Q: Which of the following is the correct equation for a tangent function with a period of $$\pi$$ and a phase shift left by $$\frac{\pi}{3}$$?

$$y = \tan(x + \frac{\pi}{3})$$

Question 1

Which of the following is the correct period for the function $$y = -4 	an\left(\frac{1}{4}xight)$$?

Accepted Answer

$$4\pi$$

Question 2

Given the function $$y = 	an\left(x + \frac{\pi}{2}ight)$$, which transformation has been applied to the parent tangent function?

Accepted Answer

Horizontal shift left by $$\frac{\pi}{2}$$

Question 3

What is the equation of the reciprocal function for $$y = \sin(x)$$?

Accepted Answer

$$y = \csc(x)$$

Question 4

For the function $$y = 2\cot\left(\frac{1}{3}x + \frac{\pi}{6}\right)$$, what is its period?

Accepted Answer

$$3\pi$$

Question 5

Which graph transformation occurs for $$y = -	an\left(x - \frac{\pi}{2}ight)$$ compared to $$y = 	an(x)$$?

Accepted Answer

Reflection over the $$x-$$axis and shift right by $$\frac{\pi}{2}$$

Question 6

If $$y = 6\csc\left(\frac{1}{2}x\right)$$, what is the amplitude of the function?

Accepted Answer

Undefined

Question 7

Which of the following is the correct equation for a tangent function with a period of $$2\pi$$ and a vertical shift down by 3 units?

Accepted Answer

$$y = 	an(x) - 3$$

Question 8

A medical device uses the function $$y = -2	an\left(x - \frac{\pi}{2}ight) - 3 to $$model sensor output. What is the vertical shift and reflection applied?

Accepted Answer

Shift down 3 units, reflect over $$x-$$axis

Question 9

Which function has vertical asymptotes at $$x = \frac{\pi}{2} + n\pi$$, where $$n is $$any integer?

Accepted Answer

$$y = 	an(x)$$

Question 10

If the graph of $$y = 	an(x) is $$horizontally compressed by a factor of 2, what is the new period?

Accepted Answer

Question 11

Which of the following is the correct equation for a cotangent function with a period of $$\pi$$ and a phase shift right by $$\frac{\pi}{4}$$?

Accepted Answer

$$y = \cot(x - \frac{\pi}{4})$$

Question 12

A sound wave is modeled by $$y = 2\sec\left(\frac{1}{3}x + \frac{\pi}{6}\right). $$What is the period of this function?

Accepted Answer

$$6\pi$$

Question 13

Which of the following is the correct reciprocal function for $$y = \cos(x)$$?

Accepted Answer

$$y = \sec(x)$$

Question 14

If $$y = 	an\left(\frac{x}{2}ight)$$, what is the period of the function?

Accepted Answer

$$2\pi$$

Question 15

Which transformation is applied to $$y = 	an(x) to $$obtain $$y = 	an\left(x + \frac{\pi}{2}ight)$$?

Accepted Answer

Shift left by $$\frac{\pi}{2}$$

Question 16

Which function has vertical asymptotes at $$x = n\pi$$, where $$n is $$any integer?

Accepted Answer

$$y = \cot(x)$$

Question 17

A chemical reaction is modeled by $$y = 6\csc\left(\frac{1}{2}x\right). $$What is the period of this function?

Accepted Answer

$$4\pi$$

Question 18

Which of the following is the correct equation for a tangent function with a period of $$\pi$$ and a phase shift left by $$\frac{\pi}{3}$$?

Accepted Answer

$$y = 	an(x + \frac{\pi}{3})$$

Question 19

If $$y = -2	an\left(x - \frac{\pi}{2}ight) - 3$$, what is the amplitude of the function?

Accepted Answer

Undefined

Graphing and Analyzing Tangent, Cotangent, Secant, and Cosecant Functions

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