Q: A quadratic function models the height $$h (in $$meters) of a projectile launched from the ground as $$h(t) = -4.9t^2$ + 20t. $$What is the maximum height reached by the projectile?

The maximum occurs at $$t = \frac{-b}{2a} = \frac{-20}{2 \times -4.9} = 2.04$$ seconds. Plugging in: $$h(2.04) = -4.9(2.04)^2$ + 20(2.04) = 20.4$$ meters

Question 1

A pharmaceutical company needs to mix two solutions: one containing 20% active ingredient and another containing 50% active ingredient. How many liters of each solution should be mixed to obtain 10 liters of a solution with 30% active ingredient? Set up and solve the system of equations.

Accepted Answer

Let $$x be $$liters of 20% and $$y be $$liters of 50%. The system is:

$$\begin{cases} x + y = 10 \ 0.2x + 0.5y = 0.3 	imes 10 \end{cases}$$

Solving gives $$x = 6.67$$, $$y = 3.33$$

Question 2

A quadratic function models the height $$h (in $$meters) of a projectile launched from the ground as $$h(t) = -4.9t^2$ + 20t. $$What is the maximum height reached by the projectile?

Accepted Answer

The maximum occurs at $$t = \frac{-b}{2a} = \frac{-20}{2 	imes -4.9} = 2.04$$ seconds. Plugging in:

$$h(2.04) = -4.9(2.04)^2$ + 20(2.04) = 20.4$$ meters

Question 3

Which of the following is the correct definition of a function?

Accepted Answer

A function is a relation in which each input is paired with exactly one output.

Question 4

A company’s profit $$P (in $$thousands of dollars) is modeled by $$P(x) = -2x^2$ + 12x - 16$$, where $$x is $$the number of units sold (in thousands). What is the minimum number of units that must be sold for the company to break even?

Accepted Answer

Set $$P(x) = 0$$:

$$-2x^2 + 12x - 16 = 0$

x$$ = \frac{-12 \pm \sqrt{$$12^2 - 4(-2)(-16$$)}}{2 	imes -2} = \frac{-12 \pm \sqrt{144 - 128}}{-4} = \frac{-12 \pm 4}{-4}$$

x = 2$ or x = 4$

Question 5

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

Accepted Answer

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

Question 6

A biologist models the population of bacteria as $$P(t) = 500e$$^{0.2t}$$, where t$ is time in hours. How long will it take for the population to double?

Accepted Answer

Set $$P(t) = 1000$$:

$$1000 = 500e$$^{0.2t}$$

2$$ = $$e^{0.2t}$$

$$\ln 2 = 0.2t$$

$$t = \frac{\ln 2}{0.2} \approx 3.47$$ hours

Question 7

Which of the following is the correct solution to the equation $$|2x - 5| = 9$$?

Accepted Answer

The solutions are $$2x - 5 = 9 or$$ $$2x - 5 = -9$$:

$$x = 7 or$$ $$x = -2$$

Question 8

A circle has center at $$(3, -2)$$ and radius $$5. $$Which of the following is its equation?

Accepted Answer

$$(x - 3)^2$ + (y + 2)^2$ = 25$$

Question 9

Which of the following is the correct simplification of $$\frac{x^2$ - 9}{x^2$ - 6x + 9}$$?

Accepted Answer

$$\frac{(x + 3)(x - 3)}{(x - 3)^2$} = \frac{x + 3}{x - 3}$$, for $$x 
eq 3$$

Question 10

A system of equations is given by $$\begin{cases} 2x + 3y = 7 \ 4x - y = 5 \end{cases}. $$What is the solution?

Accepted Answer

Solving:

$$y = 1$$, $$x = 2$$

Question 11

Which of the following is the correct property of logarithms for $$\log_b(xy)$$?

Accepted Answer

$$\log_b(xy) = \log_b x + \log_b y$$

Question 12

A matrix $$A is $$given by $$\begin{pmatrix} 2 & 3 \ 4 & 5 \end{pmatrix}. $$What is the result of $$A + B if$$ $$B = \begin{pmatrix} 1 & 0 \ -1 & 2 \end{pmatrix}$$?

Accepted Answer

$$\begin{pmatrix} 3 & 3 \ 3 & 7 \end{pmatrix}$$

Question 13

Which of the following is the correct solution to the inequality $$x^2 - 4x + 3$$ \u003c 0$$?

Accepted Answer

Factor: $$(x - 1)(x - 3) \u003c 0. $$Solution: $$1 \u003c x \u003c 3$$

Question 14

A rational function is given by $$f(x) = \frac{2x}{x - 4}. $$What is the domain of $$f$$?

Accepted Answer

All real numbers except $$x = 4$$

Question 15

Which of the following is the correct solution to the equation $$2^{x} = 16$?

Accepted Answer

$$x = 4$$

Question 16

A transformation shifts the graph of $$f(x) = x^2$ up by 3 units and right by 2 units. What is the equation of the transformed function?

Accepted Answer

$$f(x) = (x - 2)^2$ + 3$$

Question 17

A system of nonlinear equations is given by $$\begin{cases} x^2$ + y^2$ = 25 \ y = x + 3 \end{cases}. $$What are the solutions?

Accepted Answer

Substitute $$y = x + 3$$:

$$x^2 + (x$$ + $$3)^2 = 25$

x^2$ + x^2$ + 6x + 9 = 25$$

$$2x^2 + 6x - 16 = 0$

x = 2$, x = -4$; y = 5$, y = -1$

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