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This is Activity 10 from the EXPLORATIONS Book: TI InterActive! Math for High School

In this activity, students study the change in average temperature. They develop a model for average temperatures in their hometown. Students also develop an understanding of the sine and cosine functions and use them to model real world data.

Before the Activity

See the attached PDF file for detailed instructions for this activity

Print pages 83 - 85 from the attached PDF file for your class

During the Activity

Distribute the pages to the class.

Follow the Activity procedures:

Enter the Fahrenheit temperatures and months in spreadsheet

Define f(x) = a * sin(b * (x - c1) + d, g(x) = a * cos(b * (x - c2) + d, a = 1, b = 1, c1 = 0, c2 = 0, and d = 0

Graph the functions and find the maximum, minimum, and average temperature

Define d = (maximum temperature + minimum temperature)/2 and observe the effect on the graph

Observe that change in d shifts the graphs up to the middle of the data

Redefine amplitude a = (maximum temperature - minimum temperature)/2 and observe the effect on the graph

Observe that change in 'a' vertically stretches the graphs

Note that the period of the graph is 12 and redefine b = 2π/12

Observe that change in 'b' horizontally stretches the graphs

Note that horizontal shift for f(x) is c1 and for g(x) is c2

Adjust a, b, c1, and d to have a good fit and record the sine function that fits the data

Adjust a, b, c2, and d to have a good fit and record the cosine function that fits the data

Note that for this data set, d is approximately the average of the monthly temperatures, a is the temperature flux from the average, b is the number of months per year

Note that for this data set, c1 is the month that the temperature first reaches the average temperature for the year and c2 is the month that the temperature reaches the maximum temperature for the year

After the Activity

Students complete the Student Activity sheet.

Review student results:

As a class, discuss questions that appeared to be more challenging