Examining the Sine Function

By

Audrey V. Simmons

We will be examining the sine function as the coefficient values change to see what effect the change has on the graph.

Examine graphs of y = a sin(bx + c) for different values of a, b, and c.

Our first step will be to:

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

y = sin(x).

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

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.  In this case .

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.

Using the basic sine graph as our frame of reference. Let's look at 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).

Use the graphing calculator to try different values for a. Remember to try positive and negative values.

y = 2 sin x,   y = 5 sin x, and y = -3 sin x

The green graph is y=sin x. The basic sine graph will always be in green in future examples for comparison purposes.

When the values of a are positive, the amplitude is increased by a factor of the absolute value of a, and the graph follows the basic sine graph shown above. This is known as a vertical stretch. When a is less than zero, the amplitude is still increased by a factor of the absolute value of a. However, the negative value of a causes the graph to be a reflection of the basic sine graph.

Step 2: Let a=1, and c=0 and change the values for b.  Our new equation is                  now    y = sin (bx).

y = sin (2x),        y =(4x)        y = sin x

Notice that the amplitude of the graphs did not change even though the value for b was varied. The period of the graph was reduced by 1/b. There is a horizontal shrinking of the graph.  What happens when we substitute negative values for b?

By substituting negative values for b, we also see a reflection of the graph as well as a horizontal shrinking of the basic sine graph. If the absolute  value of b is greater than 1, the graph will be a horizontal shrink.

y = sin (-2x)   and y = sin (-4x)  If the absolute value of b is less than 1, the graph will be a horizontal stretch.  y = sin (-.5x)

Step 3: We will again start with our original equation y= asin(bx+c). Let a=1, b=1, and vary c.     y = sin(x+c)

y= sin (x+1)    y=sin(x+2)      y= sin (x-1)

The value of c 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.

The phase shift is the name for the movement of the graph horizontally. The phase shift is equal to the value of  .

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

·      Changes in the value of a affect the altitude of the sine graph.

·      Changes in the value of b affect the period of the graph.

·      Changes in the value of c affect the shift of the grap