Practice Test 26 - Algebra 1

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

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1. A teacher wants to describe how much time students spend on homework in a class (not predict future grades). Which variable definitions are most appropriate?
2. A bacteria culture starts with 200 bacteria and grows by 10% each hour. Write a function $P(t)$ for the population after $t$ hours. Write in the form $P(t)=a(1+r)^t$.
3. Solve using substitution: $\begin{cases} y = 2x + 1 \\ x + y = 7 \end{cases}$ Find the solution $ (x, y) $.
4. If $f$ is linear, $f(0)=4$, and $f(3)=13$, what is $f(x)$? Write in the form $f(x)=mx+b$.​
5. A school club is ordering snacks for an event. Let $x$ be the number of boxes of granola bars and $y$ be the number of boxes of fruit snacks. Constraints: - Each granola box costs $6 and each fruit snack box costs $4. - The club can spend at most $60. - They need at least 12 boxes total. - They cannot order a negative number of boxes. Which system of inequalities models this situation?
6. Factor $x^2-10x+25$ by recognizing it as a perfect square trinomial.​
7. If $f(x)=2x+1$ and $g(x)=x-3$, which expression equals $f(g(x))$?
8. A runner's distance from the starting line (in miles) after $t$ hours is given by $d(t)=6t+2$. What is the average rate of change of $d(t)$ from $t=1$ to $t=4$? Give your answer in mi/hr.
9. A population of bacteria grows according to $P(t) = 1000 \cdot 1.5^t$ where $t$ is time in hours. A chemical reaction produces a byproduct at a rate described by $Q(t) = 50t^2 + 100t$ grams after $t$ hours. At $t = 3$$ hours, $P(3) \approx 3375$ bacteria and $Q(3) = 750$ grams. If these trends continue, what will happen over a long time period?
10. A theater is choosing how many adult and student tickets to sell in advance for a small show. Let $x$ = number of adult tickets and $y$ = number of student tickets. Constraints: - The theater has at most 120 seats: $x+y\le 120$. - Adult tickets are $12 and student tickets are $8; advance sales revenue must be at least $1000: $12x+8y\ge 1000$. - At least 30 student tickets must be available: $y\ge 30$. - $x\ge 0$ and $y\ge 0$. Is the point $(70, 40)$ a viable solution?
11. Solve for $y$ in the literal equation $ax + by = c$ (treat $a$, $b$, and $c$ as constants).
12. Solve for $x$: $2(x - 3) + 5 = 15$
13. Refer to the system of inequalities graphed below. A new constraint $y \geq x - 2$ is added to this existing system. How does this change the solution region?
14. A water tank starts with 500 gallons and is drained at a constant rate of 25 gallons per minute. Express the remaining volume $V(t)$ (in gallons) as a function of time $t$ (in minutes).
15. Rewrite $\frac{2x^2+7x+1}{x+3}$ in the form $q(x)+\frac{r(x)}{x+3}$ (quotient plus remainder over divisor), where $\deg(r)<1$.​
16. A water tank is being drained at a constant rate. After 12 minutes, the tank contains 450 gallons. After 20 minutes, it contains 330 gallons. Which equation can be used to find the time $t$ (in minutes) when the tank will be completely empty?
17. Solve and check for extraneous solutions: $\frac{3}{x-1}+2=5.$
18. The quadratic function is $y=x^2+6x+5$. What is the y-intercept of $y=x^2+6x+5$? (Give your answer as a point.)
19. For $f(x) = x^3$, what is $f(-2)$?
20. A town's population is modeled by the exponential function $P(t)=800(1.03)^t$, where $t$ is the number of years since 2020. What is the annual percent growth rate?

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