Practice Test 46 - Algebra 1

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

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1. A taxi charges a base fee plus a per-mile fee. The total cost (in dollars) for $x$ miles is $C(x)=3+2x$. What is the average rate of change of $C(x)$ from $x=2$ to $x=7$? Include units.
2. Complete the square for $g(x)=x^2+8x+7$ to find the vertex and the minimum value.​
3. The function $y = 200(1.05)^t$ models the number of bacteria after $t$ hours. What is the constant percent rate of change per hour?
4. A projectile's height (in meters) is modeled as a quadratic term plus a constant initial height. Let $f(t)=-5t^2+20t$ and $g(t)=2$. Write the combined function $h(t)=(f+g)(t)$.​
5. A car's value is $\$18{,}000$ when purchased and then decreases by 20% each year. Which statement correctly describes the type of change?
6. Solve $x^2 + 2x + 5 = 0$. Express complex solutions in $a \pm bi$ form.
7. A bakery tracks the number of cupcakes left at the end of each day for a week. The function $c(d)$ gives the number of cupcakes left after $d$ days, where $d$ counts days starting at $d=1$ for Monday through $d=7$ for Sunday. Should the graph of $c(d)$ be shown as discrete points or a continuous line, and why?
8. A student claims: "Since $\sqrt{2}$ is irrational, $5\sqrt{2}$ must be rational because 5 is rational." Which response correctly evaluates the claim?
9. The profit function for a company is $P(x) = -x^2 + 12x - 32$, where $x$ is the number of items sold (in hundreds). After factoring to find break-even points, what is the maximum profit?
10. A phone plan's total monthly cost is modeled by $C = 15 + 0.10t$, where $C$ is in dollars and $t$ is the number of text messages sent in a month. What does the $15$ represent in this context?
11. In the expression $2x^3 - 5x^2 + 3x - 1$, a student claims that $-5x^2$ has two factors: $-5$ and $x^2$. Another student argues that it has three factors: $-5$, $x$, and $x$. Which analysis is more complete and why?
12. A video game has 500 active players at the start of the month. The number of active players increases by 10% each month. Which statement best describes this change?
13. A savings account balance is modeled by $A(t) = 800(1.05)^t$, where $t$ is the number of years since the account was opened and $A(t)$ is in dollars. What does $1.05$ represent in this context?
14. A sequence $(a_n)$ has domain $n=1,2,3,\dots$ and begins $9, 6, 3, 0, \dots$. Which recursive definition generates this sequence?
15. What is $x^4-16$ when factored completely? (Hint: view it as $(x^2)^2-4^2$.)
16. Show that defining $5^{1/3}$ as $\sqrt[3]{5}$ is consistent with the exponent property $(b^a)^c=b^{ac}$. Which reasoning correctly justifies the definition?
17. What is the structure of the expression $3(x-4)^2-7?$ View $(x-4)$ as a single unit.
18. A movie theater charges a $4.50 service fee plus $11.25 for each ticket purchased. Write an explicit formula for the total cost $C(t)$ (in dollars) in terms of the number of tickets $t$.
19. The zeros of $f(x) = x^2 - 4x - 5$ are needed to find the $x$-intercepts of its graph. What are the zeros?
20. A quartic polynomial is given by $P(x)=(x-2)^2(x+1)(x+3).$ Use zeros and multiplicities to sketch a rough graph. Which description is correct?​

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