IB Math SL Type II Portfolio - BMI Index

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” value of the y-minimum on the model function is 14.96, I can subtract them to get 9.95 as the value of the horizontal shift.  Since I have to shift the function to the right, I must make the “c” value negative.  Now I can substitute all these variables into the general cosine function to get my model function for the data.  

My model function is:

f(x) = 3.22*cos(0.21x – 9.95) + 18.42      

As you can clearly see, the first model function fits the data almost perfectly, but it is very unrealistic for predicting BMI’s of women, as it just keeps rising and falling; so it is virtually useless for estimating the BMI of a 30-year-old woman in the US.  This is fairly obvious when looking at the cosine graph, as you can see that a 30-year-old woman has a lower BMI than a 20-year-old woman.  This is essentially impossible unless a woman became shorter and/or lost a lot of weight.  It would even be reasonable if a 30-year-old woman had the same BMI as when she was 20, but it is much more likely that she would have a

slightly increased BMI, at the least.  I believe

that a linear function will do a much more accurate job in predicting the BMI of women in the US, but only for several years after the age of 20; probably just enough to accurately predict the BMI of a 30-year-old woman in the US.

        According to my cosine model function, the predicted BMI for a 30-year-old woman in the US is:

   f(30) = 3.22*cos{(0.21)(30) – 9.95)} + 18.42      

   f(30) = 15.61

        If this function was used to predict the BMI of a 30-year-old woman, you would be implying that a 30-year-old woman has the same BMI as a 10-year-old; which is basically physically impossible.  

However, if the linear function was used to predict the age of a 30-year-old woman, I am positive that it will be more accurate and certainly reasonable.  The linear function I obtained from technology is:

f(x) = 0.4x + 13.52

f(30) = 0.4(30) +13.52

f(30) = 25.52

        

        A body mass index of 25.52 for a 30-year-old American female seems fairly reasonable, although maybe a little too high.  According to the BMI categories, anything 25 and above is considered overweight, but with the high rate of obesity in the US this seems like a reasonable median BMI for females.  

        

The table below gives the median BMI for females aged 2-20 in London, England for the years 1995-97.

                                     (Source: )

        

This BMI data for females aged 2-20 in London fits my model to a certain extent, as it also follows the same trend.  However, you can see from the graph that the data points from London are just slightly higher.  To fix this, a simple change in amplitude and vertical shift would be sufficient.  Well, the biggest limitation on my model is that you can’t accurately predict the BMI for all ages of women in the US.