Practice Test 73 - Algebra 2

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

Let $z=4-3i$ . The complex conjugate $\overline{z}$ is obtained by reflecting the point for $z$ across the real axis. Which statement correctly describes $\overline{z}$ and its point on the complex plane?
The equation $x^2 + y^2 + 4x - 10y + 13 = 0$ can be rewritten in the form $(x - h)^2 + (y - k)^2 = r^2$ . What is the value of $h + k + r$ ?
A population study tracks two cities over time. City A has a population that can be modeled by $P_A(t) = 50000 \cdot 2^{t/10}$ , where $t$ is years since 2020. City B's population follows $P_B(t) = 45000(1.071)^t$ . After analyzing both models, which conclusion is most accurate?
Use the logarithm properties $\log_b(xy)=\log_b(x)+\log_b(y),\quad \log_b\left(\frac{x}{y}\right)=\log_b(x)-\log_b(y),\quad \log_b(x^p)=p\log_b(x)$ to simplify the expression: $\frac{\log_5(125x)}{\log_5(5)}.$
The polynomial identity $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$ can be extended to complex numbers. Using this identity, what is the complete factorization of $x^3 - 27i$ over the complex numbers?
A student council is assembling care kits using granola bars and juice boxes. Let $x$ = number of granola bars and $y$ = number of juice boxes. Constraints: - Granola bars cost $1 each and juice boxes cost $2 each; budget is at most $50: $x+2y\le 50$ . - They need at least 30 total items: $x+y\ge 30$ . - They must include at least 12 juice boxes: $y\ge 12$ . - Items must be whole numbers and non-negative. Which of the following is nonviable?
If $\log_3(2x + 1) + \log_3(x - 2) = \log_3(13)$ , which expression is equivalent to the left side of the equation after applying logarithm properties?
Consider the expression $K(x)=\frac{4}{x+2}\,(x^2-1).$ If you chunk the factors as $\left(\frac{4}{x+2}\right)$ and $(x^2-1)$ , which interpretation is correct?
Graph the trigonometric function $f(x)=2\cos(x)+3$ showing its amplitude, period, and midline.
If $p(x) = 2x^3 - 5x^2 + 3x - 7$ and $p(2) = -5$ , what can be concluded about the polynomial $q(x) = p(x) + 5$ ?
Which structural form of $6x^2 + 17x + 12$ most clearly shows why it can be factored as $(2x + 3)(3x + 4)$ ?
The geometric sequence is $3,\ 12,\ 48,\ 192,\dots$ where the first term corresponds to $x=1$ . Write an exponential function $f(x)$ that gives the $x$ th term.
Determine whether $f(x)=\dfrac{2x-1}{5}$ and $g(x)=\dfrac{5x+1}{2}$ are inverses by verifying both compositions $f(g(x))$ and $g(f(x))$ .
A bacteria culture doubles every hour. Its size after $t$ hours is proportional to $2^t$ . Rewrite $2^t$ to show an equivalent expression in terms of a half-hour growth factor applied over $2t$ half-hours.
A bacteria culture starts with 200 bacteria and grows by 10% each hour. Let $t$ be the number of hours. Which exponential function models the population $P(t)$ ?
Write both recursive and explicit formulas for the sequence $5,\ 15,\ 45,\ 135,\ \dots$ (domain: positive integers).
Use technology to solve the system using the inverse-matrix method. Write $AX=B$ and compute $X=A^{-1}B$ . $\begin{cases} x+y+z=6 \\ 2x-y+3z=9 \\ -x+4y+z=3 \end{cases}$ What is $X=\begin{bmatrix} x \\ y \\ z \end{bmatrix}$ ?
On the complex plane (real axis horizontal, imaginary axis vertical), the point corresponding to $z=2-3i$ has coordinates $(\text{Re}(z),\text{Im}(z))$ . Which ordered pair matches this point?
A 500 mg dose of a medication decreases by 12% each hour. Create an exponential decay model for the amount remaining after $t$ hours, then determine how long it takes for the amount to drop to 200 mg. Round to the nearest tenth of an hour and interpret your answer.
Find the inverse function $f^{-1}(x)$ for $f(x)=\dfrac{x-4}{3}$ .

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