**Information about function
y=sin(x):**

Range and Domain:

The range of the function are the values in the y-direction that the equation hits, which would be the interval [-1,1]. The domain is the set of all real numbers.

Amplitude:

In simple terms the amplitude is the height of the curve. To calculate the amplitude of the curve, it is half of the distance between the maximum and minimum values. For y=sin(x), the maximum value is 1 and the minimum value is -1, so the amplitude of the above curve is 1.

The Period:

The period of the sin curve is how many radians it takes to complete one cycle, or the lengh of one cycle. For y=sin(x), the period is from x=0 to x=2pi.

__Part 1: Exploring different
values of a__

Let us compare our original
graph of y=sin(x) to y=a sin(x) for different positive values
of *a, *holding *b* and *c* constant at b=c=0*.*

By comparing the red and blue
curves to the purple, we observe that as *a* increases the
height is increasing and as *a* decreases the height is decreasing.
This means that *a* is changing the amplitude of the curve.
And by our previous definition of amplitude, this means that the
maximum and minimum values are also changing. As the value of
*a* increases to 2 the maximum and minimum values also increase
to [-2,2], and as *a*** **decreases to (1/2), the maximum
and minimum values decrease to [-(1/2),(1/2)].

Now let us look at negative
values of *a *and see how *-a *affects the curve.

As we can see from the above graph, the negative
value of *a *reflects the curve over the x-axis.

__Part 2: Exploring different
values of b__

Next we will examine how *b
*affects our curve. Let us compare our original graph of y=sin(x)
to

y=sin(bx) for different positive
values of *b, *holding *a* and *c* constant at
a=1 and c=0*.*

Well, it looks like *b *is affecting the
period of the curve. Recall the period is the number of radians
it takes to complete one cycle, or the length it takes to complete
one cycle. For y=sin(x) (purple curve) the period is 2pi, but
as *b* increases to 2 (red curve), then the period of y=sin(2x)
becomes 4pi. For increasing values of *b,* the curve is stretched.
A smaller number for *b* has the opposite affect on the curve,
it shrinks the curve or makes the period smaller. For y=sin((1/2)x),
the period is pi.

__Part 3: Exploring different
values of c__

Lastly, we will compare our
original graph of y=sin(x) to y=sin(x+c) for different positive
values of *c, *holding *a* and *b* constant at
a=b=1*.*

From our observations, the
value of *c* is shifting the curve to the left by 1. So it
looks like the coefficient *c *shifts the curve to the left
or right. Now, let us look at a negative value of* c.*

When *c* is negative,
the curve is shifted to the right by the amount of *c*. In
the above graph the red curve is shifted to the right by 1.

__Conclusions from exploring
y=a sin(bx+c):__

The coefficient *a *changes
the height or amplitude of the curve and negative coefficients
of* a *reflect the curve over the x-axis.

The coefficient *b *changes
the period of the curve.

The coefficient *c* shifts
the curve to the right for negative values of *c* and shifts
the curve to the left for positive values of *c*.