**We will examine the function y = asin(bx
+ c) for different values of a, b and c. We will begin by graphing
y = sinx (a, b, c = 1). The remainder of the graphs will be compared
to this form of the function:**

**y = sinx**

**The next few graphs demonstrate what
happens when a is varied. Notice that the amplitude of
the curve (distance of the maximum and minimum points from the
x-axis) varies according to the change in this coefficient.**

**y = sin x**

**y = 3sinx**

**y = 0.5 sinx**

**Notice that when a = 3 (graphed in red),
the amplitude of the curve is 3. Likewise, when a = 0.5 (graphed
in blue), the amplitude is 0.5.**

**It is worth noting that if a is negative,
the graph is reflected across the x-axis. This is demonstrated
by the following graphs of y = 2sinx and y = -2sinx.**

**y = sinx**

**y = 2sinx**

**y = -2sinx**

**Notice that the graph of y = -2sinx (graphed
in blue) has an amplitude of 2 and is the reflection of y = 2sinx
(graphed in red).**

**We will now demonstrate the effect of
varying b while a and c remain constant. The following graphs
indicate the result of letting b = 1, 2 and 0.5 while a = 1 and
c = 0:**

**y = sin x**

**y = sin 2x**

**y = sin 0.5x**

**It is evident that changing b affects
the period of the sine function. Specifically, the period
of each function is equal to the absolute value of 2p/b. Notice that when
b = 1 (graphed in pink), the period is equal to 2p/1 = 2p, or about 6.3. When b = 2 (graphed in red), the
period equals 2p/2
= p which
is about 3.1. When b = 0.5 (graphed in blue), the period equals
2p/0.5 =
4p or about
12.6.**

**We will conclude by investigating the
effect of changing c while holding a and b constant. In the following
graphs, a and b will remain equal to 1 while c = 0, 1 and -2.**

**y = sinx**

**y = sin(x + 1)**

**y = sin(x - 2)**

**Changing c effectively shifts the graph
to the left or to the right. This phase shift is determined
by -c/b. For example, when c = 1 (graphed in red), the graph has
the same period and amplitude as when c = 0 (graphed in pink),
but has been shifted -(1/1) = -1, or 1 unit to the left. When
c = -2 (graphed in blue), the graph also has the same amplitude
and period as the graph of y = sinx, but it has been shifted -(-2/1)
= 2 or 2 units to the right.**