Practice Test 58 - Algebra 2

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

Let $z_1=7-2i$ and $z_2=1-10i$ be points on the complex plane. The distance between them is $|z_1-z_2|=\sqrt{(a-c)^2+(b-d)^2}.$ What is the distance between $z_1$ and $z_2$ ?
The parent function $f(x)=|x|$ is graphed as a V-shape with vertex at $(0,0)$ . Consider $g(x)=f(x+2)-3.$ What is the effect on the graph of $f$ to obtain the graph of $g$ ?
Determine the inverse function for $f(x)=3x-7$ .
Explain why $f(x)=x^2$ needs a domain restriction to have an inverse function. Which statement best connects this to the horizontal line test?
Let $z=2+5i$ and $w=1-3i$ . Using the property $\overline{z+w}=\overline{z}+\overline{w}$ , what is $\overline{z+w}$ ?
The pH of a solution is given by $\text{pH}=-\log_{10}([H^+])$ , where $[H^+]$ is the hydrogen ion concentration in moles per liter. If a solution has pH 3.6, create an equation and solve for $[H^+]$ . Give your answer in scientific notation (to two significant figures) and interpret it in context.
Complete the square to solve: $x^2+6x-2=0$ Write the completed-square form $(x-p)^2=q$ and then give the solutions.
A bacteria culture starts with 500 bacteria and doubles every 3 hours. After how many hours will the population first exceed 10,000 bacteria?
Consider the functions $f(x)=5x$ (linear), $g(x)=x^3$ (cubic polynomial), $h(x)=x^5$ (5th-degree polynomial), and $p(x)=2^x$ (exponential). Order these functions by eventual growth rate (for sufficiently large $x$ ).
An investment grows according to $A(t) = 5000(1.06)^t$ dollars after $t$ years. How long will it take for the investment to triple in value?
The function $f$ is one-to-one, so it has an inverse. A point on the graph of $f$ is $(7,-3)$ . Using the fact that $(a,b)$ on $f$ corresponds to $(b,a)$ on $f^{-1}$ , what is $f^{-1}(-3)$ ?
Use polynomial division to express $\frac{2x^3-x^2+0x+3}{x^2+1}$ in the form $q(x)+\frac{r(x)}{x^2+1}$ , where $\deg(r)<2$ .
Possible rational zeros of $P(x)=x^3-4x^2+x+6$ (from the Rational Zeros Theorem) are $\pm 1,\pm 2,\pm 3,\pm 6$ . Which of these candidates are actual zeros? (Test by substitution or synthetic division.)
Simplify using logarithm properties: $\log_7(49x)-\log_7(7)$ (Assume $x>0$ .)
Which statement correctly applies the Rational Zeros Theorem to $P(x)=6x^3-x^2+4x-5?$
Multiply and simplify using $i^2 = -1$ : $(3 + 2i)(1 - 4i)$ . Write your answer in $a + bi$ form.
Use polynomial long division to write $\frac{x^4 - 3x^2 + 5}{x^2 + 1}$ in the form $q(x) + \frac{r(x)}{x^2+1}$ , where $\deg(r) < 2$ .
A phone battery starts at 100% and loses 9% of its remaining charge each hour while streaming video. Create an exponential model for the battery percentage after $t$ hours, then determine how long until the battery reaches 20%. Round to the nearest tenth of an hour and interpret your answer.
A bike rental shop charges a one-time service fee of $12 plus $5 per hour. 1) Write an equation for the total cost $C$ (dollars) in terms of hours rented $h$ (hours). 2) Then choose the best axes labels and reasonable scales to graph the relationship for $0\le h\le 10$ . Which choice gives both the correct equation and an appropriate graph setup?
A bacteria culture starts with 200 cells and grows by 10% each hour. Let $t$ be the number of hours. What function models the population $P(t)$ ?

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