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..




	Home / Review Topics 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..