Practice Test 92 - Algebra 2

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

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1. All points $(x,y)$ that are $5$ units from the point $(1,2)$ form a circle. Using the distance formula, write the equation of the circle in standard form.
2. Let $z = 3 + 2i$. The complex conjugate of $z$ is $\overline{z}$. Which statement correctly describes the geometric relationship between $z$ and $\overline{z}$ on the complex plane (real axis horizontal, imaginary axis vertical)?
3. Describe the end behavior of the exponential decay function $h(x)=5(0.7)^x$.
4. The sequence $a_n$ is defined recursively by $a_1=1$ and $a_{n+1}=a_n+n$ for $n\ge 1$ (so the input $n$ is an integer term number). What is $a_5$?
5. The drug is considered effective when the concentration is at least 3 mg/L. For how many hours will the drug remain effective?
6. Which expression represents the most useful structural form of $16x^4 - 81y^8$ for finding its factors?
7. Solve the quadratic and express all solutions as complex numbers: $x^2+2x+10=0.$
8. Demonstrate that $f(x)=7-2x$ and $g(x)=\dfrac{7-x}{2}$ are inverse functions by showing both $f(g(x))=x$ and $g(f(x))=x$.​
9. Which method is most efficient for solving the equation $(x-3)^2 = 11$ and then finding the solutions?
10. For $f(x)=(x+5)^2$ with domain restricted to $x\le -5$, what is the inverse function $f^{-1}(x)$?
11. Compare the transformations of the parent function $f(x)=x^2$. Which statement is true about $y=f(2x)$ versus $y=2f(x)$?​
12. A radioactive sample has a half-life of 50 years and starts with 200 g. Create an exponential model for the mass remaining after $t$ years, then solve for how long it takes until only 25 g remain. Round to the nearest year and interpret your result in context.
13. Write the equation for the circle consisting of all points $(x,y)$ that are $6$ units from $(-1,4)$. (Use the Pythagorean Theorem/distance formula.)
14. Consider $G(x)=\frac{x^2+3x-2}{x-1}.$ Without fully simplifying, interpret the expression by viewing the numerator as a single entity. Which statement correctly describes how changing $x$ affects the structure?
15. Rewrite $\dfrac{x^4+1}{x^2+1}$ as $q(x)+\dfrac{r(x)}{x^2+1}$, where $\deg(r)<2$.
16. On the complex plane, addition corresponds to vector addition. If $z = 2 + i$ and $w = -1 + 3i$, what point represents $z+w$?
17. Factor $x^2+4$ completely over the complex numbers by rewriting it as a difference of squares. Which factorization is correct?
18. A bike-share company charges a flat unlock fee plus a per-minute rate. The total cost (in dollars) after $t$ minutes is modeled by $C(t)=1.50+0.25t$. The company allows rentals from 0 minutes up to 180 minutes, and time can be measured to the nearest second. What is an appropriate realistic domain for $C(t)$ in this context?
19. Add and write the result in standard form (sum of polynomials is a polynomial by closure): $(3x^3 + 2x^2 - x + 5) + (x^3 - 4x^2 + 3x - 2)$
20. For $f(x)=(x-1)^2$ with the restricted domain $x\ge 1$ (so the graph passes the horizontal line test on that domain), what is $f^{-1}(x)$?​

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