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.