Sinusoidal Functions of Sine and Cosine On the interval from 0 to 2, the graphs of the basic sine and cosine functions have five key points: the x-intercepts, the maximum and the minimum.

Transformations of Sinusoidal Functions Amplitude |A|: A controls vertical stretching/shrinking Period p can be in radians or degrees: p is the horizontal length to complete one cycle Horizontal shift B can be in radians or degrees: B controls horizontal stretching/shrinking Vertical shift k: +k , shift k units up –k , shift k units down Horizontal phase shift h (proportionally scaled to 2 radians or 360): –h, shift h units right +h, shift h units left where, xstart corresponds with the first point of the sine or cosine cycle

Vertical Stretching/Shrinking of Sine and Cosine Functions KEY TAKE-AWAY: x-intercepts are unchanged; multiply y-value of max/min by A.

Vertical Shifting of Sine and Cosine Functions KEY TAKE-AWAY: If A and B remain unchanged, shift entire curve up or down.

Reflections of Sine and Cosine Functions KEY TAKE-AWAY: x-intercepts are unchanged; multiply y-value of max/min by –1.

Horizontal Stretching/Shrinking of Sine and Cosine Functions What if B is not 1?

y = sin(x) y = sin(2x)

y = sin(x) y = sin(x/2)

y = sin(x) y = sin(x/3)

y = sin(x) y = sin(x-(/3)) Horizontal Shifting of Sine and Cosine Functions y = sin(x) y = sin(x-(/3))

y = sin(x) y = sin(x-(/2)) Horizontal Shifting of Sine and Cosine Functions y = sin(x) y = sin(x-(/2))

y = sin(x) y = sin(x-(3/4))