By watching the animation,
we see that the function stretches and shrinks as the values of
** a **change.

We also see that the function flattens out at a point and then flips, or reflects.

The question is where do these occurrences happen and at what point to they change?

To analyze these questions, consider a few graphs on the same plane.

y=(-3)sin x

Notice the values of ** a** change
the altitude (the highest point of the function.) of the function,
causing it to stretch and shrink. We can also see that the altitude
directly corresponds to the value of

It is obvious that when ** a**=0,
the function is y=0. This is what is causing the "flattening
out". There something special happening when

From these results we can determine the altitude
of the function y=** a** sin(

Now, what happens when the values of ** b**
change for the function y=

It looks like a slinky. We see the function's
frequency changes as well as a reflection taking place. It seems,
from the animation that the frequency is determined by the ** b**
value. Frequency is number of peaks over a given interval. The
interval that we are looking at is 2

The graph corresponds to the following functions:

Look at the following two function on the graph,
y=sin (2x) and y=sin ((-2)x). Their frequencies are two, and they are reflections
of each other. When ** b=1/2 or b=-1/2, **the frequency
is only one-half and again there is a reflection. From observing
these graphs, we can see that the frequency is the absolute value
of

What about

It looks like it is shifting across the x-axis. Let's look at several on the same plane. The graph corresponds the the following equations: y=sin x; y=sin (x+(-1)); y= sin(x+1)

We see that when ** c **is a positive
value the original function moves to the left and when

We know that if ** a** or

We see that the functions do not reflect. When
both ** a** and

Now, we have seen the effects of ** a, b,
**and