Practice Test 106 - Algebra 2

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

A small business's weekly online orders are modeled by $O(t)=250(1.04)^t$ , where $t$ is the number of weeks after a marketing campaign starts. Set up and solve an exponential equation to find after how many weeks the weekly orders first reach 400. Let $t$ = number of weeks.
A savings account is modeled by the compound interest formula $A = P(1+r)^n$ , where $P$ is the initial deposit, $r$ is the interest rate per period, and $n$ is the number of periods. If you view $(1+r)^n$ as a single entity (a growth factor), which interpretation best describes the structure of the expression?
A student has a monthly streaming budget and time limit. Let $x$ = hours of Service A watched and $y$ = hours of Service B watched. Service A costs $2 per hour and Service B costs $1 per hour. The student can spend at most $30 and watch at most 25 hours total. The student also wants at least 8 hours of Service A. Which system represents the constraints?
Sketch $f(x)=\frac{x+1}{(x-2)(x+3)}$ showing all vertical asymptotes, the horizontal asymptote, and the zero. Which option correctly identifies these features?
An investment advisor presents two options: Investment X grows according to $f(t) = 8000(0.85)^t$ and Investment Y follows $g(t) = 12000(1.15)^t$ , where $t$ represents years. A client asks about the percent change behavior of these investments. What should the advisor conclude?
A theater sells tickets for $18 each. The theater also charges a flat online processing fee of $4 per order (no matter how many tickets). Write an equation for total cost $T$ (dollars) in terms of number of tickets $n$ (tickets), and choose appropriate axes labels and scales to graph for $0\le n\le 12$ . Which choice is correct?
For the linear function $f(x)= -2x+7$ , show that over equal intervals of length $h=3$ , the change in function value is constant by simplifying $f(x+3)-f(x)$ . Which expression is correct?
Function $f$ is given by $f(x)=\left(\dfrac{1}{2}\right)^x$ . Function $g$ is represented by the table: $\begin{array}{c|cccc} x & 0 & 1 & 2 & 3\\ \hline g(x) & 5 & 10 & 20 & 40 \end{array}$ Which function has the greater output at $x=3$ ?
Distinguish terms (additive parts) from factors (multiplicative parts): In the expression $2x(x - 3)^2 + 5$ , which statement is correct?
The decibel level of a sound can be modeled by $\text{dB}=10\log_{10}(I/I_0)$ , where $I$ is the sound intensity and $I_0$ is a reference intensity. A sound has intensity $500$ times the reference (so $I/I_0=500$ ). Find the decibel level to the nearest tenth and interpret your result.
The inverse of a coefficient matrix is given. Use $X=A^{-1}B$ to solve. $A^{-1}=\frac{1}{5}\begin{bmatrix}4&-1\\-3&2\end{bmatrix},\quad B=\begin{bmatrix}5\\7\end{bmatrix}$ Which is $X=\begin{bmatrix}x\\y\end{bmatrix}$ ?
A company sells a product for $15 each. The cost to produce $q$ items is $C=7q+180$ dollars. 1) Write an equation for profit $P$ (dollars) in terms of $q$ (items). (Profit = revenue − cost.) 2) Choose appropriate axes labels and a reasonable scale to graph $P$ for $0\le q\le 60$ . Which choice is correct?
A system $A\vec{x} = \vec{b}$ has coefficient matrix $A$ with $\det(A) = 24$ . When the first column of $A$ is replaced by $\vec{b}$ , the determinant becomes $72$ . When the second column is replaced by $\vec{b}$ , the determinant becomes $-48$ . If solving using matrix inverses gives the same results as Cramer's rule, what is $y - x$ ?
Use the Rational Zeros Theorem to help factor completely: $P(x)=2x^3-3x^2-8x+12.$ (Find a rational zero by testing candidates, divide to reduce the degree, and continue factoring.)
Find the modulus $r$ and a correct argument $\theta$ (in degrees) for $z=1-i$ , then write $z$ in polar form $r(\cos\theta+i\sin\theta)$ . Use $r=\sqrt{a^2+b^2}$ and $\theta=\arctan(b/a)$ with quadrant adjustment. (Argument is measured counterclockwise from the positive real axis.)
Rewrite $\frac{x^3-5x+2}{x-3}$ in the form $q(x)+\frac{r(x)}{x-3}$ using polynomial division, where $\deg(r)<1$ .
All points $(x,y)$ that are exactly 6 units from the origin form a circle. What equation models this relationship in two variables?
What is the coefficient of $x^4 y^2$ in the expansion of $(x + y)^6$ ? (You may use Pascal's Triangle row 6: $1, 6, 15, 20, 15, 6, 1$ .)
Consider the expression $\frac{(x-3)^4 + 2(x-3)^2 - 8}{(x-3)^2 + 1}$ . Which substitution best reveals the underlying algebraic structure of this rational expression?
A gym membership costs $25 the first month and then increases by $3 each month. Let $a_n$ be the cost (in dollars) for the $n$ th month, with domain $n \in \{1,2,3,\dots\}$ . Model this situation by writing both a recursive formula and an explicit formula for $a_n$ .

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