Q: A quadratic equation models the height $h$ (in meters) of a projectile: $h = -4.9t^2 + 20t + 2$. At what time $t$ does the projectile reach its maximum height?

$t = \frac{-b}{2a} = \frac{-20}{2 \times -4.9} = 2.04$ seconds

Q: A company’s revenue $R$ is modeled by $R(x) = 2x^3 - 5x^2 + 4x - 7$, where $x$ is the number of units sold. What is the revenue when 3 units are sold?

$R(3) = 2 \times 27 - 5 \times 9 + 4 \times 3 - 7 = 54 - 45 + 12 - 7 = 14$

Question 1

A biologist is studying the growth of a bacterial colony. The population doubles every 3 hours. If the initial population is 500 bacteria, what will the population be after 12 hours? Express your answer in scientific notation.

Accepted Answer

$8.0 	imes 10^3$

Question 2

A chemical reaction produces a compound whose mass is given by $m = 4x^2 - 9$, where $x$ is the number of moles of reactant. What is the mass when $x = 3$?

Accepted Answer

$27$

Question 3

A medical researcher needs to dilute a solution. The concentration $C$ after dilution is given by $C = \frac{C_0}{1 + r}$, where $C_0$ is the initial concentration and $r$ is the ratio of added solvent to original solution. If $C_0 = 100$ mg/mL and $r = 4$, what is the final concentration?

Accepted Answer

$20$ mg/mL

Question 4

Which of the following is the correct simplification of $\sqrt{50} + \sqrt{8}$?

Accepted Answer

$5\sqrt{2} + 2\sqrt{2} = 7\sqrt{2}$

Question 5

A quadratic equation models the height $h$ (in meters) of a projectile: $h = -4.9t^2 + 20t + 2$. At what time $t$ does the projectile reach its maximum height?

Accepted Answer

$t = \frac{-b}{2a} = \frac{-20}{2 	imes -4.9} = 2.04$ seconds

Question 6

A company’s revenue $R$ is modeled by $R(x) = 2x^3 - 5x^2 + 4x - 7$, where $x$ is the number of units sold. What is the revenue when 3 units are sold?

Accepted Answer

$R(3) = 2 	imes 27 - 5 	imes 9 + 4 	imes 3 - 7 = 54 - 45 + 12 - 7 = 14$

Question 7

Which of the following is the correct factorization of $x^2 - 6x + 9$?

Accepted Answer

$(x - 3)^2$

Question 8

A pharmaceutical company uses the formula $C = \frac{2x}{x^2 - 4}$ to calculate the concentration of a drug, where $x$ is the volume in mL. For which value of $x$ is the concentration undefined?

Accepted Answer

$x = 2$

Question 9

Which of the following is the domain of the function $f(x) = \frac{1}{x - 5}$?

Accepted Answer

$x 
eq 5$

Question 10

A line passes through the points $(2, 3)$ and $(6, 11)$. What is the slope of the line?

Accepted Answer

$m = \frac{11 - 3}{6 - 2} = 2$

Question 11

Which of the following is the correct inverse of the function $f(x) = 3x + 7$?

Accepted Answer

$f^{-1}(x) = \frac{x - 7}{3}$

Question 12

A circle has the equation $x^2 + y^2 - 6x + 8y = 0$. What is the center of the circle?

Accepted Answer

$(3, -4)$

Question 13

A system of equations is given by $\begin{cases} 2x + 3y = 7 \ x - y = 1 \end{cases}$. What is the solution to the system?

Accepted Answer

$(x, y) = (2, 1)$

Question 14

A population grows according to the exponential function $P(t) = 1000e^{0.05t}$. What is the population after 10 years?

Accepted Answer

$1000e^{0.5} \approx 1648.72$

Question 15

Which of the following is the correct simplification of $\log_{2}8$?

Accepted Answer

$3$

Question 16

A radioactive substance decays according to $N(t) = N_0e^{-kt}$. If $N_0 = 500$ grams, $k = 0.1$, and $t = 20$ years, what is the remaining mass?

Accepted Answer

$500e^{-2} \approx 67.67$ grams

Question 17

Which of the following is the correct property of logarithms for $\log_{a}(xy)$?

Accepted Answer

$\log_{a}(xy) = \log_{a}x + \log_{a}y$

Question 18

A quadratic equation $x^2 - 4x + 7 = 0$ is solved using the quadratic formula. What are the roots?

Accepted Answer

$x = 2 \pm i\sqrt{3}$

Question 19

A function $f(x)$ is transformed by shifting it 3 units right and 2 units up. Which of the following represents the new function?

Accepted Answer

$f(x - 3) + 2$

College Algebra Study Guide: Fundamental Concepts, Equations, Functions, and Applications

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