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# models and graphs relating the Body Mass Index for females to their ages in the US in the year 2000

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Introduction

Anh Nhu Vu

IB

Mathemathics Standard Level 2008

Maths Coursework

This coursework will explore models and graphs relating the Body Mass Index for females to their ages in the US in the year 2000. The ages and the corresponding BMI numbers are variables and as we generate the function that models the behaviour of the graph later on, the parameters are the values of a, b, c and d in the formula of the function.

When the data points are plotted on a graph, it is interesting to see what kind of graph these points are forming. The graph BMI appears to resemble the graph of a trigonometric function. If we base on the function y=sinx to develop our model function, the type of function that models the behaviour of the graph is:

y1=a×sin [b(x+c)] +d

These are the reasons why I chose this type of function:

Primarily, the shape of the graph resembles that of the graph of the function y=sinx.

However, in comparison to the graph of the function y=sinx, the data points of the graph BMI indicates that the processes of transformation have been done:

_Vertical and horizontal stretch (the scales are represented by a and b)

_Vertical and horizontal translation (the scales are represented by c and d)

Middle

20-7.5=12.5 places and it is shifted vertically upward: 21.65-3.225=18.425 places.

Therefore we can fully develop our model function:

y1=3.225× sin [ (x-12.5)] +18.425

The graph of this function is presented below: Having developed an equation that is deduced to fit the original graph BMI, it is interesting to see how my model graph fits the points on the original graph BMI as well as to see the differences. There are clear slight differences between the graph Y1 and the graph BMI. We can see that the graph Y1 is an approximation of the original data. The table below shows how the values of y corresponding with the same values of x vary between the two graphs.

 x y value (BMI) y value  (Y1) 2 16.4 15.815 3 15.7 15.487 4 15.3 15.27 5 15.2 15.2 6 15.21 15.27 7 15.4 15.478 8 15.8 15.815 9 16.3 16.267 10 16.8 16.812 11 17.5 17.428 12 18.18 18.087 13 18.7 18.762 14 19.36 19.421 15 19.88 20.037 16 20.4 20.582 17 20.85 21.034 18 21.22 21.371 19 21.60 21.579 20 21.65 21.65

These differences can

Conclusion

Having said that, it is interesting to see some other BMI-for-age percentiles graphs that look similar to our original data BMI.

This is the data on the BMI related to ages for girls from the ages of 2 to 20 years.

SOURCE: Developed by the National Center for Health Statistics in collaboration with the National Center for Chronic Disease Prevention and Health Promotion (2000). If we go through the process of transformation again, we could also develop functions that model the behaviours of these graphs. However, it is unlikely that we can use the graph to estimate further feature of the data.

In conclusion, mathematical models in some cases can hardly be used to explore the natural tendency of certain real situations; in this case the BMI data. We can generate functions that model some behaviour of some data, but the actual data will not follow the rule of mathematical graphs. This fact is most obvious in the case of the BMI data of the females in Lebanon, in which it is very difficult to develop a function that fits the data well.

Bibliography:

http://www.emro.who.int/Publications/EMHJ/1302/article21.htm

http://www.health.vic.gov.au/childhealthrecord/growth_details/chart_girls4.htm

This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.

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