Precalculus 2/17/10

Finding Sinusoidal Functions by Hand

To write a sinusoidal functions by hand you must do four things:

1) Find the sinusoidal axis by finding the average of the highest and lowest data value

2) Find the amplitude by subtracting the highest data value by the sinusoidal axis

3)Determine the period using the formula

4)Determine any phase shift

In class we used the example of average tempertures per month from the city of Lyttelton, South Africa. The data values were: January-71, February-69, March-68, April-62, May-57, June-51, July-51, August-57, September-62, October-66, November-68, and December-69.

Finding the sinusoidal function:

1) Sinusoidal axis= 37.5

Highest data value is 64, and the lowest data value is 11. The average of 64 and 11 is 37.5

2) Amplitude= 26.5

The difference between the highest data value (64) and the sinusoidal axis (37.5) is 26.5

64-37.5= 26.5

3) Period is:

because the pattern repeats every 12 months and when we put 12 in as the variable B in the formula for the period, it reduces to this:

4) Phase shift= (x-7) because when the data values are put into a scatterplot with the Month as the x-axis adn Temperature as the y-axis, it is shaped like a cosine graph that has been shifted 7 to the right.

So once you have all of those pieces you can plug them into the equation of a sinusoidal function with A representin the amplitude, B representing the period, C representing the phase shift, and D representing the sinusoidal axis.

This is the equation of the sinusoidal function of the monthly teperatures of Lyttelton, South Africa:

This is the sinusoidal function graph of the temperature of Lyttelton, South Africa