Courrey J.
Alexander

Examine graphs of

y = a sin(bx + c)

for different values of a, b, and c.

- The
first thing I did was graphed the equation by varying
*a*and keeping*b*and*c*constant at 1.

- I have
observed that as
*b*and*c*remain constant at 1 and*a*increases by +1, the amplitude of the sine curve increases by +1. *a*=amplitude of the sine curve- Naturally,
if
*a*=amplitude of the sine curve, then I would still expect for the amplitude to increase by +1 if the value of*a*is negative. The only change I expect to see is that the sine curve will be reflected across the*x*-axis.

¤ I
have investigated the amplitude of the sine curve with both positive and
negative integers while both *b* and *c* remain constant at 1. I would like to know if I would get the same results for
positive and negative decimals. I
assume that it would only make the curves closer to each other in
proximity. I will create a graph
for -0.5£*a*£-0.1.

¤ Now
that I have investigated the behavior of *a*
with both positive and negative integers and decimals, I would like to see the behavior of the graph when *a *and *c* are
held constant at 1 as *b *varies from -10 to 10.

¤ The
graph shows that as the value of *b *approaches
0 from both sides, the sine curve behaves like a coil that expands and
contracts. The period 2P/n has
the following values:

¤ The
value of *b* apparently has something to
do with the displacement of the curve (movement from left to right). I will investigate further with *b,
*but first, I will investigate the behavior of *c* in (-10,10) as *a *and *b*
are held constant at 1.

¤ This
graph shows that as *c* approaches 0 from
both sides, the sine curve behaves like a wave going from left to right. Approaching 0 from the right causes the
curve to shift to the right.
Approaching 0 from the left, of course, causes the curve to shift to the
left. The value of *c* apparently something to do with the displacement of
the curve as well.

¤ Now
that I have investigated the behavior of *a, b, *and *c* one at a time as
the other two remained constant at 0, I would like to see the graph as all three vary from -5 to 5.

¤ As
*a, b, *and *c* vary, the sine curve behaves similarly to the graph
when *b* varies. As I stated earlier, variations in both
*b *and *c* seemed to have affected the displacement of the sine
curve. This is because the
displacement, or phase shift, is equal to . The
value of *c* alone affects the
phase angle for the sine curve *y*=*a* sin (*b*x+*c*).

¤ If
the displacement, or phase shift, has a negative value, then I need to move the
sine graph to the left of its normal position (*y*=sin *x*) by .

¤ Of course, if the phase shift has a positive value, then I need to move the sine graph to the right.