7.6 Phase Shift; Sinusoidal Curve

Fitting

The graphs of the sine and cosine functions are called sinusoidal graphs.

Theorem

For the graphs of

Find the amplitude, period, and phase shift of the following function. Graph the function.

(0,0)

-11

23

Steps for Fitting Data to Sine Function y = Asin(wx - f)+B

• Determine the amplitude A = largest data value -

2smallest data value

• Determine the vertical shift B = largest data value +

2 smallest data value

• Determine w from w=2p/T.• Determine the horizontal shift f by choosing an

ordered pair (x, y) and solving the equation y = Asin(wx - f)+B

The data on the following slide represent the average monthly temperature in Houston, Texas.

(a) Draw a scatter diagram of the data treating month as the independent variable.

(b) Find a sinusoidal function that fits the data.

(c) Graph the function found in (b) on your scatter diagram.

(d) Find the sinusoidal function of best fit.

(e) Graph the function found in (d) on the scatter diagram.

January 50February 54

March 61April 68May 75June 80July 83

August 82September 78

October 70November 61December 54

Month Average Temp (oF)

45

50

55

60

65

70

75

80

85

0 1 2 3 4 5 6 7 8 9 10 11 12 13

Month

Tem

pera

ture

We will find a sinusoidal function of the form

STEP 1: Determine the amplitude, A.

STEP 2: Determine the vertical shift, B.

Vertical Shift

STEP 3: Determine .

STEP 4: Determine the horizontal shift.

Use the first data x =1, y =50.

To use a graphing utility to find the sine function of best fit for the given data we will use the SINe REGression program to get the following: