Practice Test 42 - Algebra 1

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

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1. Complete the square to find the vertex of the parabola $f(x) = x^2 - 8x + 3$.
2. The amount of a medicine in the bloodstream is modeled by $M(t) = 60(0.9)^t$, where $t$ is time in hours and $M$ is measured in milligrams. What is the percent decay rate per hour?
3. Prove that a nonzero rational number times an irrational number is irrational. Let $r$ be a nonzero rational number and let $i$ be an irrational number. Which option gives a correct proof idea?
4. A streaming service charges a one-time sign-up fee of $12 and then $6 per month. Write an explicit formula for the total cost $C(m)$ (in dollars) after $m$ months, where $m$ is the number of months.
5. A polynomial $T(x)$ has the property that $T(x) + (2x^3 - x + 4) = 3x^3 + x^2 - 2x + 1$. What is the coefficient of $x$ in $T(x)$?
6. The Fibonacci sequence is defined by $F(1) = 1$, $F(2) = 1$, and $F(n) = F(n-1) + F(n-2)$ for $n \geq 3$. A student claims that $F(n+1) \cdot F(n-1) - F(n)^2$ always equals either 1 or -1 for any $n \geq 2$. To test this claim for $n = 5$, what calculation should be performed?
7. A machine's value is modeled by $V(t)=20000(0.80)^t$, where $t$ is in years. What is the annual depreciation rate (percent decay rate)?
8. A movie theater charges a $5 booking fee plus $12 per ticket. Let $C(n)=5+12n$ be the total cost for $n$ tickets. In this context, what is the domain of $C$?
9. A runner's average speed is computed by $v=\dfrac{d}{t}$. Which expression has correct units for speed if distance is in meters (m) and time is in seconds (sec)?
10. A function $w(d)=4d$ gives the number of widgets produced after $d$ days. A cost function $C(w)=10w+50$ gives the total cost (in dollars) to produce $w$ widgets. What does $C(w(d))$ represent?
11. A company's break-even points occur when its profit is zero. If $P(x)=-x^2+10x-16$, for what values of $x$ is $P(x)=0$?
12. A quantity grows by 5% annually, modeled by $(1.05)^t$ where $t$ is in years. Which expression is equivalent and reveals a quarterly growth factor?
13. An expression is written as $k \cdot 3^x + k \cdot 3^{x+1} + k \cdot 3^{x+2}$ where $k$ is a constant. If this expression is factored completely, what is the coefficient of $3^x$ in the factored form?
14. A model compares two quantities by division. Let $f(x)=x+2$ (linear) and $g(x)=x-3$ (linear). Find the domain of $(f/g)(x)=\dfrac{f(x)}{g(x)}$.
15. Function $f$ is given by $f(x)=x^2-4x+1$. Function $g$ is given in the table. Which function has the smaller minimum value? Table for $g(x)$: - $g(0)=5$ - $g(1)=2$ - $g(2)=1$ - $g(3)=2$ - $g(4)=5$
16. If $a^{2/3} \cdot a^{4/3} = a^2$, which property of exponents is being demonstrated, and why must this property hold for rational exponents?
17. Solve and check for extraneous solutions: $\sqrt{x+7}-2=3$
18. Rewrite $\left(\left(1.15\right)^{1/12}\right)^{12t}$ using exponent properties to express it as a single power of $1.15$.
19. A recipe scale measures flour to the nearest 1 g. You weigh flour and the display shows 253 g. Why is reporting the mass as 253.0 g inappropriate?
20. A stopwatch shows time to the nearest 0.1 second. A runner's time is recorded as 12.7 s. In a report, which value shows appropriate precision for the recorded time?​

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