Practice Test 7 - Algebra 1

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

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1. A radioactive substance has the decay function $N(t) = N_0(0.5)^{t/12}$ where $t$ is in years. After how many years will the substance be reduced to 25% of its original amount, and what is the annual decay rate?
2. Which expression is equivalent to $\frac{x^2 + 5x + 6}{x^2 + 3x + 2} \div \frac{x + 3}{x + 1}$?
3. A savings account balance is modeled by $A(t)=1500(1.04)^t$, where $t$ is time in years and $A$ is in dollars. What is the percent growth rate of the account per year?
4. What is the solution set to the equation $y=2x+1$?
5. Verify that the definition $8^{2/3} = (8^{1/3})^2$ is consistent with exponent properties. Which explanation uses the power-of-a-power rule correctly?
6. Multiply: $4x(2x^2 - 3x + 5)$
7. Show that $b^{m/n}=\sqrt[n]{b^m}=(\sqrt[n]{b})^m$ using exponent properties (assume $b>0$ and integers $m\ge 1$, $n\ge 2$). Which reasoning correctly justifies the equal representations?
8. Given $f(x)=3x+2$ and $g(x)=x-4$, find both $f(g(x))$ and $g(f(x))$.
9. Solve using the quadratic formula: $x^2 + 2x + 5 = 0$. Since the discriminant is negative, the solutions are complex. Use $i^2=-1$ and express both solutions in $a+bi$ form.​
10. What is the sum of the first 5 terms of the geometric series $2+6+18+54+162?$ (Identify $a=2$, $r=3$, $n=5$.)
11. What is the common ratio of the geometric sequence $8, -4, 2, -1, \dots$?
12. A medication concentration in the bloodstream decreases according to $C(t) = C_0(0.88)^{t/3}$ where $t$ is in hours. If the initial concentration is 400 mg/L, after how many hours will the concentration drop below 200 mg/L?
13. The expression $(8x^6)^{\frac{2}{3}}$ can be written in the form $ax^b$ where $a$ and $b$ are constants. What are the values of $a$ and $b$?
14. The perimeter of a rectangle is given by $P = 2l + 2w$, where $P$ is perimeter, $l$ is length, and $w$ is width. Solve for $l$ in terms of $P$ and $w$.
15. On a coordinate plane, graph the linear function $y = -x + 2$. Find both intercepts of the function (as points).
16. What is the maximum value of $f(x) = -x^2 + 6x + 1$? (Find it by completing the square.)
17. Function $f$ is described verbally as: "A parabola that opens upward with vertex at $(2,-5)$." Function $g$ is given by $g(x)=(x-2)^2-3$. Which function has the lower minimum value?
18. A library wants to describe how busy it is during open hours on Saturdays. Define 3–5 quantities that would be most useful for a descriptive model of busyness.
19. Which form of the function best shows the vertex of $f(x)=x^2+4x-12$ and what is that vertex? (Use completing the square.)
20. A population of rabbits doubles every 4 months. If there are currently 80 rabbits (at month 0), which function models the population $P(t)$ after $t$ months?

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