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 = -b/2a

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".

If |a| > 1, the graph of the parabola widens. This just means that the "U" shape of parabola stretches out sideways .

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

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

Answer

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


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

Answer

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

Answer

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  Identifying the vertex in vertex form  

1) What is the vertex of the following parabola:

y = (x + 3)² + 4

Answer

The vertex is the point (-3,4)


2) Find the vertex of the following parabola:

y = (x - 3)² + 4

Answer

(3,4) is the vertex.


3) What is the vertex of the parabola whose vertex form equation is

y = (x - 2)² - 3

Answer

Vertex is (2, –3)

Part III

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?

Answers

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

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 Vertex from Standard Form Equation 

The standard form equation of a parabola is

y = ax² + bx + c