H Pre Calc

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15 October, 2014

Word Count: 979

Gold Medal Heights

Over a period of 112 years, the height for the winning men's high jump, in the Olympic Games, has varied. In this paper, I will find three equations, each getting closer to the exact line, in order to be able to predict what the next years' winning jump height will most likely be. By using hand written equations, scatter plots and special formulas in the calculator each of these equations will be formed.

In the beginning, we were given many dates and various heights in centimeters. I chose the year 1932 to be my “zero”. This table shows years the Olympics were held and the winning height of that year.

Year (1932 is 0)

0

4

16

20

24

28

32

36

40

44

48

Height (cm)

197

203

198

204

212

216

218

224

223

225

236

With these points we were told to create a scatter plot.

This scatter plot shows that the dependent variable is the height and the independent variable is the year. I chose the cubic equation to be my first line that will best fit the graph because the lines created from a cubic equation seem like they would hit many of the points. The two points I chose to start at were (20, 204) and (32, 218). To create the cubic equation, I used the point (20, 204) to be x and y. I also used the point (32, 218) to be h and k. I started with the equation f(x) = a(x-h) ^3 +k. with these points the equation looked like 204 = a(20-32)^3 +218. Once, a is found, .0081, it needed to be formed on a graph. When this equation was plugged into a graphing calculator, or in this case Excel, the graph with the scattered points and the first line looks like this. I did not play with h or k in this graph to show that it was rough and that there was room for improvement. However, there are limitations to this graph: the y-axis is very large so the points are not distinct and it is not exact because it was not made in a graphing calculator.

Since the first equation was well formulated, I decided to try a cube root equation next. This equation started with f(x) = a (x-h)^(1/3)+ k. I used the same two starting points as equation 1: (20, 204) as x and y and (32, 218) as h and k. The equation turned out to be f(x) = 6.69506(x-19)^(1/3) + 202. I was not happy with the original line when I first plugged it in with the points from the scatter plot. So, I decided to change a, h, and k to see how close I could make the line. I began changing h to 24 through 33 to see where the first curve would end up. Then I changed k only slightly just to see what happens and I decreased a to make the line grow. I ended up with 21 as h, 206 as k, 5.32566 as a. Therefore having my final equation be f(x) = 5.32566 (x-21)^(1/3)+206. The graph for this line with the scatter plot points in as well looks like so.

This graph differs from the first graph because the y-axis is more narrow and shows precise points. Although the first equation looked closer since the y-axis was not so narrow, equation 2 shows a better fitting line that could possibly predict the next years height better than the equation before.

For the final equation, I was given 13 more sets of data. So all of the data combined is in the table below.

Year

-36

-28

-24

-20

-12

-4

52

56

60

64

68

Height (cm)

190

180

191

193

193

194

235

238

234

239

235

72