# Mathematic SL IA -Gold medal height (scored 16 out of 20)

by rucia (student)

Mathematics Standard Level Portfolio

Type 2- gold medal height

Candidate name: Sun Ha (Rucia). Park

Candidate number:

School: Beijing No.55 High School

Content

Introduction

Introduction

Olympic Game is a variety competition of various athletics. It is composed twice a year, summer and winter, and it is hold once four year. Because there are many athletics competitions, it’s a great opportunity to take numerical data. In this task, I will investigate the winning height for the men’s high jump in the Olympic Games. The high jump is a track and field athletics plays in which athletes jump over a horizontal bar placed at measured heights without any aid of certain devices.

I m going to use Microsoft Excel and the graphing calculator TI-84 to collect and present the data for analysis and investigation, and find out the best –fit functional curve to the original data.

The table has given the original numerical data of the height achieved by the gold medalists at various Olympic Games.

Depending on the data, I will investigate this case by setting time of Olympic for my independent variable, and setting the winning height for my dependent variable. In graphs I will show, I let x denote time of Olympic and y denote the winning height. The domain for the data is {

|1932 ≤ x ≤ 1980} and the range for the data is {

|197 ≤ y ≤ 236}.

Before graphing the data, note that the Olympic Games were not held in year 1940 and 1944 because of the wars. In addition, I will change the year values to smaller values by using the 4 years gap between the times of Olympic. I was going to set 0 value for the first data, year 1932, and plus 4 every next data, but if I do so, then the value of the times of Olympic before year 1932 will become negative number, so I decided to simplify the values from the start year of Olympic, 1896, as my starting point, 0. Then my graph will be drawn by using the data:

Here is a graph showing the data;

Figure 1 winning gold medal height in men’s high jump from1932 to 1980

These smaller values for year will make the constant values lesser in equations I am going to investigate. There are two constraints of the task. One is that the height cannot be lesser than or equal to 0, because this task is investigating the winning height of the men’s high jump in Olympic Games. The height cannot be negative number, and if the height is 0, then it means the athletes didn’t jump, and remained on the ground. The other one is that the time of Olympic Games holding cannot be before than 1896. This is because the year 1896 is the first time the Olympic was composed.

There are many types of mathematical function we have studied. There are sine, cosine functions, linear function, quadratic function, exponential function, log function, and polynomial functions. I am going to try to investigate the task by using characteristics of each type of function and find out the best-fit function model.

First of all, I will link all the points together to show a lined curve

Figure 2 the curve of linking all data points of winning gold medal heights.

The graph shows an upward sloping curve which is very irregular.

Let’s figure out best-fit function models;

Figure 3 Sine curve, y=sinx             Figure 4 cosine curve, y=cosx

To be a sine or cosine curve, the curve should have periodicity, and fixed minimum and maximum. According to the characteristics of sine and cosine functions, because the curve (figure 2) doesn’t have any periodicity and it also doesn’t have fixed maximum and minimum, therefore we cannot investigate the best-fit function model by considering sine and cosine functions.

Figure 5 the graph of a quadratic function, y=(x-4)2

The shape of the quadratic function graph is called parabola. It has an axis of symmetry, and the point where the curve turns is called vertex. Also, if the graph is concave-up, then it will have minimum point, and if the graph is concave-down, then the graph will have maximum point.

Depending on the traits of the quadratic function, and since the curve (figure 2) is up-ward sloping, it cannot be interpreted in the quadratic function form without any characteristic of the quadratic function. Thus, the quadratic function is also, not a way to investigate the task.

Let’s consider linear function;

Figure 6 the graph of a linear equation, y=x

The linear equation has a line graph. The general form of linear equation is y=ax+b. In the equation, ‘a’ is a constant which is called gradient. Linear equation can have both upward and downward sloping graphs when ‘a’ is greater than 0, and ‘a’ is lesser than 0 respectively.

However, no matter what value do x and y have, the gradient ‘a’ is the same in a linear function.

Figure 7 the graph of winning gold medal heights from year 1932 to 1980

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