We are going to examine the graph of y = **a** sin (**b**x + **c**). First, we will begin by looking at the graph of y = **a** sin (**b**x + **c**) where **a** = 1, **b** = 0, and **c** = 0.

Let's first look at the different characteristics
of the graph y = sin x. The graph passes through the origin (0,0).
The graph is continuous along the x-axis and reaches a highest
value of 1 and a lowest value of -1 on the y-axis. It also appears
that the graph repeats itself once it reaches a certain point
on the x-axis. We can correlate this information to what we know
about the sine function. The domain of the sine function is all
real numbers, while the range is [-1,1]. The sine function is
called a periodic function because it repeats itself over intervals
which are called periods. If we say that the cycle starts at the
origin (0,0), then we can see from the graph that the cycle repeats
itself when x is just greater than 6. The actual point at which
the cycle repeats itself is (2p,0). So, the period of the graph y = sin x is 2p. Comparing the
graph y = sin x to the graph y = **a** sin (**b**x + **c**),
we can see that there seems to be some differences in the graphs
when **a**, **b**, and **c** are changed. Next we are
going to see how the graph is affected when we change the **a**,
but leave the **b** and **c** constant. Let's take a look
at the graph of y = **a** sin (**b**x + **c**), where
**a** = 1/2, 1, and 2, **b** = 1, and **c** = 0.

Clearly we can see the affect **a** has
on the graph y = **a** sin (**b**x + **c**). When 0 <
**a** < 1, the amplitude of the graph decreases, causing
the slopes of the graph to appear more "flat". When
**a** > 1, the amplitude of the graph increases, causing
the slopes of the graph to appear more "steep". This
shows that changing the **a** affects the amplitude of the
graph. We can also note that all three graphs have the common
points (0,0), (p,0), (2p,0), and (3p,0). What happens when we make **a** negative? One
hypothesis is that it will cause the graph to reflect across the
x-axis. Let's see if we are correct.

It looks as though our hypothesis was correct.
Changing the **a** from positive to negative reflects the graph
across the x-axis. And if you compare the previous two graphs,
the magnitude of the amplitude is the absolute value of **a**.
The range of the function does not change when the sign of **a**
is changed. Now that we have seen the affect of changing **a**,
let's take a look at what happens to the graph when **b** is
changed, while **a** and **c** are left constant. Below
is the graph y = **a** sin (**b**x + **c**) where **b**
= 1/2, 1, and 2, **a** = 1, and **c** = 0.

As we can see from the graphs, changing **b**
affects the period. When 0 < **b** < 1, the graph "stretches"
and the period becomes larger. In this case, when **b** = 1/2,
the period doubled to 4p. When **b** > 1, the graph "squeezes"
and the period becomes shorter. In this case, when **b** =
2, the period became p. In general, the period of the graph is given by the
formula (2p
/ **b**). Note again that all three graphs have the point (0,0)
in common. What happens when we make **b** negative? Just as
changing **a** from positive to negative reflected the graph
across the x-axis, we can hypothesize that the same will hold
true when **b** is negative. Let's take a look at the graph
y = **a** sin (**b**x + **c**) where **b** = -1/2,
-1, and -2, **a** = 1,** **and **c** = 0.

Again, our hypothesis was correct. The periods
stayed the same when we changed **b** from positive to negative.
However, the graph reflected across the x-axis. So, we can further
generalize the formula for the period to (2p / |**b**|) where |**b**|
is the absolute value of **b**. So, what happens when we change
**c** while leaving **a** and **b** constant? Let's graph
y = **a** sin (**b**x + **c**) where **c** = -2, -1,
0, 1, and 2, **a** = 1, and **b** = 1.

When we change **c**, we change the phase
shift of the graph. In other words, we are shifting the graph
along the horizontal axis. Notice on the graph y = sin x, the
graph passes through the origin (0,0). Then the graph crosses
the x-axis at (p,0). When we make **c** positive, we move the graph
to the left **c** units. So the graph y = sin (x + 1) is shifted
1 unit to the left. It passes through (-1,0) and (p - 1,0). Likewise,
when we make **c** negative, we move the graph to the right
**c** units. So the graph y = sin (x - 1) is shifted 1 unit
to the right. It passes through (1,0) and (p + 1,0). Unlike **a**
and **b**, making **c** positive or negative does not reflect
it across the x-axis. It only shifts the graph left or right **c**
units.

Now that we have examined what happens to the
graph y = **a** sin (**b**x + **c**) when **a**, **b**,
and **c** are changed, is there any way to affect the vertical
change? What if we add a fourth variable, **d**, that gives
us the equation y = **a** sin (**b**x + **c**) + **d**?
Let's take a look at the graph y = **a** sin (**b**x + **c**)
+ **d** where **d** = -2, -1, 0, 1, and 2, **a** = 1,
**b** = 1, and **c** = 0.

As you can see from the graphs above, changing
**d** does, in fact, affect the vertical change. Making **d**
positive moves the graph up **d** units, while making **d**
negative moves the graph down **d** units.

To conclude, when examining the graph y = **a**
sin (**b**x + **c**), **a** affects the amplitude, **b**
affects the period, and **c** affects the phase shift. If we
add a fourth variable **d**, it affects the vertical change.