22)+Design+an+experiment+that+will+produce+sinusoidal+results+and+model+it+with+an+equation

I liked this particular section because it was cool to be able to model the things that were happening around us in real life. It taught us how to show things that happened in nature as equations and how it all is a cycle. This section starts off difficult as there are a few different things you need to learn how to find before you can start making equations but once you learn the first steps it becomes easier as the steps are almost identical for each problem.

The equations we are using to model an experiment are Y=b*sin(k (x-c))+a or Y =b*cos (k (x-c))+a b= The amplitude of the Graph k= 2π/ the period a= vertical shift c=horizontal shift

In order to model our example we must first find the values of the 5 variables. To Find b: In order to find the amplitude of our equation we must first find the average y-values of the mins and maxs of the set of data. Once we have the average of all the mins and maxs we have to find the difference between the average min and max. We then divide that number by 2 and we have the amplitude.

To Find k: In order to find the period of our equation we must first find the average differences in x-values of the mins and maxs of the set of data. Once we have the average of all the mins and maxs that is our period. We then have to divide 2π or 360 by our period and then we have our K value.

To Find a: Once you have the average y-value for the max's you take that number and subtract the amplitude and this will give you the vertical shift.

To Find c: Depending on whether you chose to make your graph in radians or degrees you will find the horizontal shift by lining up a max as 0 on x axis.

Instructional Video [] A lifeguard reports the tides throughout the day to a weather station. The weather station receives the following data: (Military) || 0 || 6 || 12 || 18 || 24 || Average min=(3+2.9+2.8)/3=2.9 Average max=(6.2+6)/2=6.1 Amplitude=(6.1-2.9)=3.2 3.2/2=1.6 k=2π/12=( π /6) Vertical shift=6.1-1.6=4.5 Horizontal shift=6
 * __Example Question__**
 * Time
 * Height of water || 3 || 6.2 || 2.9 || 6 || 2.8 ||

Final Equation: Y=1.6*(cos(π /6)(x-6))+4.5

What will the height of the water be at 15 hours? Y=1.6*(cos(π /6)(x-6))+4.5 Y=1.6*(cos(π /6)(15-6))+4.5 Y=1.6*(cos(π /6)(9))+4.5 Y=1.6*(cos(1.5π)+4.5 Y=-0.0038+4.5  Y=4.5

What time will it be when the water is 5.5 ft tall? Y=1.6*(cos(π /6)(x-6))+4.5 5.5=1.6*(cos(π /6)(x-6))+4.5 1=1.6*(cos(π /6)(x-6)) .625=( cos(π /6)(x-6)) .8957=(π /6)(x-6) X=7.71 Bill created a pendulum and began swinging it. He recorded the distance from the starting point at the max and mins. Use the data below to find an equation to model the data. Find the: Amplitude: Period: Vertical shift: Horizontal shift: Equation:
 * __ Portfolio Question __**
 * Time (seconds) || .8 || 1.5 || 2.2 || 2.8 || 3.4 || 4 ||
 * Distance from middle (cm) || 6.8 || -6.2 || 5.9 || -5.6 || 5.1 || -4.7 ||

How far from the middle will the pendulum be after 3 seconds?

At what time will the pendulum be 4 cm from the middle?