Practice Test 39 - Algebra 2

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

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1. A savings account has an annual growth factor of $1.06$. Which is the approximate monthly growth factor? (Use $1.06^{1/12} \approx 1.0049$.)​​
2. For the linear function $f(x)= -4x+9$, analyze growth over equal intervals of length $h=2$. What is $f(x+2)-f(x)$, and what does it show?
3. A one-to-one function $f$ is shown by the table, so $f$ has an inverse. Use the table to find $f^{-1}(12)$. (Look for the output 12 in the $f(x)$ column; the corresponding $x$ is the inverse value.)

   $x$	$f(x)$
   0	1
   3	6
   5	10
   6	12
   8	16

4. A medication dose starts at 120 mg in the bloodstream. Each hour, $12\%$ of the amount is eliminated. Write an exponential decay model for the amount remaining after $t$ hours and find how long it takes for the amount to drop to 40 mg. Round to the nearest tenth of an hour.
5. If $r(x) = x^3 - 7x^2 + 14x - 8$ and it is known that $x = 1$ is a zero, which statement correctly describes the complete factorization and the resulting graph behavior?
6. Use a calculator to solve the exponential equation $6\cdot 2^{0.25t}=90$ for $t$. Give the exact solution as a logarithm and an approximate value.
7. From the graph, determine if the rate of change is constant. The curve passes through $(0,0)$, $(1,1)$, $(2,4)$, and $(3,9)$ and is smoothly curved upward (not a straight line).
8. Use $i^2 = -1$ to simplify the product $(3 + 2i)(1 - 4i)$ and write the result in the form $a + bi$.
9. The system $\begin{cases} x^2 + y^2 = r^2 \\ y = x + c \end{cases}$ represents a circle and a line. If the system has no real solutions, which of the following must be true about the relationship between $r$ and $c$?
10. For the functions $f(x)=x^3$ (polynomial) and $g(x)=3^x$ (exponential), which statement is correct about when $g(x)$ exceeds $f(x)$?
11. Are the functions $f(x)=\dfrac{x-1}{x+2}$ and $g(x)=\dfrac{1+2x}{1-x}$ inverses? Verify by computing and simplifying both $f(g(x))$ and $g(f(x))$ (assume $x$ is in the domain of each composition).​
12. What is $f^{-1}(x)$ if $f(x)=3x-7$?
13. Factor $p(x)=x^2-7x+10$ and use the factors to determine where $p(x)=0$. What are the zeros?
14. Use Pascal's Triangle (row 3: $1,3,3,1$) to expand $(2x-3)^3.$
15. Compute the finite geometric sum $S_{10}=7+7\left(\frac12\right)+7\left(\frac12\right)^2+\cdots+7\left(\frac12\right)^9.$​
16. After restricting $f(x)=x^2-4x+3$ to the domain $x\ge 2$ so it passes the horizontal line test, find $f^{-1}(x)$.
17. A trigonometric function has the equation $g(x) = -2\sin(3x + \pi) + 1$. What are the amplitude, period, and midline of this function?
18. A rectangular garden has length $(x+5)$ meters and width $(x-2)$ meters. Let $A$ be the area in square meters. Create an equation for $A$ in terms of $x$, and select a reasonable graph setup (axes labels and scale) for $0\le x\le 10$.
19. Show that any exponential function $g(x)=a\cdot b^x$ (with $a\neq 0$ and $b>0$) grows by equal factors over equal intervals. For a fixed interval length $h$, simplify $\dfrac{g(x+h)}{g(x)}$ and identify what (if anything) depends on $x$.
20. The bond fund follows $B(t) = 5000(1.035)^t$ and the stock fund follows $S(t) = 3500(1.089)^t$, where $t$ is years and values are in dollars. After reviewing the projected values shown in the table, which statement about their growth characteristics is most accurate?

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