Vertex Formula
Before going to know what is the vertex formula of a parabola, let us recall what is the vertex of a parabola. The vertex of a parabola is a point at which:
 the parabola is minimum (when the parabola opens up) or maximum (when the parabola opens down).
 the parabola turns (or) changes its direction.
Let us learn the vertex formula with its derivation and a few solved examples.
What is Vertex Formula?
Formula 1
We know that the standard form of a parabola is, y = ax^{2} + bx + c. Let us convert it to the vertex form y = a(x  h)^{2} + k by completing the squares.
Subtracting c from both sides:
y  c = ax^{2} + bx
Taking "a" as the common factor:
y  c = a (x^{2} + b/a x)
Here, half the coefficient of x is b/2a and its square is b^{2}/4a^{2}. Adding and subtracting this on the right side (inside the parantheses):
y  c = a (x^{2} + b/a x + b^{2}/4a^{2}  b^{2}/4a^{2})
We can write x^{2} + b/a x + b^{2}/4a^{2} as (x + b/2a)^{2}. Thus, the above equation becomes:
y  c = a ( (x + b/2a)^{2}  b^{2}/4a^{2})
Distributing "a" on the right side and adding "c" on both sides:
y = a (x + b/2a)^{2}  b^{2}/4a + c
y = a (x + b/2a)^{2}  (b^{2}  4ac) / (4a)
Comparing this with y = a (x  h)^{2} + k, we get:
h = b/2a
k = (b^{2}  4ac) / (4a)
We know that b^{2}  4ac is the descriminant (D). Thus, the vertex formula is:
(h, k) = (b/2a, D/4a),
where D = b^{2}  4ac
Formula 2
If you feel difficult to memorize the above formula, you can just remember the formula for xcoordinate of vertex and then just substitute it in the given equation y = ax^{2} + bx + c to get the ycoordinate of the vertex.
xcoordinate of the vertex = b / 2a
Note: If you do not want to use any of the above formulas to find the vertex, then you can just complete the square to convert y = ax^{2} + bx + c of the form y = a (x  h)^{2} + k manually and find the vertex (h, k).
Solved Examples Using Vertex Formula

Example 1:
Find the vertex of y = 3x^{2}  6x + 1.
Solution:
To find: The vertex of the given equation (parabola).
Comparing the given equation with y = ax^{2} + bx + c, we get
a = 3; b = 6; c = 1.
Then the discriminant is, D = b^{2}  4ac = (6)^{2}  4(3)(1) = 36  12 = 24.
Using the vertex formula (formula 1),
Vertex, (h, k) = (b/2a, D/4a)
(h, k) =( (6) / (2×3), 24 / (4×3) ) = (6/6, 24/12) = (1, 2)
Answer: The vertex of the given parabola = (1, 2).

Example 2:
Find the vertex of a parabola whose xintercepts are (2, 0) and (3, 0) and whose yintercept is (0, 6).
Solution:
To find: The vertex of the given parabola.
Since (2, 0) and (3, 0) are the xintercepts of the given parabola, (x  2) and (x  3) are the factors of the equation of the parabola. So the equation of the parabola is of the form:
y = a (x  2) (x  3) .... (1)
Its yintercept is given to be (0, 6). Substitute x = 0 and y = 6 in the above equation:
6 = a (0  2) (0  3)
6 = 6a
a = 1
Substitute a = 1 in (1):
y = 1 (x  2) (x  3) = x^{2}  5x + 6 ... (2)
Comparing the above equation with y = ax^{2} + bx + c, we get
a = 1; b = 5; c = 6
Using the vertex formula (formula 2),
xcoordinate of the vertex = b / 2a = (5) / (2×1) = 5/2
Substitute this in (2) to find the ycoordinate of the vertex.
y = (5/2)^{2}  5 (5/2) + 6 = 1/4
Answer: The vertex of the given parabola = (5/2, 1/4)