Practice Test 15 - Algebra 2

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

A culture of bacteria starts with 300 bacteria and increases by $18\%$ every hour. Create an exponential model for the population $P(t)$ after $t$ hours, then find how many hours it takes for the population to reach 2000. Round to the nearest tenth of an hour and interpret your result in context.
An investment account grows according to $A = 1200e^{0.045t}$ where $A$ is the account value in dollars and $t$ is time in years. To find when the account reaches $2400, a student sets up the equation $1200e^{0.045t} = 2400$ and simplifies to $e^{0.045t} = 2$ . What is the next correct step to solve for $t$ ?
Show that $x=2$ and $x=-3$ are zeros of $f(x)=x^2+x-6$ by factoring $f(x)$ . Which factored form correctly reveals the zeros?
An investment account starts with $\$2{,}500$ and grows by $6\%$ each year. Model the situation with an equation and solve for how many years it will take for the account to reach $\$4{,}000$ . Let $t$ = the number of years.
A population model is $N(t)=A\,b^{ct}$ where $A$ is the initial population, $b>0$ is a growth base, $c$ is a constant, and $t$ is time. By viewing $b^{ct}$ as a single entity, which interpretation correctly identifies the independent factor with respect to $A$ ?
Graph the logarithmic function $r(x)=\log_3(x)+2$ showing the x-intercept and the vertical asymptote. (Relate your sketch to the parent function $y=\log_3(x)$ .)
Use synthetic division (more efficient than long division for $(x-c)$ ) to divide $P(x)=x^3-2x^2-5x+6$ by $(x-3)$ . Use $c=3$ and coefficients $1,-2,-5,6$ . Read the quotient from the bottom row and the remainder from the last entry (equal to $P(3)$ ). Which is correct?
A water tank contains 200 liters at time $t=0$ hours and the amount increases by 10% each hour. Write an exponential function $A(t)$ that models the amount of water (in liters) after $t$ hours.
An investment account starts with $5,000 and loses 15% of its value each month due to poor market conditions. After how many complete months will the account value first drop below $1,500?
On the complex plane, the horizontal axis is the real axis and the vertical axis is the imaginary axis. Which point represents the complex number $z=-3+4i$ ?
In finance, simple interest is modeled by $I=Prt$ , where $I$ is interest, $P$ is principal, $r$ is annual interest rate, and $t$ is time in years. Solve for $r$ in terms of $I$ , $P$ , and $t$ .
A car rental company models the total cost (in dollars) as $C(d)=45+0.30d$ , where $d$ is the number of miles driven. In reality, mileage can be measured continuously, and the rental agreement allows up to 300 miles. Should the graph of $C(d)$ versus $d$ be discrete points or continuous? Why?
Use conjugates to simplify and express in $a+bi$ form: $\frac{7-3i}{2+3i}.$
All points $(x,y)$ that are 4 units from the origin form a circle. What equation models this relationship, and how should the graph be set up so the circle is not distorted?
Function $f$ is given by the equation $f(x)=4\cdot 3^x$ . Function $g$ is described verbally as: "An exponential function with initial value $g(0)=6$ and growth factor $2$ (so each time $x$ increases by 1, the output is multiplied by 2)." Which function has the faster growth rate?
Let $f(x)=x^2$ . The graph of $g$ is shown on the coordinate plane along with $f$ . The function $g$ is of the form $g(x)=f(x+k).$ What is the value of $k$ ? (Use the graph: $f$ has vertex at $(0,0)$ and $g$ has vertex at $(2,0)$ .)
Points on the complex plane are given by $z=a+bi \leftrightarrow (a,b)$ . The midpoint of the segment from $z_1$ to $z_2$ is $\dfrac{z_1+z_2}{2}=\left(\dfrac{a+c}{2}\right)+\left(\dfrac{b+d}{2}\right)i$ , matching the coordinate midpoint formula. What is the midpoint of the segment connecting $z_1=4-i$ and $z_2=2+5i$ ?
When completing the square for $x^2 - 8x + c = 0$ to obtain the form $(x - p)^2 = q$ , the value of $q$ is expressed in terms of $c$ . If the original equation has two real solutions, which condition must be satisfied?
Let $z=-2+3i$ and $w=1+1i$ . What is the distance between them on the complex plane?
For the matrix equation (variable order $x, y, z$ ) $AX=B,\quad A=\begin{bmatrix}1&0&-2\\3&4&0\end{bmatrix},\;X=\begin{bmatrix}x\\y\\z\end{bmatrix},\;B=\begin{bmatrix}7\\-1\end{bmatrix},$ what system of equations does it represent?

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