Q: Given $$f(x) = 3x^2$ - 12x + 7$$, write $$f(x) in $$vertex form and state the vertex.

$$f(x) = 3(x-2)^2$ - 5$$,\ \text{vertex}:\ (2, -5)$

Question 1

Which of the following is the correct graph for the function $$y = 2 \sin(x) + 1$$?

Accepted Answer

A sine wave with amplitude 2, shifted up by 1 unit.

Question 2

Find the exact value of $$\cos\left(\dfrac{5\pi}{6}\right).$$

Accepted Answer

$$-\dfrac{1}{2}$$

Question 3

Solve the equation $$|2x - 5| = 9.$$

Accepted Answer

$$x = 7,\ x = -2$$

Question 4

Which of the following is the domain of the function $$f(x) = \dfrac{1}{x^2$ - 4}$$?

Accepted Answer

$$x 
eq 2,\ x 
eq -2$$

Question 5

Given $$f(x) = 3x^2$ - 12x + 7$$, write $$f(x) in $$vertex form and state the vertex.

Accepted Answer

$$f(x) = 3(x-2)^2$ - 5$$,\ 	ext{vertex}:\ (2, -5)$

Question 6

Solve for $$x$$: $$2^{x+1} = 16$.

Accepted Answer

$$x = 3$$

Question 7

Which of the following is an identity?

Accepted Answer

$$\sin^2$(x) + \cos^2$(x) = 1$$

Question 8

If $$f(x) = 2x + 3$$ and $$g(x) = x^2$, what is (f$$ \circ g)(2)$$?

Accepted Answer

$$11$$

Question 9

A car travels at a constant speed of 60 miles per hour. How long will it take to travel 150 miles?

Accepted Answer

$$2.5$$ hours

Question 10

Solve for $$x$$: $$\log_2$(x) = 5.$$

Accepted Answer

$$x = 32$$

Question 11

Which of the following is the inverse function of $$f(x) = 3x - 4$$?

Accepted Answer

$$f^{-1}(x$$) = \dfrac{x+4}{3}$$

Question 12

Which of the following is the correct property of logarithms?

Accepted Answer

$$\log_b(MN) = \log_b(M) + \log_b(N)$$

Question 13

A solution is made by mixing 10 liters of a 30% acid solution with $$x$$ liters of a 50% acid solution to obtain a 40% acid solution. What is the value of $$x$$?

Accepted Answer

$$10$$

Question 14

Which of the following is the range of the function $$f(x) = \sqrt{x-2}$$?

Accepted Answer

$$[0, \infty)$$

Question 15

If $$	an(	heta) = 3$$ and $$	heta is in $$the first quadrant, what is $$\sin(	heta)$$?

Accepted Answer

$$\dfrac{3}{\sqrt{10}}$$

Question 16

Solve for $$x$$: $$\sqrt{x+5} = x - 1.$$

Accepted Answer

$$x = 2$$ and $$x = 5 ($$but only $$x = 2 is $$valid)

Question 17

Which of the following is the equation of the line passing through $$(2, 3)$$ with slope $$-4$$?

Accepted Answer

$$y - 3 = -4(x - 2)$$

Question 18

A radioactive substance decays according to $$A(t) = A_0$ e$$^{-0.2t}$$. If the initial amount is 100 grams, how much remains after 5 hours?

Accepted Answer

$$36.79$$ grams

Question 19

Which of the following is the correct solution to the inequality $$x^2 - 5x + 6$$ \u003c 0$$?

Accepted Answer

$$2 \u003c x \u003c 3$$

Question 20

If $$\sin(\alpha) = \dfrac{1}{2}$$ and $$\alpha is in $$the second quadrant, what is $$\cos(\alpha)$$?

Accepted Answer

$$-\dfrac{\sqrt{3}}{2}$$

Comprehensive Precalculus Final Exam Study Guide

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