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?
Subtracting c from both sides:
y - c = ax2 + bx
Taking "a" as the common factor:
y - c = a (x2 + b/a x)
Here, half the coefficient of x is b/2a and its square is b2/4a2. Adding and subtracting this on the right side (inside the parantheses):
y - c = a (x2 + b/a x + b2/4a2 - b2/4a2)
We can write x2 + b/a x + b2/4a2 as (x + b/2a)2. Thus, the above equation becomes:
y - c = a ( (x + b/2a)2 - b2/4a2)
Distributing "a" on the right side and adding "c" on both sides:
y = a (x + b/2a)2 - b2/4a + c
y = a (x + b/2a)2 - (b2 - 4ac) / (4a)
Comparing this with y = a (x - h)2 + k, we get:
h = -b/2a
k = -(b2 - 4ac) / (4a)
We know that b2 - 4ac is the descriminant (D). Thus, the vertex formula is:
(h, k) = (-b/2a, -D/4a),
where D = b2 - 4ac
If you feel difficult to memorize the above formula, you can just remember the formula for x-coordinate of vertex and then just substitute it in the given equation y = ax2 + bx + c to get the y-coordinate of the vertex.
x-coordinate 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 = ax2 + bx + c of the form y = a (x - h)2 + k manually and find the vertex (h, k).
Solved Examples Using Vertex Formula
Find the vertex of y = 3x2 - 6x + 1.
To find: The vertex of the given equation (parabola).
Comparing the given equation with y = ax2 + bx + c, we get
a = 3; b = -6; c = 1.
Then the discriminant is, D = b2 - 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).
Find the vertex of a parabola whose x-intercepts are (2, 0) and (3, 0) and whose y-intercept is (0, 6).
To find: The vertex of the given parabola.
Since (2, 0) and (3, 0) are the x-intercepts 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 y-intercept 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) = x2 - 5x + 6 ... (2)
Comparing the above equation with y = ax2 + bx + c, we get
a = 1; b = -5; c = 6
Using the vertex formula (formula 2),
x-coordinate of the vertex = -b / 2a = -(-5) / (2×1) = 5/2
Substitute this in (2) to find the y-coordinate of the vertex.
y = (5/2)2 - 5 (5/2) + 6 = -1/4
Answer: The vertex of the given parabola = (5/2, -1/4)