Math Assesment (Full Marks)

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IA Math Assessment

The Olympic games is a worldwide event held once every four years, the modern Olympics started at around the 19th century and has been going on ever since. The Olympic games however were not held in the years 1940 and 1944 due to the First World War. In this assessment our aim is to consider the winning height for the men’s high jump in the Olympic games and to derive a function to properly model the given data.

Year| Height |
1932| 197|
1936| 203|
1948| 198|
1952| 204|
1956| 212|
1960| 216|
1964| 218|
1968| 224|
1972| 223|
1976| 225|
1980| 236|

Some data in our findings will be excludable because this is a real life situation so any negative values will be excluded because you simply could not high jump or even jump a negative height because it is run real and very high numbers are excluded, only realistic positive numbers will be accepted in our results and our findings in this assessment. This is a table of the data obtained in the Olympic games in men’s high jump from 1932-1980 excluding 1940 and 1944 because the Olympic games were not held on those years.

This graph shows us the heights that the athletes had jumped during the Olympic games; it shows how the variations throughout the years and how it has a significant increase year by year throughout the heights. The graph looks like it is continually increasing each four years and one of the reasons there was a significant increase is because a new jump was formed in the 1980 Olympic games. Where the heights of the athletes are represented by the x-axis and the year in which the Olympic games had taken place is represented by the y-axis. We could also take the starting year we measured the data in to be 0 meaning that 1932=0

Judging by the looks of the graph we could exclude the linear function because it will clearly not model the graph but through my understanding of what graphs look like, I can state that this graph would most probably be best fitted by a quartic function.

Here are a few functions that might model this particular graph, There are a few functions that could model this graph but for now I have chosen the cubic function and the quartic functions because the cubic function has quite a few turning points and it seems like it will fit into the equation just right because of its shape and form Cubic:

y=ax3+bx2+cx+d
203=a(4)3+b(4)2+c4+d
198=a(162)+b(16)2+c16+d
204=a203+b202+c20+d
212=a(24)3+b(24)2+c24+d

In order for us to find the values of the unknown’s in these four equations we could conduct a matrix in order to achieve the values of a, b, c and d.

y=[-0.0028645833x3+0.234375x2+-4.141666667x+216]

After obtaining such an equation we could substitute values for x and find y in order to graph the function. The value of x represents the number of years after our first year, which was 1932 and the value of y represents the height that the gold medalist had jumped in the high jump in the Olympic games that year.

Here is the graph of the cubic function compared to that of our original graph, as we can see the cubic is very accurate at certain points yet it is very off limits at the starting point for example and from the years 1960 up until 1920 or from x=28 and x=48 the graph of our function was not even close to the original graph.

With the use of technology we can plot this function on a TI-84 calculator against the original heights of the athletes and determine the regression of this function, after we do this process we find the regression to be 0.930950582; the maximum regression for any function is 1 so having a regression like this does support the face that this is an accurate function of our original height.

Quartic:
y=ax4+bx3+cx2+dx+e
203=a(4)4+b(4)3+c(4)2+d4+e
198=a(16)4+b(16)3+c(16)2+d(16)+e
204= a(20)4+b(20)3+c(20)2+d20+e
212= a(24)4+b(24)3+c(24)2+d24+e
216= a(28)4+b(28)3+c(28)2+d28+e

In order to obtain the...
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