Equation of a Parabola

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'
• If $$ a > 0 $$, the parabola opens upwards
• if $$ a < 0 $$ it opens downwards.

The axis of symmetry
• The axis of symmetry is the line $$ x = -\frac{b}{2a} $$

Picture of Standard form equation

Axis of Symmetry from Standard Form

Vertex Form of Equation

• (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).