Graphing Sinusoidal Functions Y=cos x

y = cos x Recall from the unit circle that: – Using the special triangles and quadrantal angles, we can complete a table.

 Y  Y  Y Table of Values y = cos x

Parent Function y = cos x

Domain Recall that we can rotate around the circle in either direction an infinite number of times. Thus, the domain is (- ,  )

Range Recall that –1  cos   1. Thus the range of this function is [-1, 1 ]

Period One complete cycle occurs between 0 and 2 . The period is 2 .

How many periods are shown?

Critical Points Between 0 and 2 , there are two maximum points at (0, 1) and (2 ,1). Between 0 and 2 , there is one minimum point at ( ,-1). Between o and 2 , there are two zeros at

Parent Function Key Points * Notice that the key points of the graph separate the graph into 4 parts.

y = a cos b(x-c)+d a = amplitude, the distance between the center of the graph and the maximum or minimum point. If |a| > 1, vertical stretch If 0<|a|<1, vertical shrink If a is negative, reflection about the x-axis

y = 3 cos x What changed?

y = - cos x

y = a cos b(x - c)+d b= horizontal stretch or shrink Period =. If |b| > 1, horizontal shrink If 0 < |b|< 1, horizontal stretch If b < 0, the graph reflects about the y-axis.

Tick Marks Recall that the key points separate the graph into 4 parts. If we alter the period, we need to alter the x-scale. This can be done by diving the new period by 4.

y = cos 3x What is the period of this function?

y = a cos b(x - c ) + d c= phase shift If c is negative, the graph shifts left c units (x+c)=(x-(-c)) If c is positive, the graph shifts right c units (x-c)=(x-)+c))

What changed? Which way did the graph shift? By how many units?

y = a cos b(x-c) + d d= vertical shift If d is positive, graph shifts up d units If d is negative, graph shifts down d units

y = cos x - 2 What changed? Which way did the graph shift? By how many units?

y = -2 cos(3(x-  )) +1 Can you list all the transformations?

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