Practice Test 38 - Algebra 2

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

The system $\begin{cases} ax + 2y = 6 \\ 3x + by = 9 \end{cases}$ has solution $(2, 1)$ . If the system $\begin{cases} ax + 2y = 12 \\ 3x + by = 18 \end{cases}$ is formed by doubling the right sides, what is the solution to this new system?
A car rental company charges $45 per day plus a one-time fee of $30. You have at most $300 to spend. Create an inequality and find the maximum whole number of days you can rent the car. Let $d$ = number of days.
Order the following functions by eventual growth rate (from greatest to least) as $x\to\infty$ : $\;f(x)=0.1\cdot 3^x$ (exponential), $\;g(x)=x^6$ (polynomial), $\;h(x)=50x^2$ (quadratic), $\;p(x)=100x$ (linear).
Factor $s(x)=4x^2-9$ to reveal its zeros. Then identify the x-intercepts of the graph of $y=s(x)$ .
Determine whether the system has no solution, one solution, or infinitely many solutions: $\begin{cases} 2x + y = 6\\ 4x + 2y = 12 \end{cases}$
Sketch $f(x)=\frac{x+2}{x-1}$ showing all asymptotes and intercepts. Which set of features is correct?
Simplify. This uses multiplication of rational expressions (factor, cancel, multiply), demonstrating closure under multiplication. $\displaystyle \frac{x^2+5x+6}{x^2-9}\cdot\frac{x-3}{x+2}$
What is the equation of a circle centered at $(-2,5)$ with radius $7$ ? (Use the distance formula idea that every point on the circle is $7$ units from the center.)
A science sensor records the temperature change according to $T(t)=18+2.5t$ , where $t$ is time in minutes since the start of an experiment. The experiment runs from $t=0$ to $t=40$ minutes. What is the realistic domain for $t$ ?
The complex number $w = -2 + 2\sqrt{3}i$ has modulus $4$ and argument $120°$ . What is the result of $w^3$ expressed in rectangular form?
Add the polynomials (combine like terms). The sum should be a polynomial by closure under addition: $(4x^3-2x^2+6)+( -x^3+5x-9).$
Use logarithms to solve for $x$ : $2\cdot 10^{x/3}=19$ . Express the solution as a logarithm and approximate using a calculator.
Graph $p(x)=2(x-1)^2(x+2)^3$ showing zeros with multiplicities, whether the graph crosses or touches at each zero, and the correct end behavior. (Rough sketch.)
Divide $x^3-1$ by $x-1$ and express $\frac{x^3-1}{x-1}$ in the form $q(x)+\frac{r}{x-1}$ , where $\deg(r)<\deg(x-1)$ .
Rewrite the exponential equation $10^3 = 1000$ in logarithmic form (use the common logarithm base $10$ ).
Given the graph of $f(x)=x^2$ and the graph of $g(x)=f(x)+k$ shown, the vertex of $g$ is at $(0,-3)$ . What is $k$ ?
A lab culture starts with 50 cells and grows by 20% each day. Create an exponential model for the number of cells after $t$ days, then solve for how many days it takes to reach 500 cells. Round to the nearest tenth of a day and interpret your result in context.
Which statement about long-term growth is true for the functions $f(x)=100x^2$ (quadratic) and $g(x)=1.2^x$ (exponential)?
A town has an initial population of 12,000 and grows by 3% each year. Create an exponential model for the population after $t$ years, then determine how many years it will take for the population to reach 15,000. Round to the nearest tenth of a year and interpret your result in context.
After restricting $f(x)=(x-4)^2$ to the domain $x\le 4$ so that it is one-to-one, what is the inverse function $f^{-1}(x)$ ?

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