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# Graphing Sine and Cosine Help (page 2)

based on 1 rating
By McGraw-Hill Professional
Updated on Oct 4, 2011

#### Examples

The dashed graph in Figures 13.15 through 13.18 is one period of the graph of f ( x ) = sin x , and the solid graphs are transformations. Match the equations below with their graphs.

Fig. 13.16

Fig. 13.17

Fig. 13.18

• y = 3 sin( x + π /3)
• The graph of this function is vertically stretched by a factor of 3, so we will look for a graph whose y values lie between −3 and 3. The graph will also be shifted to the left by π /3 units. The graph for this function is shown in Figure 13.16 .

• y = 3 sin( xπ /6)
• The graph of this function is also vertically stretched by a factor of three, but it is shifted to the right by π /6 units. The graph for this function is shown in Figure 13.18 .

• The graph of this function is vertically compressed by a factor of 1/2, so we will look for a graph whose y values are between −1/2 and 1/2. The graph will also be shifted to the left by π/2 units. The graph for this function is shown in Figure 13.17.

• y = 2sin( xπ /4)
• The graph of this function is vertically stretched by a factor of 2, so we will look for a graph whose y values are between −2 and 2. It will also be shifted to the right π /4 units. The graph for this function is shown in Figure 13.15.

## Amplitude, Period, and Phase Shift

Transformations of the graphs of sine and cosine have names. The amplitude is the degree of vertical stretching or compressing. The horizontal shift is called the phase shift . Horizontal stretching or compressing changes the length of the period. For functions of the form y = a sin k (xb) and y = a cos k (xb), | a | is the graph’s amplitude, b is its phase shift, and 2 π / k is its period.

#### Examples

Find the amplitude, period, and phase shift.

• y = −4 sin 2( xπ /3)
• The amplitude is | a | = | −4 | = 4, the period is 2 π / k = 2 π /2 = π , and the phase shift is b = π /3.

• y = −cos( x + π /2)
• The amplitude is | a | = | −1 | = 1, the period is 2 π / k = 2 π /1 = 2 π , and the phase shift is b = − π /2.

• The amplitude is |1/2| = 1/2. In order for us to find k and b for the period and phase shift, we need to write the function in the form y = a cos k ( xb ). We need to factor 2 from 2 x + 2 π /3.

The function can be written as cos 2( x + π /3). The period is 2 π / k = 2 π /2 = π , and the phase shift is k = − π /3.

## Sketching the Graphs of Sine and Cosine

We can sketch one period of the graphs of sine and cosine or any of their transformations by plotting five key points. These points for y = sin x and y = cos x are x = 0, π /2, π , 3 π /2 and 2 π . These points are the x -intercepts and the vertices (where y = 1 or − 1). For the functions y = a sin k ( xb ) and y = a cos k ( xb ), these points are shifted to b,

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