Q: A biologist models the population of a bacteria culture by $$P(t) = 500e$$^{0.2t}$$, where t$ is in hours. What is the population after 5 hours, rounded to the nearest whole number?

$$500e^{0.2 \times 5}$$ = $$500e^{1}$$, which is approximately $$1359$$

Question 1

Solve the quadratic equation $$3x^2 - 10x - 5 = 0$ for x$.

Accepted Answer

$$x = \frac{10 \pm \sqrt{110}}{6}$$

Question 2

Evaluate $$\sin\left(\frac{11\pi}{6}\right).$$

Accepted Answer

$$-\frac{1}{2}$$

Question 3

A Ferris wheel rotates such that the height $$h (in $$meters) of a passenger above the ground at time $$t (in $$minutes) is modeled by $$h(t) = 2\cos\left(\frac{2\pi}{3}t\right) + 5. $$What are the amplitude and period of the motion?

Accepted Answer

amplitude = 2, period = $$3$$

Question 4

Which of the following is the equation of the line passing through the point $$(-1,4)$$ and perpendicular to the line $$y = \frac{1}{3}x - 5$$?

Accepted Answer

$$y + 4 = -3(x + 1)$$

Question 5

Simplify the interval $$(-\infty, -2) \cap (-3, 7).$$

Accepted Answer

$$(-3, -2)$$

Question 6

Completely factor the expression $$27y^3$$ + $$y^3.$$

Accepted Answer

$$(3y + 1)^3$

Question 7

What is the domain of the function $$f(x) = -2\sqrt{x-2}$$?

Accepted Answer

$$[2, \infty)$$

Question 8

Given the function $$g(x) = x^2$ - 3x$$, find $$g(x + h) - g(x)$$, where $$h is an $$unknown constant.

Accepted Answer

$$2x h + h^2$ - 3h$$

Question 9

Which parent function is represented by the graph that passes through $$(0,0)$$ and $$(1,1)$$, is always increasing, and is concave up for all $$x$$?

Accepted Answer

$$y = \sqrt{x}$$

Question 10

Which of the following shows a valid simplification of $$\sqrt{(2x^2$ - 8)^2$} = 8x - 2$$?

Accepted Answer

$$|2x^2$ - 8| = 8x - 2$$

Question 11

A biologist models the population of a bacteria culture by $$P(t) = 500e$$^{0.2t}$$, where t$ is in hours. What is the population after 5 hours, rounded to the nearest whole number?

Accepted Answer

$$500e^{0.2 	imes 5}$$ = $$500e^{1}$$, which is approximately $$1359$$

Question 12

A ball is thrown upward from a height of 2 meters with an initial velocity of 10 m/s. Its height after $$t$$ seconds is given by $$h(t) = -4.9t^2$ + 10t + 2. $$After how many seconds does the ball reach its maximum height?

Accepted Answer

$$t = 1.02$$ seconds

Question 13

Which of the following intervals represents the range of the function $$f(x) = 2\cos(x) + 1$$?

Accepted Answer

$$[-1, 3]$$

Question 14

If $$f(x) = \frac{1}{x-2}$$, what is the domain of $$f$$?

Accepted Answer

$$(-\infty, 2) \cup (2, \infty)$$

Question 15

A company’s profit $$P (in $$thousands of dollars) is modeled by $$P(x) = -2x^2$ + 8x + 10$$, where $$x is $$the number of units sold (in thousands). What is the maximum profit, and for how many units sold does it occur?

Accepted Answer

Maximum profit is $$18 at$$ $$x = 2$$

Question 16

Given the system of equations: $$3x - y = 1$$ and $$2x + y = 1$$, what is the solution $$(x, y)$$?

Accepted Answer

$$(\frac{2}{5}, -\frac{1}{5})$$

Question 17

Which of the following is the correct factorization of $$x^3 + 8$?

Accepted Answer

$$(x + 2)(x^2$ - 2x + 4)$$

Question 18

A function $$f is $$defined as $$f(x) = \begin{cases} \frac{1}{2}x + 1, & x \leq 0 \ x^2$ - 1, & x \u003e 2 \end{cases}. $$What is the domain of $$f$$?

Accepted Answer

$$(-\infty, 0] \cup (2, \infty)$$

Question 19

Which of the following is the correct simplification of $$\frac{\left(\frac{x^2$ y^3$}{3y}\$$right)^2$$ \cdot 2x^2$}{y^2$}$$?

Accepted Answer

$$\frac{2x^6$ y^4$}{9}$$

Precalculus Exam Study Guide: Functions, Equations, and Graphs

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