Vertex Form

Pronunciation: /ˈvɜr.tɛks fɔɹm/ Explain

Click on the blue points in the figure and drag them to change the figure.

How can you change the figure so that the parabola opens downwards?
Manipulative 1 - Vertex Form of a 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.

    Click on the points on the sliders and drag them to change the figure. Click on the check boxes to see each step.

    Manipulative 2 - How to Graph a Quadratic Equation in Vertex Form 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.

    References

    1. McAdams, David E.. All Math Words Dictionary, vertex form. 2nd Classroom edition 20150108-4799968. pg 189. Life is a Story Problem LLC. January 8, 2015. Buy the book

    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. 12/21/2018. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/v/vertexform.html.

    Image Credits

    Revision History

    12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)
    12/17/2018: Removed broken links, updated license, implemented new markup, update to new GeoGebra app. (McAdams, David E.)
    8/7/2018: Changed vocabulary links to WORDLINK format. (McAdams, David E.)
    5/5/2011: Initial version. (McAdams, David E.)

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