Using Graphing Calculator 3.2, we can explore the continuous sine curve expressed by the equation

By examining various graphs of:

for different values of a, b, and c, we are able to see the specific impact these values have on our original curve.

In order to explore the sine curve in terms
of the coefficient **a**, we must examine **a** when b =
1 and c = 0 in our original equation. Let's first look at positive
values of **a**. What happens to the corresponding graphs as
the value of **a** changes?

The original sine curve is represented by the
gray line. Regardless of what value is chosen for our coefficient
**a**, the intersections of this graph along the x-axis will
remain x=0, x=pi, and x=2pi. By choosing different positive values
of the coefficient **a**, we can see our original sine curve
has changed only in *amplitude*, or height.

Now, what can we expect for negative values
of **a**? Let's examine the following graphs:

Again, *amplitude* of the original sine
curve has changed, as we would expect. The intersections of this
graph and the x-axis are still at x=0, x=pi, and x=2pi. The difference
with a negative value of **a **however, is our sine curve now
has a negative amplitude. In other words, our graphs are the same
as when a was a positive value, but are now reflected across the
x-axis.

To see a direct comparison between positive
and negative values of **a**, click here.

Watch the animation of our sine curve as **a**
goes from -5 to 5. Notice how different
values of **a** change the *amplitude* of our sine curve.
See how the curve appears to be growing in height in the y direction.

For this exploration, we must assuming a=1
and c=0 to examine the effect the coefficient **b** will have
on our original sine curve. Let's begin by exploring positive
values of **b**. What happens to the corresponding graphs as
the value of **b** changes?

The original sine curve is represented by the
gray line. This time, different values of our coefficient result
in different points of intersection at the x-axis. Are curves
no longer intersect the x-axis at x=0, x=pi, and x=2pi. By choosing
different positive values of the coefficient **b**, we can
see our original sine curve has changed only in *period*,
or the length of one cycle. In other words, the period for the
sine curve to make one full osscilation is no longer from 0 to
2pi.

What happens when **b** has a negative value?
Let's examine the following graphs:

Again, the *period* of the original sine
curve has changed here, as we would expect. The difference with
a negative value of **b **however, is our sine curves have
the same period as the positive values of **b** above, but
now they are reflected across the x-axis.

For a direct comparison between positive and
negative values of **b**, click here.

Watch the animation of our sine curve as **b**
goes from -5 to 5. Notice how different
values of **b** change the *period* of our sine curve.
See how the curve appears to be contracting and expanding along
the x-axis.

In this case, we are assuming a = 1 and b =
1 in order to demonstrate the affect the coefficient **c**
will have on our sine curve. First, let's look at the graphs generated
for positive values of **c**. What happens to the corresponding
graphs as the value of **c** changes?

The original sine curve is again represented
by the gray line. This time, different values of our coefficient
result in different positions of our graph along the x-axis. Are
curves no longer intersect the x-axis at x=0, x=pi, and x=2pi,
but the do have the same period. By choosing different positive
values of the coefficient **c**, we can see our original sine
curve has changed only in *phase*. In other words, the period
for the sine curve to make one full osscilation is still an inverval
length of 0 to 2pi like the original curve, but now all of the
graphs have all been shifted to the left.

What happens when **c** has a negative value?
Let's examine the following graphs:

Again, the *phase* of the original sine
curve has changed here, as we would expect. The difference with
a negative value of **c **however, is that our sine curves
have been shifted to the right this time. They all still have
the same period as the positive values of **c** above, but
now they have moved in the other direction along the the x-axis.

For a direct comparison between positive and
negative values of **c**, click here.

Watch the animation of our sine curve as **c**
goes from -5 to 5. Notice how different
values of **c** change the *phase* of our sine curve.
See how our curve appears to be moving back and forth along the
x-axis.

Return to Class Page