Practice Test 21 - Algebra 1

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

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1. A company's profit function is $P(x) = -2x^2 + 40x - 150$ where $x$ is the number of items produced (in hundreds) and $P(x)$ is profit in thousands of dollars. For what range of production levels does the company earn a positive profit?
2. Given the function $f(x)=2x-7$, what is $f(4)$?
3. Is the sequence $9, 3, 1, \tfrac{1}{3}, \dots$ arithmetic, geometric, or neither?
4. Subtract and simplify: $\frac{x}{x-1}-\frac{2}{x-1}$
5. The table shows selected values for a quadratic function $g(x)$. Based on the table, what is the vertex of this function?
6. A medication amount in the bloodstream is measured every hour: $t$ (hours): 0, 1, 2, 3 $A(t)$ (mg): 80, 68, 57.8, 49.13 Is this exponential decay, exponential growth, or neither?
7. Which describes the structure of $2^{x+3}-1$? View $(x+3)$ as a single unit.
8. A polynomial profit model is $P(x)=(x-5)(x-1)(x+2).$ The zeros represent break-even points (where profit is $0$). Identify the break-even x-values and sketch a rough graph showing where $P(x)$ is positive or negative, including end behavior.
9. Is $3\sqrt{7}$ rational or irrational? Choose the option with correct reasoning.
10. View $f(x)=2(x-1)^3+5$ by identifying the main operations and what they operate on. Which description is correct?
11. Rewrite $x^2+12x+36$ using its structure as a perfect square trinomial.
12. A rectangle has length $x$ feet and width $(x-4)$ feet. Let $A$ be the area (in square feet). What equation represents the relationship between $A$ and $x$?
13. Compute the sum of the first $8$ terms of the geometric series $1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots$ using $S_n = \frac{a(1-r^n)}{1-r}$. (Here $a=1$, $r=\tfrac{1}{2}$, $n=8$.)
14. Given $q(x)=x-2x^2$, evaluate $q(2)$.​
15. A ball is thrown upward from a platform 4 feet high with an initial upward velocity of 24 ft/s. Its height (in feet) after $t$ seconds is modeled by a quadratic function with gravity $-16t^2$. What is the formula for the height function $h(t)$?
16. A science lab grows a bacteria culture modeled by $N(t) = 300(1.10)^t$, where $t$ is in hours and $N(t)$ is the number of bacteria. What does 1.10 represent in this context?
17. A phone battery's remaining charge is modeled by $C(t)=100(0.95)^t$, where $t$ is measured in hours. What percent rate of change does this represent per hour?
18. Solve for $x$: $\frac{x}{3} + 2 = 7$
19. A piecewise function models the elevation (in meters) of a hiker along a trail: $E(x)=\begin{cases} 2x & \text{for } 0\le x\le 4,\\ - x+12 & \text{for } 4< x\le 10. \end{cases}$ On what interval is the elevation decreasing?
20. What is the minimum value of the function $g(x)=2x^2+12x+7$? (Use completing the square.)

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