In this write-up, we will explore the sin function
y= *a* sin (*b*x + *c*) and evaluate how different
values of *a, b*, and *c* transform the graph.

The *a* value:

First, let's take a look at the graph y = sin
(x) where *a*=1, *b*=1, and *c*=0.

From the graph, we can tell that the range
on the y-axis goes from -1 to 1. Thus, giving us an amplitude
of 1. Let's look at this graph with *a* values of 4 and -2.

As you can see from the graphs above, both
functions go through the origin. One difference between the graphs
is the range on the y-axis, also called the domain. The *a*=4
function, y = 4 sin (x), has a domain of 4 to -4. The *a*
= -2 function, y = -2 sin (x), has a domain of -2 to 2. Let's
see if the domain continues to be *a* to -*a* by testing
other *a* values.

As we can see from the graph, the* a*
value determines the domain of* a* to -*a *thus giving
us an amplitude of *a*. By analyzing the graphs above, we
see that when *a*=3 the amplitude is 3, when *a*=1.5
the amplitude is 1.5, when *a*=(4/5) the amplitude is 4/5,
and when *a*=-1.5 the amplitude is 1.5. So, as *a* increases
the amplitude increases for the positive values of *a*. For
the case of *a*=0 there is no graph. Furthermore, as *a
*increases the amplitude decreases for the negative values
of *a*.

By looking at the yellow(*a*=1.5) and
green(*a*=-1.5) functions, we can observe that the *-a*
value reflects the graph over the x-axis.

The *b* value:

Now, let's explore the *b *value. First,
let's set *a*=1 and *c*=0. Now, let's examine the graphs
of *b*=0, *b*=1, *b*=2.

Well, it appears that our *b*=0 function
does not have a graph. This graph is a bit misleading because
sin(0)=0. So, this function would actually have a graph of (0,0),
which is not seen on our graph. As we look at the *b*=1 function,
y=sin(x) and the *b*=2 function, y = sin(2x), we see that
the range of a cycle differs. The *b*=1 function has a range
of about 6 and the *b*=2 function has a range of about 3
for a complete cycle. The term given to the cycle range is a period.
From these graphs, it appears that as *b* values increase
the period decreases. Let's test our hypothesis.

According to the graph, the function y=sin
(3x) has a period of approximately 2 and the function y=sin(6.5x)
has a period of approximately 1. Thus, as the *b* value increases,
the period decreases for positive values of *b*. Next, let's
test some negative *b* values.

Oops, our hypothesis did not hold true for
the negative values of b. By looking at the graph above, when
*b*=(-25/4) in the purple graph, the period is around 1,
when *b*=-3.5 in the red graph, the period is approximately
1.8, and when *b*=-1, in the gray graph, the period is about
6.4. So, as *b* increases, the period increases for negative*
b *values.

The *c* value:

Finally, we will explore the *c* values
for y=*a* sin (*b* x + *c *). First, let's look
at positive *c* values letting *a*=1 and *b*=1.
We will begin with *c*=1, 2.5, 10/3.

By looking at the first graph in this write-up,
you can see that when *c*=0, the graph crosses the origin,
(0,0). Now, by looking at the above functions, when *c*=1,
the function crosses the x-axis at -1, when *c*=2.5 the function
crosses the x-axis at -2.5, and when *c*=(10/3) the function
crosses the x-axis at (-10/3). Therefore, *c* shifts the
function -*c* units on the x-axis for positive values of
*c* and *c*=0. Now, let's explore negative values of
*c*.

In our first function, y = sin(x-1), where
*c*=-1, the graph shifts one unit to the right on the x-axis.
In the second function, y = sin(x-3), where *c*=-3, the graph
shifts three units to the right on the x-axis. In the third function,
y = sin(x-11/2), where *c*=-11/2, the graph shifts 11/2 units
to the right on the x-axis. The shift on the x-axis is called
a phase shift. From our graphs, we can conclude that the *c*
value shifts the graph of y = sin(x) -*c* units on the x-axis
for all values of *c*.

To conclude our write-up let's look at what
we found to be true about the *a, b, *and *c, *values
for y = *a* sin(*b*x + *c*). First, we found that
the* a* value determines the domain of* a* to -*a
*thus giving us an amplitude of *a. *Second, we found that as the *b*
value increases, the period decreases for positive values of *b. *We also
found that as *b* increases, the period increases for negative*
b *values. Third, we found that the *c* value shifts the
graph of y = sin(x), -*c* units on the x-axis for all values
of *c*. We call this the phase
shift.