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.

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 Case	Example	Description
   y = a((h + 1) - h)2 + k	y = -1((2 + 1) - 2)2 + 1	Substitute h + 1 in for x.
   y = a(h - h + 1)2 + k	y = -1(3 - 2)2 + 1	Simplify the innermost parenthesis.
   y = a · 12 + k	y = -1 · 12 + 1	Simplify the remaining parenthesis.
   y = a · 1 + k	y = -1 · 1 + 1	Simplify the exponent.
   y = a + k	y = -1 + 1	Simplify the multiplication.
   y = k + a	y = 0	Simplify 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 Case	Example	Description
   y = a((h - 1) - h)2 + k	y = -1((2 - 1) - 2)2 + 1	Substitute h + 1 in for x.
   y = a(h - h - 1)2 + k	y = -1(1 - 2)2 + 1	Simplify the innermost parenthesis.
   y = a(-1)2 + k	y = -1(-1)2 + 1	Simplify the remaining parenthesis.
   y = a · 1 + k	y = -1 · 1 + 1	Simplify the exponent.
   y = a + k	y = -1 + 1	Simplify the multiplication.
   y = k + a	y = 0	Simplify 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. https://www.allmathwords.org/en/c/completethesquare.html.

Cite this article as:

McAdams, David E. Vertex Form. 5/13/2019. All Math Words Encyclopedia. Life is a Story Problem LLC. https://www.allmathwords.org/en/v/vertexform.html.

Image Credits

• All images and manipulatives are by David McAdams unless otherwise stated. All images by David McAdams are Copyright © Life is a Story Problem LLC and are licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

Revision History

5/13/2019: Changed equations and expressions to new format. (McAdams, David E.)
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.)