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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".

  • 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).

  • 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


    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

      Identifying the vertex in vertex form  



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

    Answer


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


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


    Answer
    Part II

    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
    2) If a parabola's equation is  y =  3(x+3)2 +4, what is its vertex? Which way does it open?
    Answers
    3) A parabola has the equation y = -22(x - 9)2 + 5. What is its vertex? Which way does the parabola open?
    Answer