A parabola is (mathematically speaking) a quadratic function, which looks like this.
y = ax 2 + bx + c.
where a, b and c are constants. (If three points on the curve are kn…own, then a, b and c can be found.) The gradient, then, can be found by differentiation:.
dy/dx = 2ax + b.
A parabola has one maximal or minimal point, where the gradient is zero..
2ax + b = 0.
x = -b/2a.
Use the original function to find the corresponding value of y:.
y = a(-b/2a) 2 + b(-b/2a) + c.
= b 2 /4a - b 2 /2a + c.
= c - b 2 /4a.
So the coordinates of your turning point are.
( -b/2a , c - b 2 /4a ).
This result can also be derived by completing the square. (MORE)