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'

the role of A in standard form

The axis of symmetry

Picture of Standard form equation parabola opens upwards or downards
Axis of Symmetry from Standard Form Standard form axis symmetry

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
    Vertex Form Axis
  • 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
Problem 2

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

Parabola
Problem 3

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

Parabola

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)

Parabola
Problem 4.2

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

(3,4) is the vertex.

Parabola
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.

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