Equation of a Parabola

Standard Form and Vertex Form Equations

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

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(x-h)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".

Practice Problems

Vertex and Direction-Vertex Form Equation

Problem 1

The parabola's vertex is the point (1,1).

Identifying the vertex in vertex form

Problem 4.1

The vertex is the point (-3,4)

Problem 4.2

(3,4) is the vertex.

Problem 4.3
vertex is (2, –3)

Part II

Problem 5.1

The vertex is (3,4) and it opens upwards since a is positive( it is 2), it opens upwards.

Problem 5.2

Vertex = (-3, 4), and it opens upwards since a is positive.

Problem 5.3

Vertex = (9, 5) and since a is negative (it is -22), it opens downwards.