Parabola Question!?

Question

Find the coefficients a, b, and c such that the parabola ax2 + bx + c passes through the points (0,6), (2,−2), and (3,0).

sorry..i kno its extremely easy but i haven't been brushing up my math...how would you carry out this question? any help would be greatly appreciated!! thx!

Answer 1

y = ax² + bx + c
6 = c

y = ax² + bx + 6

- 2 = 4a + 2b + 6
0 = 9a + 3b + 6

4a + 2b = - 8
9a + 3b = - 6

- 12a - 6b = 24
18a + 6b = - 12---------ADD
6a = 12
a = 2

8 + 2b = - 8
2b = - 16
b = - 8

ANSWER
a = 2 , b = - 8 , c = 6

Answer 2

There are several ways to do this. Probably the most straightforward way is to plug the three points given into the equation

y = ax^2 + bx + c,

and solve for a, b, c (a system of three linear equations). So, you would have

6 = a*0 + b*0 + c
-2 = a*4 + b*2 + c
0 = a*9 + b*3 + c

or

c = 6
4a + 2b + c = -2
9a + 3b + c = 0

Solve this for a, b, and c.

Another way is to use the Lagrange interpolation polynomial, which is complicated to write down, but if you look in Wikipedia you will see the formula for it. There is yet another way using derivatives and Taylor polynomials. And more ways, besides.

Answer 3

the vertex of the parabola is at (2, -2)

if you have a parabola f(x) = ax² + bx +c in general then for this problem we have

f(2) = -2
f(0) = 6
f(3) = 0

the equations we have are then

a*2² + b*2 + c = -2
a*0² + b*0 + c = 6
a*3² + b*3 + c = 0

from a*0² + b*0 + c = 6 we get c = 6

so now we have
a*2² + b*2 + 6 = -2
a*3² + b*3 + 6 = 0

which reduces to
4a + 2b = -8
9a + 3b = -6

using the first equation we get b = -4 - 2a
substitute into the second equation
9a + 3 * (-4 - 2a) = -6
9a -12 - 6a = -6
3a = 6
a = 2

use this in either equation to get b
4a + 2b = -8
8 + 2b = -8
2b = -16
b = -8

so the equation for the parabola is f(x) = 2x² - 8x + 6

Answer 4

Do you mean the parabola has the equation:

y = a x^2 + b x + c ?

In that case, for each (x,y) pair, plug the x and y into the equation, giving three equations with a, b and c as the unknowns. Solve the system of equations.

Answer 5

Plug in the points:

c = 6
4a + 4b + c = -2
9a + 3b + c = 0

reduces to:

4a + 4b = 4
9a + 3b = -6

you can solve this for a and b. GL.

Answer 6

I've been through too many ups and downs with parabolas. I'm finished with them.