IB Math SL Type II Portfolio - BMI Index

IB Math SL Type II Portfolio

January 14, 2010

In this portfolio, we will look at the median BMI indexes for females between the ages of two and twenty in the US in the year 2000; and from this data, a model function will be determined.  Body mass index (BMI) is a measure of a person’s body fat, and is calculated by this formula:

Looking at the data table we can see that there are two variables:  one being the age (in years), and the other being the BMI.  The independent variable is the age and the dependent variable is the BMI, since the BMI’s of the females depend on what age they are.  The parameters are the weight and height of the females, which both directly affect the BMI.  After using software to graph the data points, I believe the graph’s behavior is best modeled by the cosine function, since it has a wave-like shape and is periodic.  Next, I will create an equation that fits the graph by examining the parts that make up a cosine function.  First, let’s take a look at the general cosine function:  f(x) = a*cos(bx + c) + d.  To find the amplitude (a), I will simply take the y-maximum (21.65) and subtract the y-minimum (15.20) and then divide by 2.  This gives me 3.22 for the amplitude.  Then to find the period I will have to see the length of one cycle, which is when the function goes from the maximum to the minimum and back to the maximum.  By looking at the graph, I determine that one cycle is 30.  I then take 2 and divide by 30 to get the period, since there are 2 radians in one cycle.  This gives me 0.21 for the period.  Then to find the vertical shift I will simply add the amplitude to the y-minimum:

3.22 + 15.20 = 18.42.  Finally, to find the horizontal shift (c) I must see how much I need to shift the graph so that it is aligned with the model function.  I can do this by entering the model function I have created, leaving out the “c” value, into a graphing calculator.  Then I can compare it to the scatter plot and determine how much I have to shift the function, by looking at the “x” values of the y-minimums of both curves.  Since the “x” value of the y-minimum on the scatter plot is 5 and the “x” ...