= P (x²-nx-mx+mn)
= P (x²+x(-n-m)+mn)
This formula will turn to a quadratic formula after solving it, and then it would be the equation for the parabola.
For example for the second parabola if you plot in the lowest and highest X coordinate in the bounce it should come out like this: = P (x-6) (x-12)
Then if you solve the equation it should turn out like: y= P (x²-18x+72)
Value P basically is the value that influence the curve so it’s a very important value in this equation. In order to find out P we have to take point from the table for example 9 then to find P we have to : f(9) =P (9²-18(9)+72=13.21
= P = 13.21/-9
= P = -1.46778(5d.p)
Therefore the equation should be y= -1.4677…..8 (x²-18x+72)
And if you type the equation on to the computer program it comes out to be basically on the points, it’s not all on the points because the value has been round to 5 d.p.:
Then for the third bounce, basically I use the same formula to find the equation for the curve so it turns out to be like: = P (x-12) (x-18)
= P (x²-30x+216)
f (15) = P (15-12) (15-18)
10.48 = P (3) (-3)
10.48/-9 = P
1.2(1d.p)= P
f (17) = P (17-12) (17-18)
7.43 = P (+5) (-1)
7.43/-5 = P
-1.486 = P
When you type the equation in to the computer program the curve comes out like this:
This tells us that the two point is only a approximate value of the equation, in my prediction according to the equation: f (17) = -1.486 (x²-30x+216) and f (15) = -1.2 (x²-30x+216) I predict that the curve will have a equation like : -1.165 (x²-30x+216). On the computer it appears as: ( the yellow line)
And for the little three points at the back the equation can be find by using the formula too and because there is only three points we can predict the value of M and N because there is a pattern to a bounce, it is landing basically every 6 second so the M and N values are : = P (x-18) (x-24)
= P (x²-42x+432)
To find P : f (20) = P (20²-840+432)
6.44/-8 = P
-0.805 = P
f (19)= P (19²-798+432)
2.89/-5= P
-0.578 = P
On the graph it occurs to be like:
I predict that the curve for the last little part of the graph should be approximately around y= -0.805 (x²-42x+432). But that is just a prediction because the information given is to less to make a more precise prediction.
In conclusion, according to my investigation the equations are quadratic formulas, and using the equation y= P (x-m) (x-n) you can find the quadratic form and if you plot the values in the quadratic form then you can find value P. then after that the quadratic formula of the curve or parabola will appear. Also the value of P and M and N can be changed to change the parabola or curve.