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# Transformations
of the Graphs of the Sine and Cosine

### Vertical Shift

The graph of _{ }can be obtained by shifting
the graph of

graph of vertically units. If
is
positive, the graph is shifted units upward and
if is negative, the
graph is shifted_{ } units downward.
The number is called the vertical shift.

### Amplitude

The graph of is the same as
the graph of stretched vertically
by a factor of , if. If , then the graph is
stretched and reflected across the horizontal axis. The number is called the
amplitude of the function. Note that the amplitude is always positive.

### Phase Shift

The graph of _{ }can be obtained by shifting
the graph of

graph of horizontally units. If is positive, the
graph is shifted_{ } units right and
if_{ } is negative, the
graph is shifted units left. The
number is called the phase shift.

### Period

The graph of _{ }can be obtained by horizontally
stretching or shrinking the graph of . A horizontal shrink
by a factor of k occurs if k >1 and a horizontal shrink by a
factor of k occurs if k < 1. Since the sine function has period
, the function
completes one
cycle as varies from 0
to . That is, for . Solving this
inequality for_{ }, we
get . So this function completes
one cycle as t varies from 0 toand has period .

Note: all of the above also can be applied
to the cosine function.

## Summary

The sine and cosine
functions

have amplitude

, period , phase shift
and vertical shift .

One complete cycle of
the graph occurs on the interval _{.}

### Example

Find the amplitude,
period, phase shift and vertical shift of the function

.

Graph one complete cycle.

### Solution

Comparing our problem
to the general form , we see that

a = 1, b = 3, k = 2, and . This tells us that the amplitude is
3, the period is , the phase shift is , and the vertical shift is 1. One complete
cycle of the graph occurs on the interval One complete cycle of the graph is
shown in Figure 1.

Figure 1

The __interactive examples__
below allow you to see more graphs of for different values of
the constants a, b, c and k..