Completed by: Harsh Patel

Student Number: 643984

IB number:

Teacher: Mr. Persaud

Course Code: MHF4U7-C

Due Date: November 16th, 2012

Introduction

This report will investigate the winning heights of high jump gold medalists in the Olympics. The Olympics composed of several events evaluating physical strength of humans. Every Olympic game brings different numerical value for a height that a person can jump, allowing for different data collection. First of all, the data ranging from the years 1932 to 1980 will be portrayed on a graph for observation. From thorough investigation, the best curvature or line that fits the trend of the data will be selected to represent the relation. The function will be algebraically approached using the numerical values of the data points. Through the use of technology, the parameters will be plugged in to illustrate that function electronically. The function will be observed and any limitations with the model will be indicated. The function might need refinement according to any outliers/ anomalies, also to better represent the trend. The model function and a given function by the programme will be compared. Since the Olympics were not held during the years 1940 and 1944 due to WWII, this function will help determine the possible outcomes of the gold medal heights during those years. Also, a prediction of the years 1984 and futuristic year 2016 will be made according to the function created manually. Furthermore, the range of data will be extended to evaluate the effectiveness of the created function for extrapolated points. Analysis of the extrapolated points will be discussed with special reference of significant fluctuations. Lastly, the self created function that was modelled for the first set of data will be discussed for modifications in conjunction with the extrapolated data to create a more accurate representation of overall trend.

Considering Data

Table 1 Gold metal heights achieved at various Olympic Games.

Table 1 Through the use of Spreadsheet, a table is produced to represent the data points in a chart

In this case, time is the independent variable and the height is the dependant variable in Table 1. The height of the Gold medalist is measured in the scientific unit, centimetre (cm). The time is represented years as the Olympic Games are held every 4 years.

Plotting data points

Using the values from the data given in table 1, a visual representation is created using technology to plot the points into a graph (Figure 1).

Figure 1

Figure 1 shows the relation of the gold metal heights achieved during the years from 1932 to 1980 using Graphmatica.

Specific variables need to be set to represent the different unknown values in the graph. First of all, let t represent the year of Olympic Games. Subsequently, let h represent the height jumped by gold medalist.

The grid range for the graph (figure 1) begins from 1915 to 1995 on the horizontal axis and 185 to 245 on the vertical axis. A set domain and range needs to be created to determine the constriction of the graph. The specific domain for the data is {tϵℝ |1932 ≤ x ≤ 1980}) and accordingly, the range is {hϵℝ |197 ≤ y ≤ 236}.

The graph (figure 1) is based on the given data, where the x and y values are located a great distant away from the origin. This disables the understanding of where the graph is located in relation with the origin. However, the graph is chosen to be represented in this manner because it helps the viewer visualize the relation between the year and the height jumped.

Analysing to discover which function best models the behaviour of the graph

There are various mathematical functions that can accurately model the behaviour of the graph (figure 1). Functions with various curvatures are considered to discover which one would visually be the best to representation of the correlation in the data.

Parabola – general equation y= a (x- h) 2 + k

Figure 2

Figure 2 shows the shape of the quadratic function, y= x2

A model f(x) = x2 has a vertex point, and the values of y are successively increasing as the x value increases from the vertex. The slope of a quadratic function increases rapidly, causing this curve (figure 2) to not accurately represent the correlation of the data (figure 1).

Cubic function- general equation f (x) = ax3 +bx2 +cx + d