Expert: Sherry Wallin - 4/22/2009

Question
Hello.  
I am trying to produce a spreadsheet that will calculate to peak of a parabolic curve (convex up) using 4 points.  A previous (unknown) employee of my firm had done one using three points, but for this exercise i need to be able to use four points.  Is there a way of expanding the three point formula to include the fourth point.  Any help would be much appreciated.  Thanks

Answer
Hi Philip~
    Yes there is a way to use 4 points to generate the equation of a parabola, although it is not necessary. The general/standard equation of a parabola is y = ax^2 + bx + c. Note there are two variables and 3 coefficients/constants. Suppose for example the points (1,2) and (-1,2), and (0,0) are known to be on the parabola. Plug in each and see what you get: 2 = a + b + c; 2 = a - b + c; 0 = 0
2 = a + b + c
2 = a - b + c
----------------   add the two equations
4 = 2a + 2c    ->  divide all by 2
2 = a + c but from 0 = a*0^2 + b*0 + c we get the point (0,0) so we know that 2 = a + 0 or a = 2. Knowing that a = 2 and c = 0 we have
y = 2x^2 + bx. Again we know that (1,2) is on the graph so we have
2 = 2*1^2 + b -> b = 0 so our equation is that y = 2x^2.
Notice I didn't use a fourth point however you could have as many points as you want as long as they all satisfy the equation when all is said and done.

I may not have answered exactly what you are looking for but perhaps this will serve as an exercise for you to see if you can make 4 points work.

Food for thought. You know that 2 points determine a line so certainly you can show that 3 points are on the line, right? And how? You simply show that the 3rd point satisfies the equation. Likewise with a 2nd degree equation (1 more degree than a line), you need 3 points to determine the parabola but any 3 of the 4 will generate the equation you are looking for providing all the points are on the parabola.

Math Prof