   SEARCH HOME Math Central Quandaries & Queries  Question from Tiffany, a student: Find the equation of a parabola that passes through the points (-2,3), (-1,1) and (1,9) Hi Tiffany.

You can do this using simultaneous equations. Assuming the parabola is up-down, it has the form:

y = Ax2 + Bx + C

where A, B and C are the co-efficients.

So you can substitute in (x, y) three times and you will have three equations with three unknowns:

For the first point (-2, 3) :
3 = A(-2)2 + B(-2) + C

this reduces to
3 = 4A - 2B + C

solve for C and we have:
C = 2B - 4A + 3 (call this equation i)

Now for the second point (-1, 1):
1 = A(-1)2 + B(-1) + C
C = B - A + 1 (call this equation ii)

Before we do the third point, let's combine these equations, eliminating C:
2B - 4A + 3 = B - A + 1
and solve for B this time:
2B - B = -A + 1 + 4A - 3
B = 3A - 2 (call this equation iii)

Okay, now let's do the third point (1, 9) and solve for B
9 = A(1)2 + B(1) + C
B = 9 - A - C (call this equation iv)

but remember that we can use equation i to substitute for C in equation iv:
B = 9 - A - (2B - 4A + 3)
B = 6 + 3A - 2B
3B = 6 + 3A
B = 2 + A (call this equation v)

Now we can combine equations iii and v, because both equal B and have only A's on the other side:
3A - 2 = 2 + A
2A = 4
A = 2

So we've solve for A. Now use that in equation v to find B:
B = 2 + (2)
B = 4.

Now use A and B in equation ii to find C:
C = (4) - (2) + 1
C = 3.

So our answer should be this: y = 2x2 + 4x + 3.

Let's test the answer by making sure it works for each of the points. Put the value of x into the equation and see if you get the right value for y:

(-2, 3) test:
2(-2)2 + 4(-2) + 3
= 8 - 8 + 3
= 3. Good.

(-1, 1) test:
2(-1)2 + 4(-1) + 3
= 2 - 4 + 3
= 1. Good.

(1, 9) test:
2(1)2 + 4(1) + 3
= 2 + 4 + 3
= 9. Good.

Tiffany, you can notice that if we had instead assumed that the parabola opened left or right, we would have used the same steps, but following the model x = Ay2 + By + C. We would have got different values for A, B and C, but it would have also satisfied the question. So both answers would be acceptable. Notice, though, that if two points were on the same vertical line (same x value), then the parabola would HAVE to be right-left opening (ie. x = Ay2 + By + C). If two points were on the same horizontal line, it would HAVE to be up-down opening (y = Ax2 + Bx + C). If you have both situations at the same time, or all three points are on on any line (vertical, horizontal or oblique), there is no parabola that exists satisfying the three points.

Cheers,
Stephen La Rocque.     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.