1. Phase Shifts of Sinusoidal Graphs

2. We will graph sine functions of the following form: The amplitude A =  A  The period T = 22  The phase shift comes from setting what is in the parenthesis equal to 0 and solving it.  x -  = 0 x =   The phase shift =  

3. Steps for graphing: Determine the amplitude Determine the period Mark 4 intervals with last being the period Determine the phase shift Put a point for max, zero or min for each 1/4 Shift each point by the phase shift

4. Steps for graphing: Determine the amplitude Determine the period Mark 4 intervals with last being the period A = 3 Determine the phase shift Mark max, zero or min for each 1/4  Period is , so each interval is  /4 sine would start at (0,0) but is shifted right by Shift each interval by the phase shift

5. Curve Fitting If the scatter diagram of observed data looks like a sinusoidal function, we'll use the following sine function to model the observed data: We'll need to determine A, ,  and B. Let's look at some data and go through the steps of how to find the constants.

6. Month, x Ave Temp °F Jan, 129.7 Feb, 233.4 Mar, 339.0 Apr, 448.2 May, 557.2 Jun, 666.9 Jul, 773.5 Aug, 871.4 Sep, 962.3 Oct, 1051.4 Nov, 1139.0 Dec, 1231.0 Here is the data for average monthly temperatures in Denver, Colorado. Average over many years does not vary much from year to year so will repeat each year. The scatter plot shows 2 years. We'll build a model using:

7. How to find A Month, x Ave Temp °F Jan, 129.7 Feb, 233.4 Mar, 339.0 Apr, 448.2 May, 557.2 Jun, 666.9 Jul, 773.5 Aug, 871.4 Sep, 962.3 Oct, 1051.4 Nov, 1139.0 Dec, 1231.0 21.9 We have the amplitude but need to vertically shift so we'll now find B.

8. How to find B Month, x Ave Temp °F Jan, 129.7 Feb, 233.4 Mar, 339.0 Apr, 448.2 May, 557.2 Jun, 666.9 Jul, 773.5 Aug, 871.4 Sep, 962.3 Oct, 1051.4 Nov, 1139.0 Dec, 1231.0 21.9 Good vertically but we see that we also need a horizontal stretch since the period isn't right so we'll now find .

9. How to find  Month, x Ave Temp °F Jan, 129.7 Feb, 233.4 Mar, 339.0 Apr, 448.2 May, 557.2 Jun, 666.9 Jul, 773.5 Aug, 871.4 Sep, 962.3 Oct, 1051.4 Nov, 1139.0 Dec, 1231.0 21.9 Okay---so the period looks right but we need a horizontal phase shift so we'll find . Since this cycles yearly we know the period is 12.

10. How to find  Month, x Ave Temp °F Jan, 129.7 Feb, 233.4 Mar, 339.0 Apr, 448.2 May, 557.2 Jun, 666.9 Jul, 773.5 Aug, 871.4 Sep, 962.3 Oct, 1051.4 Nov, 1139.0 Dec, 1231.0 We'll substitute in a known x and y and solve for  in the equation we've built so far. Subtract this Divide by this What angle has a sine value of -1? This angle must equal

11. Month, x Ave Temp °F Jan, 129.7 Feb, 233.4 Mar, 339.0 Apr, 448.2 May, 557.2 Jun, 666.9 Jul, 773.5 Aug, 871.4 Sep, 962.3 Oct, 1051.4 Nov, 1139.0 Dec, 1231.0 21.9 Hey---Not Bad!