Suppose that we have an equation y=ax^2+bx+c whose graph is a parabola with vertex (3,2), vertical axis of symmetry, and contains the point (1,0). What is (a, b, c)?
If the vertex is (3,2) and it contains the point (1,0).....then x = 1 is a root
So...by symmetry......(5, 0) is also a root
So....we can find "a" thusly
0 = a(1 - 3)^2 + 2
-2 = a(-2)^2
-2 = 4a
a = -2/4 = -1/2
And by Vieta......
The sum of the roots -b/a
So
1 + 5 = - b / ( -1/2)
6 = 2b
b = 3
And the product of the roots = c/a
So
5*1 = c /(-1/2)
5 = -2c
c = -5/2
So
(a,b , c) = ( -1/2, 3, -5/2)
Here's the graph : https://www.desmos.com/calculator/yd99xbrpsg