1. The General

2. What happens to the graph of a sine function if we put a coefficient on the x. y = sin 2x y = sin x It makes the graph "cycle" twice as fast. It does one complete cycle in half the time so the period becomes .

3. What do you think will happen to the graph if we put a fraction in front? y = sin 1/2 xy = sin x The period for one complete cycle is twice as long or 4  22 44

4. So if we look at y = sin Bx the B affects the period. The period P = This will be true for cosine as well. What is the period of y = cos 4x? This means the graph will "cycle" every  /2 or 4 times as often y = cos 4x y = cos x

5. What will the graph look like if we put a multiple of  as the coefficient? y = sin 2  x The period for one complete cycle is y = sin x 111

6. absolute value of A is the amplitude Period is 2  divided by B Phase shift is C when in the form x – C Vertical translation is D

7. 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 C Shift each point vertically by D amplitude = absolute value of A period = 2  /B divide the period by 4 to know how long the intervals are phase shift = C The function must be written in the form shown to use this. If it is not, you can algebraically modify it.

8. 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 and up 1 Shift each interval by the phase shift Factor out a 2 from the stuff in parenthesis to get the right form Shift each point vertically by D D = 1

9. Many physical phenomena can be modeled with sine waves. the swinging of a pendulum radio and television waves light and sound waves These are often described in terms of how often they cycle. This is called the frequency and is the reciprocal of the period.

10. 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, B, C and D. Let's look at some data and go through the steps of how to find the constants.

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 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:

12. 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 D.

13. How to find D 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 B.

14. 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 Okay---so the period looks right but we need a horizontal phase shift so we'll find C Since this cycles yearly we know the period is 12.

15. How to find C 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 C 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

16. 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! So how does our curve fit the data?