Practice Test 14 - Algebra 1

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

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1. View the expression $A=\pi r^2+2\pi rh$ by identifying a common factor involving $r$. Which describes the structure of $A$ in terms of factors involving $r$ (without fully simplifying)?
2. A runner completes $3\ \text{miles}$ in $24\ \text{min}$. Find the runner's average speed in miles per hour (mi/hr), showing unit conversion. (Use $60\ \text{min}=1\ \text{hr}$.)
3. A function $f$ with domain $n=0,1,2,3,\dots$ is defined recursively by $f(0)=3$ and $f(n+1)=f(n)+5$. What is $f(3)$?
4. A phone plan charges a $40 monthly fee plus $0.10 per text message. The function $C(n)=40+0.10n$ gives the monthly cost (in dollars) for $n$ text messages. Which values of $n$ are realistic for this situation?
5. In the expression $3a(b+4)^2,$ interpret it as the product of $a$ and a factor not depending on $a$. Which factor does not depend on $a$?
6. A town's population is modeled by $P(t) = 25{,}000(1.02)^t$, where $t$ is the number of years since 2026. What is the initial value of the population in this model?
7. A taxi charges a flat fee of $4 plus $2 for each mile traveled. Does this situation involve a constant rate of change, and if so what is the rate (in $$/mile)?
8. Use factoring to analyze $h(x)=2x^2-10x+12$. Identify the zeros (x-intercepts) and the axis of symmetry.
9. A school fundraiser sells tickets for $15 each. The venue charges a fixed rental fee of $300 plus $8 per person attending. If $x$ represents the number of tickets sold, which system correctly models the constraints for the fundraiser to break even or make a profit?
10. Describe the end behavior of the function $f(x)=-3x^3+2x$.
11. A student claims: "$16^{1/4}=2$." Which reasoning best supports this claim using exponent properties (not just computation), assuming we want $(b^a)^c=b^{ac}$ to extend to rational exponents?
12. At what $x$-value does $f(x)=x^2-12x+5$ reach its minimum? (Find this by completing the square.)
13. Solve the linear-quadratic system algebraically (substitution method). The solution(s) represent the intersection point(s) of the line and the parabola: $\begin{cases} y = x + 1 \\ y = x^2 - 2x + 1 \end{cases}$
14. Verify whether the given point is a solution to the system by checking both equations: $\begin{cases} y = x^2 - 2x \\ y = 2x \end{cases}$ Is $(4,8)$ a solution?​
15. Given the polynomial in factored form $P(x)=(x-1)(x+3)(x-4)$, identify the zeros and use them to sketch a rough graph. Which option correctly lists the x-intercepts and the end behavior (left/right) of the graph?
16. A school cafeteria wants a descriptive model of how much food is wasted during lunch each day (to summarize what is currently happening, not to predict future waste). Which set of variables is most appropriate to track and define?
17. Which reasoning correctly explains why the sum of two rational numbers is always rational? (Recall: a rational number can be written as $\frac{p}{q}$ where $p,q$ are integers and $q\ne 0$.)
18. A theater is selling two ticket types. Let $x$ be the number of adult tickets and $y$ be the number of student tickets. Constraints: - Adult tickets cost $12 and student tickets cost $8. - The theater wants revenue of at least $400: $12x+8y\ge 400$. - They can sell at most 45 tickets total: $x+y\le 45$. - $x\ge 0$ and $y\ge 0$. Which system of inequalities matches these constraints?
19. A company's weekly profit (in thousands of dollars) depends on the number of ads $x$ it runs and is modeled by $P(x)=-2(x-3)^2+18$, for $0\le x\le 6$. On what interval is $P(x)$ increasing?
20. Is point $(2,1)$ in the solution region of the inequality $y < x - 2$?

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