The equation of a parabola can be expressed in either standard or vertex form as shown in the picture below.
Standard Form Equation
The standard form of a parabola's equation is generally expressed:
$ y = ax^2 + bx + c $

The role of 'a'
 The axis of symmetry is the line $$ x = \frac{b}{2a} $$
The axis of symmetry
Picture of Standard form equation
Axis of Symmetry from Standard Form
Vertex Form of Equation
The vertex form of a parabola's equation is generally expressed as: y = a(xh)^{2}+k (h,k) is the vertex as you can see in the picture below
 If a is positive then the parabola opens upwards like a regular "U".
 If a is negative, then the graph opens downwards like an upside down "U".
 If a < 1, the graph of the parabola widens. This just means that the "U" shape of parabola stretches out sideways . Explore the way that 'a' works using our interactive parabola grapher.
 If a > 1, the graph of the graph becomes narrower(The effect is the opposite of a < 1).
Practice Problems
Vertex and DirectionVertex Form Equation
Part I
The parabola's vertex is the point (1,1).
Identifying the vertex in vertex form
The vertex is the point (3,4)
(3,4) is the vertex.
vertex is (2, –3)
Part II
The vertex is (3,4) and it opens upwards since a is positive( it is 2), it opens upwards.
Vertex = (3,4), and it opens upwards since a is positive.
Vertex = (9, 5) and since a is negative (it is 22), it opens downwards.