Gold Medal Heights

By: Petrel Liu

Olympic game is a sport even that held internationally once every 4 years, different country will have athletes gather around in one place to compete each other in a specific sport. The purpose of this math portfolio is to consider the winning heights for the men’s high jump in the Olympic Games. This math portfolio will investigate the relationship between data points and functions of the graph. The function will be developed multiple times to have an accurate result. This math portfolio will be separated into 4 sections; each section has a further investigation in function. In the 1st section I will be graphing the data that is given and develop a function to support the graphed data. In the 2nd section I will be compare the developed function to the original data, and then improve my function based on the result. In the 3rd section an accurate function will be developed using technology, and I will compare the function of technology to my modeled function, also predicting and estimating the data that did not given using my modeled function. In the 4th section an investigation will be held to check if the functions that I have developed match the additional points, and a better function will be developed after the additional data have been added. The table below gives the height (in centimeters) achieved by the gold medalist at various Olympic games. Year| 1932| 1936| 1948| 1952| 1956| 1960| 1964| 1968| 1972| 1976| 1980| Height (cm)| 197| 203| 198| 204| 212| 216| 218| 224| 223| 225| 236|

There were no Olympic games on 1940 and 1944 due to the world war. Therefore the data point does not exist. During war time athletes will not be able to get together to compete due to the war. In 1948 the jumper’s height have decreased compare to the year of 1936, I assume that the high jumpers had lost their practicing time during the war. Therefore the data point will be lower than previous year. Section I:

In this section we will investigate the data that is given, the most accurate way to investigate a set of data is to graph the function and find the accurate equation to represent the graph. The equation will let you further investigate by identifying the behavior of the graph. This section will be separated into 2 steps the first is to graph the data and the 2nd step is to find a model function to represents the graph. Step 1: Graph the data

If the number is too big, a computational error would be big as well. In order to reduce the chances of error I scaled the numbers down to make it at the same ratio, so it will not affect the portion of the graph. | Equation| Variable| Example|

Year | Y= 1900 + 4(x)| Y= year X= Reduced number| 1932= 1900 + 4(8) X=8| Height | Y= 100(x)| Y= Height X= Reduced number| 192=100(1.97) X=1.97|

This is the data that has been scaled down followed by the equation that I listed above. Year| 8| 9| 12| 13| 14| 15| 16| 17| 18| 19| 20| Height (m)| 1.97| 2.03| 1.98| 2.04| 2.12| 2.16| 2.18| 2.24| 2.23| 2.25| 2.36|

Step2:Find an equation

The graph show above shows a structure that is generally increasing, however some point of the graph shows in a decreasing structure for example when the data point of x=9 to the data point of x=12 the graph is in a decreasing structure. The point of the graph is generally in an increasing structure therefore I will be using a linear function to model the data by finding the line of the best fit.

I know that my function will be in linear form which is y=ax+b I am giving the y value and x value according to the data. I will find the value of “a” and “b” by solving a system of equation that made up by 2 points that I choose from the data which is (8, 1.97) and (19, 2.25), the points are on the line of the best fit and it will show the general structure of the data. Set a system of equation:...