Mathematics help needed for quadratic equation

My exam is tomorrow and I can't remember how to do this problem and it's not in my notes.

2) Find the equation of the quadratic function y = ax^2 + bx + c that passes through the points (1,5), (-1,3), (2,9).

The professor says the answer is y = 3x^2 - 5 on this paper but I don't understand how he gets the answer.

It involves matrices.
He, for some reason, plugs in x = 1, 0, -1 and gets y = -2, -5, -2.
He makes a matrix:
1 1 1 -2
0 0 1 -5
1 -1 1 -2

this gives the answer but how did he solve for the y values?

Nah, the professor is talking bollocks.

Put those points into y = ax^2 + bx + c and you have
5=a+b+c
3=a-b+c
9=4a+2b+c *

so c=5-a-b

Therefore 3=a-b+(5-a-b)
So b=1

From * 7=4a+c
a=4-c so c=3
and so a=1

The equation is x^2 + x + 3. Forget matrices, not needed here.

Using a graphing calculator for the function y = f(x) = 3*x^2 - 5 does not show that the graph goes through (1,5), (-1,3), and (2,9) . Is there another possible answer?

Try the substitution of (1,5) in the original y = a*x^2 + b*x + c to get 5 = a*1^2 + b*1 + c.
Then substitute the point (-1,3) to get 3 = a*(-1)^2 - b + c
Then substitute the point (2,9) and solve 3 equations in 3 unknowns.