Practice Test 44 - Algebra 2

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

Convert the complex number $-2+2i$ from rectangular form $a+bi$ to polar form $r(\cos\theta+i\sin\theta)$ . Use $r=\sqrt{a^2+b^2}$ and $\theta=\arctan\!\left(\frac{b}{a}\right)$ with the correct quadrant adjustment. (Angle in degrees; $\theta$ is the argument measured counterclockwise from the positive real axis.)
The function $f(x)=\sqrt{x-2}$ can be used to model the side length (in meters) of a square when $x$ represents the measured area (in square meters) plus 2. What is the realistic domain for $x$ ?
Condense to a single logarithm: $3\ln(u)+\ln(v)-2\ln(w)$ (Assume $u>0$ , $v>0$ , and $w>0$ .)
Simplify using the inverse relationship between exponentials and logarithms: $\log_3(3^8)$ .
An investment account starts with $1200 and earns $6\%$ interest compounded annually. Model the situation with an exponential equation and solve for how many years $t$ it will take the balance to reach $2000. Let $t$ = time in years.
Using the fact that $(a+bi)(a-bi)=a^2+b^2$ , compute $(2+5i)(2-5i)$ .
The geometric sequence $3,\ 6,\ 12,\ 24,\dots$ defines values of a function at integer inputs with $g(1)=3$ , $g(2)=6$ , $g(3)=12$ , etc. Write an exponential function $g(x)$ that extends this pattern.
A rectangle has an area of 96 cm $^2$ . Its length is 4 cm more than its width. Write and solve a quadratic equation to find the rectangle's width. Let $w$ = the width in cm.
Does the table show a constant rate of change? Use $\Delta y/\Delta x$ for each interval. Table (equal $x$ -intervals of 1): $x$ : 1, 2, 3, 4 $y$ : 2, 6, 12, 20
A community garden is planning crops. Let $c$ = acres of corn and $s$ = acres of soybeans. Constraints: - Total land available: at most 60 acres: $c+s\le 60$ . - Water available: at most 100 acre-ft. Corn uses 2 acre-ft per acre; soybeans use 1.5 acre-ft per acre. - At least 10 acres of corn must be planted. - Nonnegativity: $c\ge 0$ , $s\ge 0$ . Which system correctly represents all constraints?
Identify zeros and their multiplicities for $p(x)=x(x-2)^3(x+1).$ Also state whether the graph crosses or touches the x-axis at each zero and give the end behavior.
Let $f(x)=x^2$ . Compare the transformations $g_1(x)=2f(x)$ and $g_2(x)=f(2x)$ . Which statement is correct?
On the complex plane, the difference $z_1-z_2$ represents the vector from $z_2$ to $z_1$ , and its modulus gives the distance: $|z_1-z_2|=\sqrt{(a-c)^2+(b-d)^2}$ . What is the distance between $z_1=2+3i$ and $z_2=-1+7i$ ?
Given the explicit formula $a_n=9+4(n-1)$ , write an equivalent recursive definition (domain: positive integers).
Consider two investment accounts. Account A starts with $500 and increases by $75 each month. Account B starts with $200 and doubles every 3 months. After how many months will Account B first exceed the value predicted by the linear growth pattern if Account A were to maintain a constant monthly increase?
A car rental company charges a fixed fee plus a per-day fee. The total cost is modeled by $C = 45d + 60$ , where $C$ is in dollars and $d$ is the number of days rented. In the equation $C = 45d + 60$ , what does the 60 mean in context?
Use technology to solve $AX=B$ using the inverse method $X=A^{-1}B$ . $A=\begin{bmatrix}1&1&0\\0&1&1\\1&0&1\end{bmatrix},\quad X=\begin{bmatrix}x\\y\\z\end{bmatrix},\quad B=\begin{bmatrix}3\\5\\4\end{bmatrix}$ What is $X$ ?
A signal is represented by the complex number $z=\sqrt{3}+i$ in rectangular form $a+bi$ . Convert it to polar form $r(\cos\theta+i\sin\theta)$ , where $r=\sqrt{a^2+b^2}$ and $\theta$ is the argument measured counterclockwise from the positive real axis. (Angle in degrees.)
Which rational function has a horizontal asymptote at $y = -2$ and vertical asymptotes at $x = 1$ and $x = -4$ ?
In geometry, the volume of a sphere is $V=\dfrac{4}{3}\pi r^3$ . Solve for the radius $r$ in terms of $V$ .

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