4.5 Graphs of Sine and Cosine Functions

The amplitude of y = a sin x and y = a cos x represents half the distance between the maximum and minimum values of the function (the distance from the maximum to the midline or the distance from the minimum to the midline).
The amplitude = 
The period of y = a sin bx and y = a cos bx, given b is a positive real number, is equal to 


The Graph of Sine

The period is 2π and the range is [-1,1].
As shown in the graph, the sine function is odd.
y = a sin (bx - c) + d
amplitude = 

period  = 

phase shift (shift left/right) = c

midline shift (shift up/down) = d

The Graph of Cosine

The period is 2π and the range is [-1,1].
As shown in the graph, the cosine function is even.
y = a cos (bx - c) + d
amplitude = 

period  = 

phase shift (shift left/right) = c

midline shift (shift up/down) = d