Practice Test 17 - Algebra 1

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

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1. A sequence is defined by $h(1) = 4$ and $h(n+1) = \frac{1}{2}h(n) + 6$ for $n \geq 1$. If this pattern continues indefinitely, what value does $h(n)$ approach as $n$ becomes very large?
2. The ideal gas law is $PV = nRT$, where $P$ is pressure, $V$ is volume, $n$ is the number of moles, $R$ is the gas constant, and $T$ is temperature. A chemist needs to find the number of moles when pressure decreases by half and volume doubles, while temperature remains constant. If the original number of moles was $n_0$, what is the new number of moles?
3. If $R(x) = x^3 - 2x + 5$ and $S(x) = 3x^2 + x - 4$, which expression represents $R(x) - S(x)$?
4. A streaming service charges according to $C = 8n + 20$, where $C$ is the total cost (in dollars) and $n$ is the number of months. What does it mean that the y-intercept is 20?
5. The value of a laptop over time is shown by $V(t)=1200(0.90)^t$, where $t$ is the number of years since purchase. Identify the constant percent change per year.
6. How many real solutions (intersection points) does the linear-quadratic system have? $\begin{cases} y = x + 3 \\ y = x^2 + 2x + 5 \end{cases}$
7. Solve for $x$: $\frac{2x - 1}{4} = 3$
8. In the perimeter formula for a rectangle, $P = 2l + 2w$, solve for the length $l$ in terms of $P$ and $w$.
9. A gym charges a membership fee plus a fixed cost per visit. The total cost $C$ (in dollars) after $v$ visits is $C = 25 + 4v$. Is the rate of change of cost with respect to visits constant? If so, what is the rate?
10. A gasoline container holds 3.5 gallons. Convert this to liters. (Use $1\text{ gal}=3.785\text{ L}$.)
11. A taxi charges a $4 starting fee plus $2 per mile. Let $m$ be miles traveled and $C$ be total cost (in dollars). Which equation represents this relationship?
12. A bank account balance is modeled by $A(t)=1000(1.01)^{12t}$, where $t$ is in years. What is the monthly interest rate?
13. Solve by completing the square: $x^2+2x+5=0$. What are the solutions?
14. Solve using elimination: $\begin{cases} 2x + 3y = 12 \\ 2x - y = 4 \end{cases}$ What is the solution $(x, y)$?
15. Consider $g(x)=x^4$ (polynomial) and $h(x)=2^x$ (exponential). Which statement best describes what happens as $x$ gets large?
16. Let $f(x)=x-4$ and $g(x)=x^2$. Evaluate $f(g(-3))$.
17. Verify by multiplying back that the following rewriting is correct. Which expression equals $\frac{x^2 + 6}{x - 1}$ in the form $q(x) + \frac{r(x)}{x - 1}$ with $\deg(r) < 1$?
18. The graph shows the temperature $T$ (in degrees Fahrenheit) as a function of time $h$ (in hours after midnight). Estimate the average rate of change in temperature from 6 AM to 2 PM.
19. An exponential function has initial value $a = 6$ and triples each time $x$ increases by 1. Write the function in the form $y = a \cdot b^x$.​​​
20. For $f(x)=\sqrt{x}$ and $g(x)=x+15$, evaluate $f(g(1))$.​

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