21)+Given+a+set+of+data,+calculate+the+amplitude,+period,+and+shifts+then+model+the+data+with+an+equation.


 * For all working on this for the portfolio, i messed up and didn't do the right objective originally, so don't pay attention to the example problems in the video***

While __working__ with this chapter during the year, I found it very interesting to find the graph of a sine wave and to calculate it using new terms and values. Being able to use this for finding data with real world applicantions, such as average temperatures, heart beats and many other real world situations. This section wasnt relatively too hard to understand, but a few areas i found to be tough was remembering the change in period from __degrees__ and radians, along with remembering when the graph is stretched or is shrunken in the amplitude and the phase shift. __**Parent Equations**__:

__**Cosine**__: Y = a + b cos (k (t - c) algebraically find each variable) || the amplitude is always a postive number, can never be negetive. If b > 0, then the entire calculated graph is stretched vertically. If b < 0, then the graph is streched horizontally along the plane. || Take the highest maximum and the lowest minimum. Subtract the max from the min and then divide by 2. || horizontial axis of the graph. If "c" is positive, then the graph is shifted that many units to the right. If "c" is negative, then the graph is shifted that many units to the left. (Realize that "c" is already negative in the parent equation, so only having it be apositive number will change the sign. || Divide the phase shift by the period. For example, 2 cos (pi/4x - 4) - 7. Divide the 4 by pi/4 to get a phase shift of 16/pi. || consecutive relative maximums/minimums, before the progress of the graph is repeated (frequency). It also stretches or shrinks depending on the value of "k". If "k" is greater than 1, then the graph shrinks horizontally. If "k" is less than 1, then the graph stretches vertically. For __degrees__, the period is represented as 360/k, but in radians, the period is representedas 2 π/k. || Divide 360 or 2 π by "k" || calculated graph shifts that many points up vertically on the plane. If "a" is a negetive number,the graph shifts that many points down vertically on the plane. || Take the highest maximum and the lowest minimum. Add both the max and the min together and then divide that number by 2. ||
 * __Sine__**: Y = a + b sin (k (t - c))
 * Name || Variables in Equation || Meaning Within Equation || Caculations (How to
 * Amplitude || b || The amplitude is represented as "b" in the equation. *Note that
 * Phase Shift || c || The Phase Shift is represented as "c" in the equation. This is the shift along the
 * Period || kt || The period of the graph shows the range of x values between 2
 * Vertical Shift || a || The vertical shift is represented as "a" in the equation. If "a" is a positive number,the entire

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The lowest average temperature in January in Barcelona, Spain was 26 degrees. The highest average temperature for the year was 94 degrees in July. Calculate the amplitude, phase shift, vertical shift, and period for the past year. Remember that no month can equal 0 for this equation.
 * __Example Problem__**

1. Realize that the period is 12 months, and January is the first month and June is the sixth month because of the demand in the question. 2. Calculate the amplitude by taking the highest maximum and the lowest minimum.Then, subtract the max from the min and then divide by 2. (94-26 = 68 / 2 = 34) Amplitude is 34. 3. Calculate the vertical shift by taking the highest maximum and the lowest minimum and add both the max and the min together and then divide that number by 2. (94 + 26 = 120 / 2 = 60) The vertical shift is 60. 4. Next is the period. Since 12 months is the period, leave it was 360 (or 2π for radians) over 12 (360/12) 5. Lastly have the phase shift calculated. The easiest way to accomplish this is just have it written as X - 7 for a cosine equation to have January be x = 1 (first month) and July being the seventh month (x = 7). 6. Put all components together. Y = 34 cos (360/12(X - 7)) + 60

__**Video**__: []

The average temperature in Milan, Italy has gotten reletively hot within the last year. Last year alone, the temperature rose to record highs. The average temperature in January topped out at 40 degrees. In addition, the highest average temperature for the year clocked out at 100 degrees in July. Calculate the amplitude, phase shift, vertical shift, and period for the past year. Remember that no month can equal 0 for this equation.
 * __Portfolio Problem__**: