Math

Introduction
The purpose of this project is to create a model function in order to find the median body mass index of US women in the year 2000. In order to understand this project you will have to be able to understand and work properly with a cubic function (function in the form of), which will be used to represent each model equation created. You’ll also be working with matrices, which aregoing to be used to represent the linear equations. The formula used for matrices is: , where A represents the matrix of coefficients, x the matrix of variables, and c the matrix of constants. In order to make it visually understandable, graphs and tables are going to be included too. In order to be able to know how accurate my model is, I’ll be using the % change formula:

Body mass index (BMI)is a measurement of one’s body fat. It s calculated by taking one’s weight (kg) and dividing by the square of one’s height (m).
Age (yrs) | BMI |
2 | 16.40 |
3 | 15.70 |
4 | 15.30 |
5 | 15.20 |
6 | 15.21 |
7 | 15.40 |
8 | 15.80 |
9 | 16.30 |
10 | 16.80 |
11 | 17.50 |
12 | 18.18 |
13 | 18.70 |
14 | 19.36 |
15 | 19.88 |
16 | 20.40 |
17 | 20.85 |
18 | 21.22 |
19| 21.60 |
20 | 21.65 |
The table below gives the median BMI for females of different ages on the US in the year 2000

Using technology, plot the data points on a graph. Define all variables used and state any parameters clearly.



Graph A shows the median BMI for females of different ages in the US in the year 2000. In order to create this graph, I used the Grapher applicationin my computer. I plugged in the different x and y values that were given to me in Table 1. As you can see, the x-axis represents the females’ age in years and the y-axis their BMI. According to the outcome, after turning two years old, their BMI starts decreasing, and when they reach the age of 20, it continues to increase, but at a lower range.

What type of function models the behavior ofthe graph? Explain why you chose this function. Create an equation (a model) that fits the graph.
After looking carefully and analyzing Graph A, I came to the conclusion that it has the shape of a polynomial of degree three (cubic function): , where a is nonzero; and a, b, c, and d are real numbers. Since the curve is the other way around, you can conclude that the formula you’re facing isnegative. At the beginning the curve is decreasing, and when it reaches the point (5, 21), it starts to increase, creating the shape you’re able to see (can also represent a sinusoidal function).

In order to create my model equation, I chose four different points from Table 1: (5, 15.20), (10, 16.80), (15, 19.88), and (20, 21.65). I plugged in the x and y-values in the formula , solved it, and cameout with four new equations.

Equation 1 Equation 2



Equation 3 Equation 4



After getting the four equations, I needed to find the a, b, c, and d values. In order to do so, I used the matrices equation: , where A represents the matrix of coefficients, x the matrix of variables, and c the matrix of constants.

After plugging in the values, I came up with:Since I had to find the matrix of the variables, I had to ostracize x: . As you can see, according to the formula, in order to find the parameters, I had to find first the inverse of A, or . To do this, I plugged in the values in the graphing calculator.

Now that I have its inverse, I have to multiply it times the y-values.

After finding the value of each of my parameters, I was able to createmy model function (taking into account that there might be some rounding errors):

On a new set of axes, draw your model function and the original graph. Comment on any differences. Refine your model if necessary.





Graph B shows both, my model and original equations. According to the graph, and taking into account the...