Q: Solve $$4\cos(2\theta) - 2 = 0$$ for $$\theta \in [0, 2\pi].$$

$$\theta = \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}$$

Question 1

Which of the following is the simplified form of the expression $$\frac{(4xy$$^{-1}$$)^{\frac{3}$${2}}}{2xy}$, assuming all variables represent positive real numbers?

Accepted Answer

$$4x^{\frac{1}$${2}}$$y^{\frac{5}$${2}}$$

Question 2

What is the domain of the function $$f(x) = \frac{\sqrt{x-5}}{4}$$?

Accepted Answer

$$[5, \infty)$$

Question 3

Convert $$50^\circ to $$radians.

Accepted Answer

$$\frac{5\pi}{18}$$

Question 4

Which of the following functions represents a reflection of $$y = \sqrt{x}$$ across the $$x-$$axis?

Accepted Answer

$$-\sqrt{x}$$

Question 5

Find the intersection of the intervals $$(-4, 8] \cap (-\infty, 8).$$

Accepted Answer

$$(-4, 8)$$

Question 6

Which of the following is a factorization of the expression $$9x^6$$ - $$y^2$$?

Accepted Answer

$$(3x^3$$ - $$y)(3x^3 + y$$)$$

Question 7

A parabola $$g(x) is a $$transformation of $$f(x) = x^2$ and is reflected, shifted right by 1, and up by 3. Which expression represents g(x$$)$$?

Accepted Answer

$$g(x) = -(x - 1)^2$ + 3$$

Question 8

Solve the equation $$2x^2 + 4x - 1 = 0$ for x$.

Accepted Answer

$$x = -2 \pm \frac{\sqrt{6}}{2}$$

Question 9

Which of the following is the equation of a line perpendicular to $$3x + 6y = 0$$?

Accepted Answer

$$y + 2 = -\frac{1}{3}(x - 4)$$

Question 10

Evaluate $$	an\left(\frac{5\pi}{6}ight).$$

Accepted Answer

$$-\frac{1}{\sqrt{3}}$$

Question 11

A trigonometric graph has amplitude $$3$$ and period $$\pi. $$Which equation could represent this graph?

Accepted Answer

$$y = 3\sin\left(x\right)$$

Question 12

Solve the equation $$x = \frac{y + 2}{y - 1}$$ for $$y.$$

Accepted Answer

$$y = \frac{x + 2}{x - 1}$$

Question 13

Determine whether the function $$g(x) = \sin(x) + 4x^6$ is even, odd, or neither.

Accepted Answer

The function $$g is $$neither odd nor even.

Question 14

Assume all variables represent positive real numbers. Which of the following equalities is true?

Accepted Answer

$$\left(\frac{x}{y-1}\$$right)^{-1}$$ = \frac{y}{x}$$

Question 15

A function $$f is $$graphed and appears symmetric about the origin. Which statement is true?

Accepted Answer

The function $$f is an $$odd function.

Question 16

Given $$\sin(	heta) = \frac{2}{\sqrt{5}}$$ and $$	heta \in [\frac{\pi}{2}, \pi]$$, what is $$\cos(	heta)$$?

Accepted Answer

$$-\frac{1}{\sqrt{5}}$$

Question 17

A line passes through $$(2, -1)$$ and is parallel to $$y = 2x + 3. $$What is its equation in slope-intercept form?

Accepted Answer

$$y = 2x - 5$$

Question 18

Solve $$4\cos(2	heta) - 2 = 0$$ for $$	heta \in [0, 2\pi].$$

Accepted Answer

$$	heta = \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}$$

Question 19

Which of the following is the correct factorization of $$x^3$$ + $$6x^2 - 4x - 24$ by grouping?

Accepted Answer

$$(x + 6)(x^2$ - 4)$$

Precalculus Exam 1: Step-by-Step Study Guidance

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