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Examining
the Sine Function

(Assignment 1)

**by**

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Let us examine the sine function as the coefficient values change to see the effects these changes have on the various graphs.

Given the graph y = a sin (bx + c) with variables of a, b, and c.

Our first step is to :

Look at the basic sine graph when a=1, b=1, and c=0

Notice that the** Domain** is the set of real numbers, and the **Range** is [-1,1]. The **Period** for the sine function is 2¹. The graph of the sine function continues indefinitely.

The **amplitude **is the distance from the axis to the highest or lowest
point. Or it is half the distance from the highest to the lowest point. In this
case the amplitude is 1. Let us use during this example the variable ÔaÕ in
demonstrating the **amplitude**. Currently
a = 1.

The **period** is the time it takes for the graph to make one complete **cycle** or in other words, the amount of time it takes for the
graph to begin repeating. Let us use the variable ÔbÕ in conjunction with
the adjusting the **period**. In this case
2¹.

In our example the sine wave **phase** is controlled through variable ÔcÕ, initially let c = 0.

Continue to use the basic sine graph
as our frame of reference. Let us examine what happens to the graph under the
following guidelines.

**Step 1:** a sin (bx +c)

Let b=1, c=0, and vary the values
of a. Our new equation becomes y=a sin(x).

Let us use the graphing calculator
to examine the effects of varying the values for ÔaÕ, remembering to use both
positive and negative values.

The blue graph is y=sin x. Y = sin (x), the basic sine graph will always be in blue in future
examples for comparison purposes.

Notice that when the value for
variable ÔaÕ is positive, the **amplitude**
increases by a factor of the absolute value of ÔaÕ, and the graph emulates the
y=sin x original graph
demonstrated above. This is known as a **vertical stretch**. Similarly, when the variable ÔaÕ is negative, the **amplitude** still increases by the same factor, the absolute value of
ÔaÕ.

However, the negative value of ÔaÕ
causes the graph to be a reflection across the x-axis.

**Step 2: **Now we are examining the effects of variable ÔbÕ. Let a=1,
and c=0 and change the values for variable ÔbÕ. Our new equation
is now y = sin (bx).

Notice that the **amplitude** of the graphs does not change even though the value of
variable ÔbÕ was varied. When b>1 the **period** of the graph is changed to 2¹/b resulting
in a horizontal shrinking of the graph. When 0 < b < 1 then the **period** is still
changed to 2¹/b, however, the graph is now stretched.

This leaves the question: What
happens when negative values are substituted for variable ÕbÕ?

By substituting negative values
for variable Ô**bÕ, **notice there is a
reflection across the x-axis for our two graphs as well as horizontal change of
the basic sine graph.

**Step 3: **Let us again start with equation y= asin(bx+c). Let a=1,
b=1, and vary c, resulting in y = sin(x+c)

The value of variable ÕcÕ moves
the sine graph to the right or the left. When c > 0, the graph moves to the
left. When c < 0, the graph moves to the right.

This horizontal movement is called
the **phase shift**. The phase shift **appears** to be equal to the value of ÔÐcÕ. To be sure let us
check what with a change to variable ÔbÕ simultaneously.

This
shows that the phase shift is effected by ÔbÕ. Thus, the **phase shift** is actually the result of
Ô-c/bÕ.

In summary, given the equation y =
a sin (bx +c) the following are true:

- Changes
in the value of variable Ô
**aÕ**effects the**altitude**of the sine graph.

- Changes
in the value of variable Ô
**bÕ**effects the**period**of the graph.

- Changes
in the value of variable Ô
**cÕ**in conjunction with the value of variable**ÔbÕ**together effects the**phase****shift**of the graph.