Q: Given the graph below, which of the following could be the equation for $$k(x)$$? (Refer to the fourth image provided) The graph is a tangent function with vertical asymptotes at $$x = -\frac{3\pi}{4}, \frac{\pi}{4}, \frac{5\pi}{4}$$, etc.

$$y = \tan\left(8x - \frac{\pi}{2}\right)$$

Question 1

Given the graph below, which equation best represents the function shown?

(Refer to the first image provided)

A sinusoidal function is graphed with a maximum at $$y = 10$$, a minimum at $$y = 0$$, and a midline at $$y = 5. $$The period is $$8$$ units.

Which equation matches the graph?

Accepted Answer

$$y = 5 + 5\cos\left(\frac{\pi}{4}x\right)$$

Question 2

Consider the function $$h(	heta)$$ shown below.

(Refer to the second image provided)

What are the amplitude and period of $$h(	heta)$$?

Accepted Answer

Amplitude: $$8$$, Period: $$2\pi$$

Question 3

Which of the following equations matches the polar graph shown below?

(Refer to the seventh image provided)

The graph is a circle centered at $$(3, 0)$$ with a radius of $$3.
$$

Accepted Answer

$$r = 3 + 3\cos	heta$$

Question 4

Given the graph below, which of the following best describes the vertical asymptotes of the function?

(Refer to the third image provided)

The graph is a tangent function with vertical asymptotes at $$x = -2\pi, 0, 2\pi, 4\pi, \ldots.
$$

Accepted Answer

$$x = 2k\pi$$, where $$k is an $$integer

Question 5

Given the graph below, which of the following could be the equation for $$k(x)$$?

(Refer to the fourth image provided)

The graph is a tangent function with vertical asymptotes at $$x = -\frac{3\pi}{4}, \frac{\pi}{4}, \frac{5\pi}{4}$$, etc.

Accepted Answer

$$y = 	an\left(8x - \frac{\pi}{2}ight)$$

Question 6

The ellipse shown below has its center at $$(-2, -3)$$ and vertices at $$(-2, -6)$$ and $$(-2, 0). $$What is the equation of this ellipse in standard form?

(Refer to the fifth image provided)

Accepted Answer

$$\frac{(x+2)^2$}{9} + \frac{(y+3)^2$}{16} = 1$$

Question 7

Given the piecewise function graphed below, what is $$\lim\limits_{x 	o 0} f(x)$$?

(Refer to the sixth image provided)

Accepted Answer

$$-1$$

Question 8

Given the function $$f(x) = \sin x$$ and $$k(x) is a $$horizontal translation of $$f(x) by$$ $$-\pi$$ units and has twice the period of $$f(x). $$Which of the following defines $$k(x) in $$terms of $$f$$?

Accepted Answer

$$k(x) = f\left(\frac{x+\pi}{2}\right)$$

Question 9

A ramp forms a triangle with angles $$A = 27^\circ$$, $$B = 78^\circ$$, and side $$c = 19$$ feet. Using the Law of Sines, what is the length of side $$a$$ opposite angle $$A$$?

Accepted Answer

$$8.8 ft$$

Question 10

Given the complex number $$z = -5 - 5i$$, what is its polar form?

Accepted Answer

$$5\sqrt{2}\left[\cos\left(\frac{5\pi}{4}\right) + i\sin\left(\frac{5\pi}{4}\right)\right]$$

Question 11

If $$	an	heta = 15$$ and $$	heta is in $$Quadrant I, what is $$\sin 2	heta$$?

Accepted Answer

$$\frac{15}{32}$$

Question 12

Given $$\sin\alpha = \frac{4}{5}$$ and $$\cos\beta = \frac{12}{13}$$, both in Quadrant I, what is $$\sin(\alpha + \beta)$$?

Accepted Answer

$$\frac{56}{65}$$

Question 13

Which of the following vectors is orthogonal to $$\vec{v} = 20\vec{i} - 8\vec{j}$$?

Accepted Answer

$$\vec{w} = 15\vec{i} - 6\vec{j}$$

Question 14

Find the rectangular coordinates of the point with polar coordinates $$\left(8, -\frac{2\pi}{3}\right).$$

Accepted Answer

$$(-4, -4\sqrt{3})$$

Question 15

Given the ellipse $$\frac{(x+2)^2$}{9} + \frac{(y+3)^2$}{16} = 1$$, which of the following statements is true?

Accepted Answer

The center is $$(-2, -3)$$ and the major axis is vertical with length $$8.$$

Question 16

Evaluate $$\lim\limits_{x 	o 2^+} f(x)$$ for the function graphed below.

(Refer to the sixth image provided)

Accepted Answer

$$3$$

Question 17

Which of the following is equivalent to $$\cos^2$ x - \sin^2$ x$$?

Accepted Answer

$$\cos 2x$$

Question 18

Solve for all $$	heta in$$ $$[0, 2\pi)$$ where $$2\cos	heta = -\cos	heta.$$

Accepted Answer

$$	heta = \pi$$

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