Practice Test 12 - Algebra 2

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

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1. The formula for the period of a pendulum is $T = 2\pi\sqrt{\frac{L}{g}}$, where $T$ is period, $L$ is length, and $g$ is gravitational acceleration. An engineer needs to determine what length produces a specific period. Which expression correctly gives $L$?
2. Consider the quadratic equation $ax^2 + bx + c = 0$ where $a \neq 0$. When this equation is transformed by completing the square to the form $a(x - p)^2 = q$, which expression correctly represents $q$ in terms of $a$, $b$, and $c$?
3. Two polynomial functions and one exponential function are shown: - Quadratic: $g(x)=2x^2$ - Polynomial (degree 4): $h(x)=0.01x^4$ - Exponential: $p(x)=2^x$ Which statement about their long-term behavior is correct?
4. Consider the nested expression $H(x)=2(3x+1)^3.$ If you treat $(3x+1)^3$ as a single entity, which statement correctly describes the role of the outer factor $2$?
5. Use the table to approximate the solution to $f(x)=g(x)$, where $f(x)=e^x$ and $g(x)=4x$. Based on the values below, in which interval does the solution lie? Values: - At $x=2.1$: $f(x)\approx 8.17$, $g(x)=8.40$ - At $x=2.2$: $f(x)\approx 9.03$, $g(x)=8.80$
6. Two payment plans are described below. Plan 1: You pay a $10 sign-up fee plus $5 per month. Plan 2: You pay $5 the first month, $10 the second month, $20 the third month, and $40 the fourth month. Which plan has a constant rate of change in total cost with respect to months?
7. A gym charges a one-time signup fee of $60 plus $18 per month. You have at most $240 to spend. Create an inequality for this constraint and find the maximum number of whole months you can afford. Let $m$ = number of months.
8. On the complex plane, addition corresponds to vector addition. If $z_1=2+i$ and $z_2=1+3i$, what point represents $z_1+z_2$?
9. Convert between exponential and logarithmic forms. Which logarithmic equation is equivalent to $2^6=64$?​
10. Use the logarithm properties $$\log_b(xy)=\log_b(x)+\log_b(y),\quad \log_b\left(\frac{x}{y}\right)=\log_b(x)-\log_b(y),\quad \log_b(x^p)=p\log_b(x)$$ to evaluate without a calculator: $\log_3\left(\frac{81}{9}\right).$
11. In chemistry, the ideal gas law is $PV=nRT$, where $P$ is pressure, $V$ is volume, $n$ is moles, $R$ is a constant, and $T$ is temperature. Solve for $T$ in terms of $P$, $V$, $n$, and $R$.
12. Apply the Rational Zeros Theorem to help factor completely: $P(x)=2x^3-9x^2+7x+6.$ (Find a rational zero, factor it out, then factor the remaining quadratic.)
13. Solve using completing the square: $x^2+2x+5=0.$ (Your completed-square form should look like $(x-p)^2=q$.)
14. Consider polynomials of the form $g(x) = ax^3 + bx^2 + cx + 12$ where $a$ is a positive integer. For which value of $a$ would the Rational Zero Theorem yield the fewest possible rational zero candidates?
15. A bacteria culture grows according to the pattern shown in the coordinate plane. During which time period does the bacteria population demonstrate a constant rate of change?
16. Identify the amplitude, period, and midline of the trigonometric function $f(x)=-3\sin(2x)+1$.
17. Consider the expression $E = x^2(x+1)-4(x+1).$ Chunk by viewing $(x+1)$ as a single entity. Which statement best describes the structure revealed by this chunking?
18. Let $f(x)=mx+b$ with $m\neq 0$. Consider sampling at integer inputs $x=0,1,2,3,\dots$. Which statement correctly connects the sampled values to a sequence type, and why?
19. A gym charges members according to the linear cost model $C = 12m + 35$, where $C$ is the total cost in dollars and $m$ is the number of months of membership. In this model, what does the parameter $35$ represent in context?
20. The polynomial $h(x) = 2x^3 + bx^2 + cx - 15$ has $\frac{3}{2}$ as one of its zeros. Using the Rational Zero Theorem, which of the following could be another rational zero of $h(x)$?

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