Practice Test 107 - Algebra 2

19 of the 20 questions below are in this test; the other one can't be scored yet. Answer them in any order, then submit to see your score and every explanation.

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1. Find the sum of the finite geometric series $3 + 6 + 12 + 24 + 48 + 96$. (This is a geometric series, meaning you are finding the sum of terms.)
2. A bacteria culture starts with 200 bacteria and doubles every 3 hours. Create an exponential model for the population $P(t)$ after $t$ hours, then solve for when the population first reaches 10,000 bacteria. Round to the nearest tenth of an hour and interpret your result.
3. Solve $f(x)=g(x)$ approximately by finding where the graphs intersect: $f(x)=\dfrac{x+1}{x-1}$ and $g(x)=2$. Give the solution(s).
4. For the rational function $f(x)=\frac{x^2+x-6}{x-2},$ what is the correct description of the discontinuity at $x=2$ and the end behavior asymptote?​
5. For the exponential function $g(x)=6\cdot\left(\tfrac{1}{2}\right)^x$, analyze growth over equal intervals of length $h=2$. What is $\dfrac{g(x+2)}{g(x)}$, and what does it show?
6. To compare methods, you could use long division, but synthetic division is more efficient for a divisor of the form $(x-c)$. Use synthetic division to divide $x^3+x^2-5x+2$ by $(x-1).$ What is the quotient and remainder? (Remainder Theorem check: remainder $=P(1)$.)
7. View the common structure by chunking: $x^2(x+1) - 4(x+1).$ Treat $(x+1)$ as a single entity. Which statement best describes the structure revealed by this chunking?
8. For a quadratic $ax^2+bx+c=0$ with real coefficients, the discriminant $b^2-4ac$ determines the type of solutions: $>0$ two distinct real, $=0$ one repeated real (multiplicity 2), $<0$ two complex conjugates. Regardless of type, there are exactly 2 complex solutions counting multiplicity. For $x^2-2x+5=0$, which choice correctly gives the solutions in $\mathbb{C}$?
9. The expression $\frac{e^{2t} - 3e^t + 2}{e^t + 1}$ appears in a differential equation solution. Which substitution most effectively transforms this into a standard rational function form?
10. Use structure to factor by grouping. Which factorization is correct for $6x^2+9x+4x+6$?
11. A company's revenue is $R=15q$ dollars and its cost is $C=8q+200$ dollars, where $q$ is the number of items produced. What equation models profit $P$ (in dollars) in terms of $q$, and what axes labels and a reasonable scale would you use to graph $P$ for $0\le q\le 100$?
12. The concentration of a medication in the bloodstream follows $C(h) = 8(0.7)^h$ mg/L, where $h$ is hours after administration. A new formulation has the model $C(h) = 12(0.7)^h$. How do the parameters compare in terms of their practical effects?
13. Perform the operation and simplify to lowest terms: $\frac{2}{x-3}-\frac{1}{x+3}$ (Assume $x \neq 3, -3$.)
14. The table gives values of a one-to-one function $f$, so $f$ has an inverse. Use the table to determine $f^{-1}(-4)$.
15. The functions $u(x) = 3^x + 1$ and $v(x) = 5 - 2x$ intersect at point Q. If a student uses successive approximations and finds that the x-coordinate of Q is between 1.2 and 1.3, which inequality best represents this approximation process?
16. A coordinate plane shows two functions: - Function 1 is a straight line passing through points $(0,1)$ and $(4,9)$. - Function 2 is a curved graph passing through points $(0,1)$, $(2,5)$, and $(4,17)$. Which statement is true about constant rate of change?
17. Function $e(x) = x^2 - 4x + 7$ and function $f$ is given in the table. Which statement correctly compares the minimum values and their locations for these functions?
18. What is the coefficient of $x^2 y^3$ in the expansion of $(x+y)^5?$ (You may use Pascal's Triangle.)
19. Evaluate $\log_2(32)$ using the inverse relationship $b^y=x \Leftrightarrow \log_b(x)=y$.
20. Let $z_1=-6+2i$ and $z_2=2-4i$ be points on the complex plane. The midpoint is $\dfrac{z_1+z_2}{2}$ (averaging real and imaginary parts, as in coordinate geometry). What is the midpoint?

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