Graphs Of Sine and Cosine Functions

Graphs of Sine and Cosine Functions

Y=cos x

Y=sin x

 

The following two equations:

Y=d+a sin(bx-c)

And

Y=d+a cos(bx-c)

can experience various transformations as the functions change.

Amplitude

The amplitude of y= a sin x and y= a cos x respresents half of the distance between the max and min values and the equation below can be used:

Amplitude= IaI , where the absolute value of a is taken.

Increasing or decreasing the value of a will either vertically shrink or stretch the graph.

Example #1:

Consider the values:

Y=sin x

Y=2sin x

Y=1/2sin x

Y=cos x

Y=2cos x

Y=1/2cos x

Period

The period of a function of y=a sin bx and y= a cos bx can be found by the equations:

Period= 2 /b

Examples:

 

Shifting of Sine and Cosine graphs

Below you can see both the original graph of y =sin(x) and the graph of the translation

y = sin(x) + 1

y = (1/2)Cos 3x

Identify each before you graph:

Amplitude = 1/2
Period = 2p/ 3
Maximums  are at the beginning point  (0, 1/2) and
End point (2p/ 3, 1/2)
minimum point at ( p/3, -1/2)
Zeros at ( p/ 6, 0)  and ( p/ 2, 0)

y = -2 Sin (p/2)x

 Amplitude = | -2 | = 2

Period = 2p/ (p/ 2) = 4
Note that this graph is a reflection about the x-axis.  This interchanges the maximum and minimum values.
                        zeros : (0, 0), ( 2, 0), ( 4, 0)
                        minimum :( 1, -2)
                        maximum : ( 3, 2)