Thread: Equation of a porabola

Hi I was hoping someone could help me with a math problem I am confused about. If you could, please use simple steps. Thank you

Here is what the problem looks like:

The graph of a parabola passes through the three points (0,3), (1,2), (2,3). Find the values of a, b, c and write the equation of the parabola in the form y = ax^2 + bx +c.

Originally Posted by Troy S

Hi I was hoping someone could help me with a math problem I am confused about. If you could, please use simple steps. Thank you

Here is what the problem looks like:

The graph of a parabola passes through the three points (0,3), (1,2), (2,3). Find the values of a, b, c and write the equation of the parabola in the form y = ax^2 + bx +c.

This is a simultaneous equations problem. You will get three equations and three unknowns. Here's how:

For (0,3) => x = 0, y = 3
so y = ax^2 + bx + c becomes
3 = a(0)^2 + b(0) + c
=> c = 3
ah, so we found c from the first, yah! This means we will only need to solve 2 equations. This will turn out easier than i thought.

For (1,2) => x = 1, y = 2
so y = ax^2 + bx + c becomes
2 = a(1)^2 + b(1) + c .........remember, c=3
=> 2 = a + b + 3
=> a + b = -1 ...........(1)

For (2,3) => x = 2, y = 3
so y = ax^2 + bx + c becomes
3 = a(2)^2 + b(2) + c .........remember, c = 3
=> 3 = 4a + 2b + 3
=> 4a + 2b = 0 ..............(2)

So we found c, to find a and b we need to solve the system:
a + b = -1 ................(1)
4a + 2b = 0 ..............(2)

=> 2a + b = 0 ...............(2)/2
......a + b = -1 ..............(1)
=> a = 1

but a + b = -1
=> 1 + b = -1
=> b = -2

so our parabola is:
y = x^2 - 2x + 3

y = a(x - p)^2 + q
= a(x - 1)^2 + 2 [Take the turning point of the parabola and insert it into the p and q values]

= a(x^2 - 2x + 1) + 2

Set (2;3) in the equation. x = 2 ; y = 3

3 = a((2^2) - (2 x 2) + 1) + 2
3 = a(1) + 2
3 = a + 2
a = 1

Thus y = x^2 - 2x + 3


Hope this helps.

If i might give you a tip, physically draw the parabola when you are given info like this. It really helps you to understand it and you will see the bigger picture

Thank you, both of you.