## 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