## y = a sin (bx + c)

First, let b = 2 and c = 1, and let's look at the graph of
y = a sin (2x + 1) when a = 1.

Now, overlay the graph when a = 2.

Notice that the amplitude has changed to 2.

What happens when a is negative? Let's compare the graphs when a = 2 and a = -2.

The amplitude is still 2, but the graph is flipped.

Here are the graphs when a = 0.5 and when a = -0.5.

Once again, the amplitude has been changed to `a`. When
`a` is negative, the graph is flipped. Thus, we see that
the value of `a` determines the amplitude and the concavity
of the graph of y = a sin (bx + c).

For the next example, let a = 3 and c = -1, and vary the value of b. First, let b = 1.

Now, overlay the graph when b = 2.

Notice that within one period of the graph when b = 1, there are two periods when b = 2.

Let b = -2,

The graph still has two cycles within b = 1 (not shown), but it has been shifted to the right.

Let b = 0.5 (the green curve),

there are two periods of the graph when b = 1 contained in one period of the graph when b = 0.5. In other words, y = 3 sin (0.5 x -1) cycles one-half (0.5) time within y = 3 sin (x -1).

Here is the graph when b = -0.5.

Notice that the negative value for b has again shifted the
graph to the right.

Lastly, let a = 1 and let b = 1. So,

We are familiar with the graph when c = 0.

Let's see what happens when c = 1 (red), 2 (green), and 3 (blue).

Notice that the graph shifts to the left and crosses the y-axis when y = - c.

What happens when c is negative? Let c = -1 (dull green), -2 (red), and -3 (bright green).

Again, the graph crosses the y-axis when y = - c.

Here, c = 0.5 (red) and c = - 0.5 (green).

Thus, varying the value of c shifts the graph to the left and
right and the y-intercept is - c.

By looking at y = 18 sin (-7x + 5.4) we know that the amplitude is a = 18, there are 7 ...