Quadratic Equation from Coordinates?

May 2010
1
0
Hi guys, is there any way of figuring out an equation just using coordinates?
for example if you had the points: (32, 6), (48, 9) and (96, 18)
is there any way of manipulating those coordinates to form an accurate equation? (without trying to form a line of best fit and then using trial and error to get it as close as possible?)


thanks.
 

Prove It

MHF Helper
Aug 2008
12,883
4,999
Hi guys, is there any way of figuring out an equation just using coordinates?
for example if you had the points: (32, 6), (48, 9) and (96, 18)
is there any way of manipulating those coordinates to form an accurate equation? (without trying to form a line of best fit and then using trial and error to get it as close as possible?)


thanks.
Yes. Recall that a general quadratic is of the form

\(\displaystyle y = ax^2 + bx + c\).


Using the three points, you get

\(\displaystyle 6 = a(32)^2 + b(32) + c\)

\(\displaystyle 9 = a(48)^2 + b(48) + c\)

\(\displaystyle 18 = a(96)^2 + b(96) + c\).


Simplify each equation, then solve the three equations simultaneously for \(\displaystyle a, b, c\).