Here we will examine the graphs of y=a sin(bx + c) for different values of **a**, **b**, and **c**.
Let us first look at the effects of changing **a** while **b** and **c** remain constant.

We can see that the amplitudes are changing, if we look more
closely at the graphs of y= 1 sin (1x + 1) and 4 sin (1x + 1) we will observe the following.

Notice that the amplitude is
greater with the larger value of **a**, and corresponds with its value on the
y-axis. So, what might we expect
to see with negative values for **a**?

For negative values of **a**, the
amplitude is negative and still corresponds with its value on the y-axis. From these observations we can conclude
that the change in **a** determines the type of vertical
expansion of the function.

Now let us observe the effects of changes in **b** on the
graph.

Notice the graphs have
different periods. If we look at
the graphs when **a** and **c** are constant and **b**=1, 4, and ½ we can take a closer look at how the periods are changing with
the change in **b**.

The change in the period can
be determined with the formula **, therefore we can see why ****0 < b > 1** would yield a horizontal
expansion (shown when b = ½) and **b > 1** would yield a horizontal compression.

Lastly, letŐs observe changes
in **c**.

Here we can see that the
graphs are translated horizontally.
If we take a closer look

We can see that the graph
shifts to the left for **c>0**,

and shifts to the right for **c<0**.

Based on the observations we
have made what do you think would happen if we changed the original equation to y=a sin(bx + c) +
d. We have already demonstrated
changes in amplitude, period, and horizontal expansion/compression, so we could
logically assume that changes in **d** will result in a vertical
expansion/compression.

LetŐs see.

We can see that when **d>0** the graph
shifts up,

and for values of **d<0** the graph shifts down.

So for

**y=a
sin(bx + c) + d**

|a| = amplitude

** **= phase
shift

c
= horizontal expansion/compression

d
= vertical expansion/compression