Vertex Form

Pronunciation: /ˈvɜrtɛks fɔrm/ Explain

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)
Manipulative 1: Vertex form of the quadratic equation. Created with GeoGebra.

The vertex form of a quadratic equation is f(x) = a(x - h)2 + k . This form is called the vertex form because the point (h,k) is the vertex of the parabola described by the equation. Click on the blue sliders in manipulative 1 and drag them to change the figure.

Graphing a Quadratic Equation in Vertex Form

An advantage of transforming a quadratic equation into vertex form is that it is easier to graph. Since the vertex can be read from the equation, it can quickly be identified and plotted.

  1. Step 1 In the equation y = -1(x-2)^2 + 1, the vertex is (2,1).
  2. Step 2 Plot (h+1,k+a).
    General CaseExampleDescription
    y=a((h+1)-h)^2 + ky= -1((2+1)-2)^2 + 1Substitute h+1 in for x.
    y=a(h-h+1)^2 + ky= -1(3-2)^2 + 1Simplify the innermost parenthesis.
    y=a(1)^2 + ky= -1(1)^2 + 1Simplify the remaining parenthesis.
    y=a*1 + ky= -1*1 + 1Simplify the exponent.
    y=a + ky= -1 + 1Simplify the multiplication.
    y=k+ay=0Simplify the addition and subtraction.
    So, (h+1,k+a) is a point on the graph.So, (3,0) is a point on the graph.Plot the point.
    Table 1: Plot (h+1,k+a).
  3. Step 3 Plot (h-1,k+a).
    General CaseExampleDescription
    y=a((h-1)-h)^2 + ky= -1((2-1)-2)^2 + 1Substitute h+1 in for x.
    y=a(h-h-1)^2 + ky= -1(1-2)^2 + 1Simplify the innermost parenthesis.
    y=a(-1)^2 + ky= -1(-1)^2 + 1Simplify the remaining parenthesis.
    y=a*1 + ky= -1*1 + 1Simplify the exponent.
    y=a + ky= -1 + 1Simplify the multiplication.
    y=k+ay=0Simplify the addition and subtraction.
    So, (h-1,k+a) is a point on the graph.So, (1,0) is a point on the graph.Plot the point.
    Table 1: Plot (h-1,k+a).
  4. Step 4 Sketch the parabola.

    Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)
    Manipulative 2: Graphing the vertex form of the quadratic equation. Created with GeoGebra.

    Converting a Quadratic Equation from Standard Form to Vertex Form

    Often it is easier to sketch a quadratic equation by converting it to vertex form. This can be done using complete the square.

    More Information

    • McAdams, David E.. Complete the Square. allmathwords.org. All Math Words Encyclopedia. Life is a Story Problem LLC. 12/13/2009. http://www.allmathwords.org/en/c/completethesquare.html.

    Cite this article as:

    McAdams, David E. Vertex Form. 8/7/2018. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/v/vertexform.html.

    Image Credits

    Revision History

    8/7/2018: Changed vocabulary links to WORDLINK format. (McAdams, David E.)

All Math Words Encyclopedia is a service of Life is a Story Problem LLC.
Copyright © 2018 Life is a Story Problem LLC. All rights reserved.
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License