Assignment I

By:  Sharren M. Thomas

The exploration is to examine graphs of   y = a sin (bx + c) for different values of a, b, and c.

Let's first begin by examine the graph when a= 1; b = 1; and c = 0.  Thus we have the sine graph with an amplitude of 1, x-intercepts occur at πn where n is an integer, y-intercept of 0, and period of 2π.

Now when a = 2; b = 1; and c = 0.  We have the graph of

Notice that the graph y = sin x has been vertically stretched by a factor of 2.  Each of the original y-coordinates of each point has been multiplied by 2.  This is the largest y-coordinate of the graph, the amplitude of the graph.  The period is still 2π.

See below the graph of y = sin x for a = 1, 2, and 3.  Notice that the graph of y = sin x is vertically stretched, but the period is not effected.

Click below to see what happens for different values of , a, the amplitude, when a is increased from 1 to 10, including all rational numbers in between  Press play after you press below.

Click

Let's examine what happens for a = -1.

Below examine the graph of

The graph y = sin x has been reflected through the x- axis.  Notice the amplitude for both graphs is still 1 and the period is still 2π.

Click below to examine the graph y = a sin x for negative values of a.  Notice that the graph

y = sin x will be reflected through the x - axis then vertically stretched as was for the positive values of a.

Then press play.

Now, let's examine the graph when a = 1; b =1, 2, & 3; and c = 0.  Thus we have the sine graph with amplitude of 1 but the graph

As you can see the original graph of y = sin x is being horizontally compressed by a factor of 1/b.  Notice that this causes a change of the period.  The period for the graphs

y = sin (bx + 0) is 2π / b.

Now, let's examine the graph when a = -1; b =-1, -2 & -3; and c = 0.  Thus we have the sine graph with an amplitude of 1, but once again the period is changing or the graph y = sin (-x) is being horizontally compressed.

As you can see the original graph of y = sin x has been reflected across the x-axis and then the same as for the positive values of b, the graph is being compressed by a factor of 1/b.

Now, let's examine the graph y = a sin (bx+ c) when a = 1; b = 1; and c > 0

Notice that the graph has been shifted to the left.  Each new graph has a phase shift -c/b.

Now, let's examine the graph y = a sin (bx+ c) when a = 1; b = 1; and c < 0.