# 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..