Practice Test 20 - Algebra 1

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

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1. Explain why the sum of two rational numbers is always rational. Which reasoning is correct? Let $a=\frac{p}{q}$ and $b=\frac{r}{s}$ where $p,q,r,s$ are integers and $q\ne 0$, $s\ne 0$.
2. Graph the quadratic function $y=x^2-2x-3$ on a coordinate plane. What is the vertex?
3. If $f(x)=2x+3$, $g(x)=x^2$, and $h(x)=x-1$, find $f(g(h(x)))$.
4. Write both the explicit formula and the recursive formula for the arithmetic sequence $4,\ 9,\ 14,\ 19,\ \dots$.
5. A movie theater wants a descriptive model of concession sales during the last weekend. Which variable definitions are most appropriate and measurable?
6. You time a runner using a stopwatch that displays tenths of a second. The recorded time is 12.7 s. Later you compute a speed using this time, but first you want to record the time itself. What is an appropriate level of precision for reporting the time?
7. Given $P(x)=-2(x+2)(x-1)(x-3),$ use the zeros to sketch a rough graph. Which statement correctly gives the x-intercepts and end behavior?
8. Simplify the expression $\frac{(3^t)(3^4)}{3^2}$ using properties of exponents.​​
9. A plant's height (in cm) after $t$ weeks is given by $h(t)=t^2+2$. Is the rate of change of height with respect to time constant? (Linear vs. nonlinear)
10. Use the factored form $P(x)=(x-4)(x-1)(x+2)$ to construct a rough sketch. Which choice correctly matches the zeros (x-intercepts) and indicates that the graph crosses at each intercept (no repeated factors)?
11. A streaming service charges $10 per month plus a one-time setup fee of $5. Let $m$ be months and $C$ be total cost (in dollars). What is an appropriate scale to graph this relationship if you want to show from 0 to 6 months?
12. Verify that $(4,1)$ is a solution to $x-y=3$. Substitute: $4-1=3$.
13. A streaming service charges a base fee plus a cost per movie rented. The total cost $C$ (in dollars) for renting $n$ movies is $C = 3n + 12$. What does the parameter $3$ represent in this context?
14. The cost function for producing $x$ widgets is $C(x) = 0.1x^2 + 5x + 100$. A manufacturer increases production from 20 widgets to 40 widgets. What is the average rate of change in cost per widget over this interval?
15. Consider the function expression $g(x)=\big(x^2+1\big)^2.$ What are the main parts of this expression if you view it as an outer operation applied to an inner part?
16. A student claims that since $\sqrt{9} = 3$, then $9^{1/2} = 3$, and since $\sqrt[3]{8} = 2$, then $8^{1/3} = 2$. The student concludes that $9^{1/2} \cdot 8^{1/3} = 3 \cdot 2 = 6$. Which statement best evaluates this reasoning?
17. Consider the piecewise function $f(x) = \begin{cases} 2x + 1 & \text{if } x < 0 \\ x^2 & \text{if } x \geq 0 \end{cases}$. What is $f(-2) + f(2)$?
18. A ball's rebound heights form a geometric sequence: after the first bounce it rises to $10$ ft, and each bounce reaches $80\%$ of the previous height. What is the sum of the heights of the first 5 bounces (in feet)? This is the geometric series $10+10(0.8)+10(0.8)^2+10(0.8)^3+10(0.8)^4$ with $a=10$, $r=0.8$, $n=5$.​
19. Given $f(x) = 7 - 2x$, what is $f(5)$?
20. Consider the quartic polynomial $P(x)=(x-2)(x+2)(x-1)(x+1).$ Identify all zeros and use them to sketch a rough graph, including correct end behavior.

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