standard and vertex thumb

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 and Vertex Form picture

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
Picture of Standard form equation parabola opens upwards or downards
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
    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".

Practice Problems

Vertex and Direction-Vertex Form Equation

Part I
Problem 1

What is the graph of the following parabola y = (x–1)² + 1?

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

parabola picture
Problem 2

What is the graph of the following parabola y = –(x–1)² + 1?

Problem 3

What is the graph of the following parabola y = (x+2)² –3?

Identifying the vertex in vertex form

Problem 4.1

What is the vertex of the following parabola: y = (x + 3)² + 4

The vertex is the point (-3,4)

Problem 4.2

Find the vertex of the following parabola: y = (x - 3)² + 4

(3,4) is the vertex.

Problem 4.3

What is the vertex of the parabola whose vertex form equation is y = (x - 2)² - 3

vertex is (2, –3)

Part II

Problem 5.1

What is the vertex of a parabola with the following equation:
y=  2(x-3)2+4 ? Does the parabola open upwards or downwards?

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

Problem 5.2

If a parabola's equation is  y =  3(x+3)2 +4, what is its vertex? Which way does it open?

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

Problem 5.3

A parabola has the equation y = -22(x - 9)2 + 5. What is its vertex? Which way does the parabola open?

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

back to Parabola Home next to Converting between Forms