Tuesday, February 26, 2013

4.5 Graphs Of Sine And Cosine Functions



When using a calculator,make sure that you are in radians when graphing!!!!

Sine

When y=sin x



  • The range of the function is [-1,1] 
  • The domain of the function is negative infinity to infinity.
  • the period is 2╥ 
The sine graph is reflected about the origin.  This is further proof that sine is an odd function.


Cosine

When y=cos x


 
  • The range of the function is [-1,1] 
  • The domain of the function is negative infinity to infinity.
  • the period is 2╥
 The cosine graph is reflected about the y-axis.  This is further proof that cosine is an even function.

Changes To The Graphs

y=a sin(bx-c) +d
y=a cos(bx-c) +d

Changes In A

Changes in a affect the amplitude

The amplitude is half the distance between the maximum and minimum values of the function
The larger a is, the larger the amplitude will be.
When a is negative the graph is reflected about the x-axis
This is the graph of y=2 sin x

Changes In B

Changes in b affect the period
If  b is between 0 and 1  the period will be greater than 2╥
The graph is horizontally stretched.
If b is greater than one the period is less than 2╥ 
The graph is horizontally compressed.
If b is negative than we can use sin(-x)=-sin x and cos(-x)=cos x to figure out what the new equation and graph look like.


 This is the graph of y=(-4x) compared to the graph of y=sin x

Changes In C And D

Changes in c and d have to do with horizontally and vertically translating the graphs.
When c is positive the graph is horizontally translated to the left
When c is negative the graph is horizontally translated to the right
When d is positive the graph is vertically translated to the right
When d is negative the graph is vertically translated to the left

This graph shows how each variable changes the graph.


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