Practice Test 22 - Algebra 1

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

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1. Derive the finite geometric series formula starting from $S_n = a + ar + ar^2 + \cdots + ar^{n-1},$ by multiplying by $r$, subtracting to cancel middle terms, and solving for $S_n$ (assume $r \ne 1$). Which final expression is correct?
2. A city's population grows linearly according to $P_1(t) = 50000 + 2000t$, while a nearby suburb's population grows exponentially according to $P_2(t) = 8000(1.08)^t$, where $t$ is years after 2020. In 2025 ($t = 5$), the city has 60,000 people and the suburb has approximately 11,755 people. The mayor claims that since the city grows by 2000 people per year while the suburb only grows by 8% annually, the city will always have a larger population. Is this reasoning correct?
3. Multiply: $3x(2x^2 - 5x + 4)$
4. A student claims that $\frac{2x^3 + 5x^2 - 3x + 1}{x^2 + 3} = 2x + 5 + \frac{-9x - 14}{x^2 + 3}$. To verify this claim, which calculation would be most efficient?
5. Use the table to determine which statement best describes the eventual behavior of functions $r(x)$, $s(x)$, and $t(x)$ as $x$ increases without bound.
6. A balance increases by 5% annually. Which expression is equivalent to $(1.05)^t$ but written to show a quarterly growth factor?
7. Use polynomial division to rewrite $\frac{2x^3+3x^2-5x+1}{x-2}$ in the form $q(x)+\frac{r(x)}{x-2}$, where $\deg(r)<\deg(x-2)$.​
8. A car's fuel efficiency is measured as 28 miles per gallon. If gasoline costs $3.20 per gallon, what are the appropriate units for the expression $\frac{\text{distance traveled}}{\text{cost of gasoline used}}$?
9. A fitness tracker estimates calories burned during a walk using $C = 60w + 20$, where $C$ is calories and $w$ is the number of miles walked. What do the parameters 60 and 20 represent in this context?
10. Function $f$ is given algebraically by $f(x)=2x-3$. Function $g$ is shown by the table below. Table for $g$: - when $x=0$, $g(x)=1$ - when $x=2$, $g(x)=5$ Which function has the larger y-intercept?
11. A gym charges a monthly membership fee plus a cost per class. The total cost $C$ (in dollars) for taking $c$ classes in a month is modeled by $C = 8c + 35$. What does the 35 represent in this context?
12. What is the constant term in $-6x^3+2x-8$?
13. Complete the square to rewrite $f(x)=x^2-6x+11$ in vertex form, then identify the vertex and axis of symmetry.​
14. Factor $2x^2 + 7x + 3$ and use the factors to find the zeros.
15. 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)?​
16. A gym membership costs $50 to join plus $30 per month. After 6 months of membership, the monthly rate decreases to $25. Which function represents the total cost $C(m)$ after $m$ months of membership?
17. Find three points on the graph of $2x+y=6$.
18. A school cafeteria wants a descriptive model of how much food is thrown away each day (not to predict next month). Which set of variables is most appropriate to track and define?
19. Which direction should be shaded for the inequality $y < 3x + 2$ (using the boundary line $y=3x+2$)?
20. A student wants to describe how their study time relates to their quiz results in the past unit (not to predict future scores). Which set of variable definitions best captures this relationship?

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