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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