# 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 CaseExampleDescription
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 CaseExampleDescription
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.
5.  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.

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

• 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.

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.

### 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.)