## The Sine Graph

#### An investigation of y = a sin (bx+c) for different values of a, b, and c.

The graphs of functions defined by y = sin x are called sine waves or sinusoidal waves. Notice that the graph repeats itself as it moves along the x-axis. The cycles of this regular repeating are called periods. This graph repeats every 6.28 units or 2 pi radians. It ranges from -1 to 1; half this distance is called the amplitude. So the graph below has a period of 6.28 and an amplitude of 1.

We will investivage different values for the amplitude and period as well as phase shift which appears to set the graph at a different place on the x axis.

### Amplitude

Let's look at what happens to the graph with different values of a. On the left below is x= 2 sin x and y = 6 sin x on the same graph with y = sin x. Notice how high and how low the graph goes; this is called the range. What do you think will happen when the sign of a is changed to a negative? Look at the graph on the right below to see y = -3 sin x and y = -5 sin x on the same graph with y = sin x. What happens to the graph as a changes?

.

### Period

Now let's look at the period. See how the cycle repeats every 6.28 units (2 pi). Looking at the left graph, notice what happens y = sin (1)x is changed to y = sin 2x. There are two periods in the space where there was one. That means periods occur twice as often or we say they are one-half as long. Does this one look as if it could be 3.14 or pi.? Now look at the graph on the right below. Here y = sin x is overlaid with y = sin 1/2 x. What happens here? Make a conjecture about what happens to the graph y = sin bx as b is varied.

### Phase Shift

Look again at the equation y = a sin (bx + c). Notice that we have varied a, the amplitude, and b, the period. The last variation in this equation will be c. In the first equation, y = sin x, c is equal to zero. Look at the graph on the left to see that curve as well as the curve of the equation y = sin (x + 2). Notice that the new curve is shifted two units to the left of the original one. See the graph on the right to find out what happens when y= sin x is overlaid by y = sin (x-3). Can you see that the graph is shifted three units to the right? Make a conjecture about variations of c.

### Variations of a, b, and c

What will happen if all three constants are varied at the same time? Look below left at a graph of
y = sin x and overlay it with y = 4 sin (2x +1). Notice that the new graph has an amplitude of 4, the period is 3.14 or pi, and the phase shift is 1 unit to the left. What will happen if all the coeffiecients are negative? Look at the graph on the right to see y = -3 sin (-1/2 x -1). Explain what is happening here. Can you now look at equations of the form y= a sin (bx + c) and predict what the curve will look like?

The End