Practice Test 101 - Algebra 2

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

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1. Subtract and simplify. Rational expressions behave like rational numbers: find an LCD, subtract the numerators, and simplify. $\displaystyle \frac{x+1}{x^2-4}-\frac{3}{x-2}$
2. Find $x$ in $4\cdot 2^{3x}=70$. Express your answer as a logarithm (base 2) and approximate using technology.
3. Simplify $\log_7(7^x)$ using the inverse relationship between exponentials and logarithms.
4. Expand the expression using logarithm properties: $\ln\left(\frac{(ab^2)^3}{\sqrt{c}}\right)$​
5. Construct a linear function from the arithmetic sequence $4,\ 9,\ 14,\ 19,\ \dots$ where the first term corresponds to $x=1$ (so $f(1)=4$). Write $f(x)$ in the form $f(x)=mx+b$.
6. Function $f$ is given algebraically by $f(x)=\dfrac{x+4}{x-1}$. Function $g$ is described verbally as: "$g$ is a rational function with a vertical asymptote at $x=-3$." Which function has a vertical asymptote farther from the origin on the x-axis?​
7. For the linear function $f(x)=3x+2$, verify using equal intervals of length $1$ that the function grows by equal differences. Compute $f(0),f(1),f(2),f(3)$ and the successive differences, and identify the correct conclusion.
8. Let $z=2+5i$ and $w=1-3i$. Find $\overline{(z+w)}$.
9. Consider $M(x)=7\bigl(2x-3\bigr)^2.$ If you view $(2x-3)$ as a single entity, which statement correctly describes how $M(x)$ changes when only the constant factor $7$ is changed (with $(2x-3)^2$ unchanged)?
10. The scientist models the biomass recovery using $B(t) = 15(1.4)^t$ for the first phase, where $t$ is years after the fire and $B$ is in tons per hectare. Based on this model and the graph shown, what can be concluded about the ecosystem's recovery pattern?
11. Use the inverse relationship to simplify $\log_3(3^8)$.
12. Tom attempts to solve $\begin{cases} x - 3y = 7 \\ 2x + y = 4 \end{cases}$ by elimination. He multiplies the second equation by 3 and adds to get $7x = 19$, so $x = \frac{19}{7}$. What error did Tom make?
13. A function $f(x) = \sqrt{16 - x^2}$ represents the upper semicircle of radius 4 centered at the origin. To create an invertible function that includes the point $(2\sqrt{2}, 2\sqrt{2})$, which domain restriction should be used?
14. A rectangular garden has a fixed perimeter of 40 ft. Let $L$ be the length (ft) and $W$ be the width (ft). Write an equation relating $L$ and $W$, then choose axes labels and a reasonable scale to graph the relationship for $0\le L\le 20$.
15. Find the center and radius of the circle given in standard form: $(x+6)^2+(y-1)^2=49.$
16. Simplify using the inverse property $\log_b(b^x)=x$: $\log_3(3^8)$.
17. Which identity allows factoring the expression $9m^2-25n^2$ directly?
18. In the expression $4x^2 + 3x$, is the number $4$ best described as a term, factor, or coefficient?
19. A town's population is modeled by $P(t)=18{,}000(1.015)^t$, where $t$ is the number of years since 2026. In this model, what does the base $1.015$ represent?
20. If $\log_{10}(x)=2$, find $x$ by converting to exponential form.

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