The Vertex of a Parabola
The vertex of a parabola is the point where the curve changes direction.
â€¢ If the parabola opens up, the
vertex is the lowest point on
the graph. This is called the
minimum.
â€¢ If the parabola opens down,
the vertex is the highest point
on the graph. This is called
the maximum. 

A vertical line drawn through the vertex is
called the axis of symmetry.
If you fold the graph along the axis of
symmetry one side of the graph will lie on
top of the other.
We can use the following procedure to find the vertex of a parabola.
Procedure â€” To Find the Vertex of a Parabola y = Ax^{2} + Bx + C
Step 1 Find the xcoordinate. It is given by the formula
.
Step 2 Find the ycoordinate. It is found by substituting the
xcoordinate into y = Ax^{2} + Bx + C and then simplifying.
Note:
It can be shown that the ycoordinate of
the vertex of a parabola is given by
Example 1
Find the vertex of the parabola: y = x^{2}  8x + 12
Solution
Here, A = 1, B = 8, and C = 12.
Step 1 
Find the xcoordinate.

x 


Substitute 1 for A and 8 for B. 
x 


Simplify. 
x 
= 4 
Step 2 
Find the ycoordinate.
Substitute 4 for x.
Simplify. 
y y
y
y 
= x^{2}  8x + 12 = (4)^{2}
 8(4) + 12
= 16  32 + 12
= 4 
So, the vertex of y = x^{2}  8x + 12 is (4, 4).
Note:
When a parabola has two xintercepts, the
xcoordinate of the vertex always lies
halfway between the xintercepts.
Here the xintercepts are x = 2 and x = 6. The xcoordinate of the vertex, 4, is
halfway between 2 and 6.
Example 2
Find the vertex of the parabola: f(x) = 2x^{2}  12x  18
Solution
Here, A = 2, B = 12, and C = 18.
Step 1 
Find the xcoordinate.

x 


Substitute 2 for A and 12 for B.

x 


Simplify. 
x 
3 
Step 2 
Find the ycoordinate.
Substitute 3 for x.
Simplify. 
f(x) y
y
y 
= 2x^{2}  12x  18
= 2(3)^{2}  12(3)  18
= 2(9) + 36  18
= 0 
So, the vertex of = 2x^{2}  12x  18 is (3, 0).
Note:
When a parabola has one xintercept, the
xintercept is the xcoordinate of the
vertex.
Here the xintercept is x = 3 and the
xcoordinate of the vertex is x = 3.
