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 $
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The role of 'a'
- The axis of symmetry is the line $$ x = -\frac{b}{2a} $$
The axis of symmetry
Picture of Standard form equation
![parabola opens upwards or downards]()
Axis of Symmetry from Standard Form
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Vertex Form of Equation
The vertex form of a parabola's equation is generally expressed as: y = a(x-h)2+k- (h,k) is the vertex as you can see in the picture below
![parabola opens upwards or downards]()
![]()
- 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 Direction-Vertex 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.
