Question 1

Given the function $$y = 3 \sin(2x - \pi)$$, what is the amplitude, period, and phase shift of the function?

Accepted Answer

Amplitude: $$3$$, Period: $$\pi$$, Phase shift: $$\frac{\pi}{2} to $$the right

Question 2

For the function $$y = 4 \cos(2x + 3\pi) - 1$$, what is the vertical shift and the range of the function?

Accepted Answer

Vertical shift: $$-1$$, Range: $$[-5, 3]$$

Question 3

A temperature model for a city is given by $$y = 22.1 \sin\left(\frac{\pi}{6}x - \frac{2\pi}{3}\right) + 52.1$$, where $$x is $$the month. What is the period of this function?

Accepted Answer

$$12$$

Question 4

If a sinusoidal function modeling daylight hours in Boston is $$y = 3.105 \sin\left(\frac{2\pi}{365}x - 0.44\right) + 12.175$$, what is the amplitude and what does it represent in this context?

Accepted Answer

Amplitude: $$3.105$$, representing the maximum deviation from the average daylight hours

Question 5

Given $$y = A \sin(\omega x - \phi) + B$$, which parameter controls the phase shift and how is the direction of the shift determined?

Accepted Answer

$$\phi$$ controls the phase shift; positive $$\phi$$ shifts right, negative $$\phi$$ shifts left

Question 6

A sinusoidal function $$y = 5 \sin(\omega x - \phi) + 10$$ models the average monthly rainfall in a city, with a maximum of $$15$$ inches and a minimum of $$5$$ inches. What are the values of $$A$$ and $$B$$?

Accepted Answer

$$A = 5$$, $$B = 10$$

Question 7

If the period of a sinusoidal function is $$T = 8$$, what is the value of $$\omega in$$ $$y = A \sin(\omega x)$$?

Accepted Answer

$$\omega = \frac{2\pi}{8}$$

Question 8

A sinusoidal function $$y = 2 \cos(\pi x - \frac{\pi}{2}) + 4$$ models the height of a tide in meters, where $$x is $$time in hours. What is the phase shift and in which direction?

Accepted Answer

Phase shift: $$\frac{1}{2}$$ units to the right

Question 9

You are given the data for average monthly temperatures in a city. To fit a sine function $$y = A \sin(\omega x - \phi) + B to $$the data, what is the correct order of steps?

Accepted Answer

Find $$A$$, then $$B$$, then $$\omega$$, then $$\phi$$

Question 10

A sinusoidal function $$y = 6 \sin(\frac{\pi}{4}x - \frac{\pi}{2}) + 2$$ models the number of daylight hours in a city, where $$x is $$the month. What is the maximum number of daylight hours predicted by this model?

Accepted Answer

$$8$$

Question 11

Suppose you are given a scatter plot of average monthly temperatures and you want to fit a sine function. If the data repeats every 12 months, what should the period $$T be in $$your model?

Accepted Answer

$$T = 12$$

Question 12

Given $$y = 4 \cos(2x + 3\pi) - 1$$, what is the phase shift and in which direction?

Accepted Answer

$$\frac{3\pi}{2}$$ units to the left

Question 13

A sinusoidal function $$y = A \sin(\omega x - \phi) + B is $$used to model the number of daylight hours in a city. If the maximum is $$16$$ hours and the minimum is $$8$$ hours, what is the value of $$B$$?

Accepted Answer

$$B = 12$$

Question 14

If the period of a sinusoidal function is $$T = 365$$ days, what is the value of $$\omega in$$ $$y = A \sin(\omega x)$$?

Accepted Answer

$$\omega = \frac{2\pi}{365}$$

Question 15

A sinusoidal function $$y = 10 \sin(\frac{\pi}{6}x - \frac{\pi}{2}) + 50 is $$used to model the average monthly temperature in a city. What is the average temperature predicted by this model?

Accepted Answer

$$50$$

Question 16

If a sinusoidal function has amplitude $$7$$, period $$24$$, and phase shift $$3$$ units to the right, which of the following is a possible equation for the function?

Accepted Answer

$$y = 7 \sin\left(\frac{2\pi}{24}x - \frac{2\pi}{8}\right)$$

Phase Shift and Sinusoidal Curve Fitting in Trigonometric Functions

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