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IB Math SL Type II Portfolio

1/12/13

Ms. Rozell

Gold Medal Heights

Objective: To consider the winning height for the men’s high jump in the Olympic Games

This table shows the heights achieved by the gold medal winners in the High Jump in numerous Summer Olympic Games throughout the 20th century 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|

(Note: Olympic Games were not held in 1940 and 1944.)

Task One:

Graph 1: This graph shows the correlation between the year and the height of the gold medal winners in the Olympic. The x-axis on this graph presents the years that the data was collect while the y-axis presents the height of the gold winner of that current year. This graph was graphed using the program Graphical Analysis.

Constraints: During the years 1940 and 1944, the Olympic was cancelled due to World War II. Due to this, the data point does not continue according to its intervals of 4 years. It can be assumed that the high jumpers lost their practicing times during the World War, limiting their ability to improve on their skill, making the highest jumper in the year 1948 when the Olympic resumed lower than that of the previous year. Because of this the first two data points can be eliminated from the calculation of the equation due to the fact that the trend seems to start over at the year 1948.

Task Two:

The function that models the behavior of the graph the best is a linear function since the graph trends to correlate in a straight line as shown by the line of best fit.

Calculation: To calculate the line of best fit, I split data the data points in half as shown by the gray line, and I then chose a point to be the median of both sides as shown below.

Graph 2: Shows the midline splitting the data into two sections and the points that are chosen on both sides to represent the median.

An equation was then found that went through both points. This equation was then used as the line of best fit for the data. The equation y=mx+b was used to calculate the line of best fit since the trend of that data seemed to be linear.

Graph 3: Shows the two points in comparison to the rest of the data By using the coordinates of the two points, the equation of the line of best fit was found as shown below. Slope=(y2- y1) / (x2- x1)

Slope=208.84-228.24(1953.96-1973.28)

Slope=-19.4(-19.32)

Slope=1.0

With the slope, I now find the y-intercept of the line of best fit y=1.0x+b

208.84=1.0 1953.96+b

b= -1,745.12

y=1.0x-1,745.12

Task Three:

Graph 4: Shows the line of best fit as found by using the two median points

As seen in the graph shown above, the line of best fit goes through the middle of all the data. Since this line was calculated using the two points the line goes through both points. The first median point is an accurate representation of the left side of the data since the data points are in a straight line in correlation. However the second point was not quite accurate due to the fact that in years 1972 and 1976 there was not much improvement in the height of the high jumpers so the points that are used to calculate the median is quite off. This model does not work for the years after 1980 because there are limitations to how high a person can jump since in modern society we cannot yet overcome the forces of gravity. Since this model depicts that the years after 1980 there will be a steady increase in the height of the gold medal heights, this model is then false after about 10 years after 1980. This model also depicts that before the year 1952, there is a steady decline in the heights of the gold medal heights which is not true as seen in the years 1932 and 1936. This linear model is only valid in range of the years 1932 to about 1990 which is in the range of the data given.

Task Four:

Graph 5: Shows both models (linear and logarithmic) in...