and graph the polynomial function $p(x)=x^3-x^2-6x$ showing zeros, the $y$-intercept, and end behavior. (A rough sketch is sufficient.)
3. The distance between two complex numbers $z$ and $w$ on the complex plane is $|z-w|$. What is the distance between $z=3+4i$ and $w=1-i$?​
4. Let $f(x)=x^2$ be the parent function. Which description matches the transformations from $f(x)$ to $g(x)=2f(x-1)+3$?
5. The matrix equation $\begin{bmatrix}2&-1\\3&4\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}5\\7\end{bmatrix}$ represents what system of linear equations?
6. Add and simplify to lowest terms. Since rational expressions are closed under addition, the sum is also a rational expression (for allowed $x$ values). Add: $\displaystyle \frac{2}{x-3}+\frac{5}{x+2}$
7. Divide and simplify to lowest terms. (Use "multiply by the reciprocal," like fraction division.) $\frac{x^2-1}{x^2+4x+3}\div\frac{x-1}{x+1}$ (Assume $x \neq -3,-1,1$.)
8. Use polynomial identities over complex numbers to factor $x^4+4$ completely over $\mathbb{C}$. (Hint: $x^4+4=(x^2)^2- (2i)^2$.) Which option gives a correct complete factorization into linear factors?
9. Which function eventually grows fastest as $x\to\infty$? $A(x)=0.01x^{10}$ (polynomial), $B(x)=1.1^x$ (exponential), $C(x)=500x$ (linear), $D(x)=3x^2$ (quadratic).
10. Rewrite the exponential equation $10^3=1000$ in logarithmic form using $b^y=x \Leftrightarrow \log_b(x)=y$.
11. A rectangular patio has an area of 96 square feet. Its length is 4 feet longer than its width. Write and solve an equation to find the patio's width. Let $w$ = the width (in feet).​
12. Function $f$ is described verbally as: "An exponential function with initial value $5$ and growth factor $1.5$." Function $g$ is given algebraically by $g(x)=4\cdot 2^x$. Which function has the larger $y$-intercept?
13. A streaming service has 5000 subscribers and grows by 12% each month. Model the number of subscribers after $t$ months and solve for when it will reach 10,000 subscribers. Let $t$ = number of months.
14. Function $f$ is given algebraically by $f(x)=-2x^2+8x-1$. Function $g$ is described verbally as: "$g$ is a downward-opening parabola with vertex at $(1,6)$." Which function has the larger maximum value?
15. Let $z_1 = 5 + 3i$ and $z_2 = 2 - 7i$, where $i^2 = -1$. What is $z_1 + z_2$ in standard form $a + bi$?
16. A quadratic equation has the form $ax^2+bx+c=0$ with real coefficients. In $\mathbb{C}$, every quadratic has exactly 2 solutions counting multiplicity (two distinct real, one repeated real with multiplicity 2, or two complex conjugates). For $x^2+4x+4=0$, use the discriminant $b^2-4ac$ to predict the solution type and state the solutions counting multiplicity. Which statement is correct?
17. Consider the expression $x^2(x+1) - 4(x+1).$ By viewing $(x+1)$ as a single factor (a chunk), which statement correctly describes the structure you can see?
18. A student claims that $h(x) = \sqrt{x + 4}$ and $k(x) = x^2 - 4$ are inverse functions because $h(k(5)) = h(21) = 5$ and $k(h(5)) = k(3) = 5$. Which statement best describes the error in this reasoning?
19. Express $\frac{2x^4-3x^3+x^2+4}{x^2+2}$ as $q(x)+\frac{r(x)}{x^2+2}$ using polynomial division, where $\deg(r)<2$.
20. A town's population is modeled by $P(t)=18{,}000(1.02)^t$, where $P$ is the population and $t$ is time in years. In the function, interpret the parameter $1.02$.

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