The Graphs of Sine Function
by
Avijit Kar
In this exploration I will examine
the effects of 
and 
on the graph of 
. I will start with the basic Sine function
.
Graph 1
The Sine function has a period of 
and amplitude of 1.
The Sine function is also an Odd function since it’s symmetric with respect to
the Origin.
Now let’s examine how the sine
function transforms as 
and
changes. First I will
explore the effects of 
.
Let’s look at the graph of 
for 
Graph
2/Animation1
Here is the amplitude
(i.e. the height) of the Sine function. So, for 
, the Sine function graph will reflect against the x-axis
line but the height will remain the same. Below is an animation of , where 
.
Graph
3/Animation 2
Now let’s examine
the effects of changing 
on
the graph of 
. First we will the examine the graph of
the fuction for 
.
Graph 4/Animation 3
By looking at the graph and the
animation, we can see that the period of the Sine function changes for
different values of . We know that has a period of . Meaning, the graph will repeat itself after that interval.
Now as we change , and as gets larger, the
period of Sine function gets smaller. This is for the fact that, period of 
is 
.
Now I will look at the effect
of changing on 
function. The
following graphs are for ,where -2
.
Graph
6/Animation 4
We can see from graph and animation
that, the graph shifts horizontally as changes. Thus we can
conclude that the Sine graph shifts horizontally (to the left if c
, and to the right if 
) for different values of . This is known as phase shift of Sine Curve. If 
, then the phase shift is calculated by the formula 
.
Finally, let’s look at some
example of sine function with differ values of and . The basic function is also shown on each
curve for reference.
In summary, depending on the values
of and
, the graph of 
either change
amplitude and/or reflects, dilates and/or contracts, and shifts horizontally to
the graph of . To be more precise, through this exploration, we can
conclude that 
affects the amplitude, 
affects the period, and affects the phase
shift of the Sine graph.