This portfolio is an investigation into how the median Body Mass Index of a girl will change as she ages. Body Mass Index (BMI) is a comparison between a person’s height (in meters) and weight (in kilograms) in order to determine whether one is overweight or underweight based on their height. The goal of this portfolio is to prove of disprove how BMI as a function of Age (years) for girls living in the USA in 2000 can be modeled using one or more mathematical equations. This data can be used for parents wanting to predict the change in BMI for their daughters or to compare their daughter’s BMI with the median BMI in the USA.

The equation used to measure BMI is:

The chart below shows the median BMI for girls of different ages in the United States in the year 2000:

The values “height (m)” and “weight (kg)” can be discarded when trying to analyze this data because it is not consistent that a girl will have a fixed weight to height or vice versa when they are a certain age. For example, the median BMI for a 10 year old girl is 16.80. With only the data provided, one cannot isolate either the height variable or the weight variable using the formula for calculating BMI.

Using the computer program TI InterActive!™ to graph a scatterplot using the data, the resultant graph is as follows:

One of the parameters of this data are that all x and y values are positive, because one cannot have a negative value for one’s weight, height or Age. The scope of this question can be extended to ages beyond the maximum value given by the data, and could conceivably not end until the maximum possible lifespan of a human female (unknown). If this data was a record of the extremes rather than the median BMI of a population, the y values could lower than the window setting of 12 for this graph, as well as much higher than the window setting of 24.

If a curve was fitted to this data, the domain should be: {}, those values being the lowest and highest ages given in the table of data.

The lowest y-value on the scatter plot is the data point at 5 years (15.20 BMI); the highest y-value on the scatter plot is the data point at 20 years (21.65 BMI).

Using those values, the range of a curve fitted to this data should be: {}

Judging from the distribution of data, the pattern created looks very much like that of a sinusoidal function. Based on this assumption, Finding the values of a, b, c and d in the function should result in a graph that fits the data points.

Assuming that the point 21.65 is the maximum of the sinusoidal curve and using 15.20 as the minimum, one can find the value for d using the formula:

d = 18.425

d = 18.4

The line of symmetry in a sinusoidal function is equal to its vertical displacement (d) value. Knowing that a sine function begins by curving up from the line of symmetry, the horizontal phase shift (c) value can be approximated by looking at what the x value is where the y value is ≈ 18.425 and the graph is curving up. From looking at the graph: