SOLUTION: You are interested in the maximum height the skier will reach after becoming airborne at (7,4). A series of photos of a jumper performing his jump shows the skier at the points (7,

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Question 975336: You are interested in the maximum height the skier will reach after becoming airborne at (7,4). A series of photos of a jumper performing his jump shows the skier at the points (7,4) (9,7) (14,5.75) and (15,5). Find the equation of this parabolic path in height vs time.
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
Two points determine a straight line.
Three points determine a parabolic path.
We have four points.
We could use some statistics software to determine the parabola that most closely fits the data.
We can also use three of those points to determine a quadratic function and see if the fourth point would fit.

USING THE FIRST THREE POINTS:
The parabolic path will be determine by a quadratic function of the form
y=ax%5E2%2Bbx%2Bc .
For (7,4) , substituting system%28x=7%2Cy=4%29 , gives us the equation
4=49a%2B7b%2Bc .
For (9,7) , substituting system%28x=9%2Cy=7%29 , gives us the equation
7=81a%2B9b%2Bc .
For (14,5.75) , substituting system%28x=14%2Cy=5.75%29 , gives us the equation
5.75=196a%2B14b%2Bc .
We can solve system%284=49a%2B7b%2Bc%2C7=81a%2B9b%2Bc%2C5.75=196a%2B14b%2Bc%29 to find a , b , and c .
We find system%28a=-0.25%2Cb=5.5%2Cc=-22.25%29 .
That gives us y=-0.25x%5E2%2B5.5x-22.25 as the equation for the parabolic path .
For that equation, the maximum would happen at
x=-b%2F2a=-5.5%2F%282%2A%28-0.25%29%29=-5.5%2F%28-0.5%29=11 ,
with y=-0.25%2A11%5E2%2B5.5%2A11-22.25=-30.25%2B60.5-22.25=8 ,
so the maximum height the skier would reach is estimated as 8 .
Also, for x=15 , the equation y=-0.25x%5E2%2B5.5x-22.25 would give us
y=-0.25%2A15%5E2%2B5.5%2A15-22.25+=-0.25%2A225%2B82.5-22.25=-56.25%2B82.5-22.25=4 , meaning that the skier would go through point (15.4) rather than 15,5).
Is there a typo, or did they mean for the four points not to fit perfectly into a quadratic function?

USING STATISTICS TO FIND THE BEST FIT:
Plugging points (7,4) (9,7) (14,5.75) and (15,5) into a spreadsheet a linear regression gives us a best-fit parabolic path as
y=-0.2096x%5E3%2B4.6872x-18.426 ,
which predicts a maximum of y=7.78%28rounded%29 at about x=11.18 .