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Amplitude (a):

The value of ∣a∣ determines the height of the wave from its midline (the horizontal line defined by y=c).

If a>0, the sine or cosine wave opens upwards; if a<0, it opens downwards.

Example: For a=3, the amplitude is 3, meaning the function oscillates 3 units above and below the midline.

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Frequency (k):

The value of k affects the period of the function. The period T is calculated using the formula:

A larger value of ∣k∣ results in a shorter period (more oscillations in a given interval), while a smaller value results in a longer period.

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Phase Shift (d):

The value of d shifts the graph horizontally. The phase shift can be found by setting x−d=0, leading to a shift of d units to the right. If d is negative, it shifts to the left.

Example: For d=2, the graph shifts 2 units to the right.

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Vertical Shift (c):

The value of c shifts the entire graph vertically. The midline of the wave moves to y=c.

If c>0, the graph shifts upward; if c<0, it shifts downward.

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Determine the amplitude, the period, the phase shift, and the vertical translation for each function with respect to y = sin x. Then, sketch a graph of the function for two cycles.

y =2 sin (x +3) - 2

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Determine an equation for each sine function. Check by graphing.