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Vertex Form of a Quadratic Equation

Convert to vertex form of a quadratic equation by completing the square

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Vertex Form of a Quadratic Equation

Vertex Form of a Quadratic Equation 

Probably one of the best applications of the method of completing the square is using it to rewrite a quadratic function in vertex form. The vertex form of a quadratic function is

This form is very useful for graphing because it gives the vertex of the parabola explicitly. The vertex is at the point .

It is also simple to find the intercepts from the vertex form: just set and take the square root of both sides of the resulting equation.

To find the intercept, set and simplify.

Finding the Vertex and Intercepts of Parabolas 

Find the vertex, the intercepts and the intercept of the following parabolas:

a)

Vertex: (1, 2)

To find the intercepts,

The solutions are not real so there are no intercepts.

To find the intercept,

b)

To find the intercepts,

To find the intercept,

To graph a parabola, we only need to know the following information:

  • the vertex
  • the intercepts
  • the intercept
  • whether the parabola turns up or down (remember that it turns up if and down if )

Graphing Parabolas 

1. Graph the parabola given by the function .

To find the intercepts,

To find the intercept,

And since , the parabola turns up.

Graph all the points and connect them with a smooth curve:

License: CC BY-NC 3.0

2. Graph the parabola given by the function .

To find the intercepts,

Note: there is only one intercept, indicating that the vertex is located at this point, (2, 0).

To find the intercept,

Since , the parabola turns down.

Graph all the points and connect them with a smooth curve:

License: CC BY-NC 3.0

Example

Example 1

Graph the parabola given by the function .

To find the intercepts,

The intercepts are and .

To find the intercept,

Since , the parabola turns up.

Graph all the points and connect them with a smooth curve:

License: CC BY-NC 3.0

Review 

Rewrite each quadratic function in vertex form.

For each parabola, find the vertex; the and intercepts; and if it turns up or down. Then graph the parabola.

Review (Answers)

To view the Review answers, open this PDF file and look for section 10.7. 

RE

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Vocabulary

Intercept

The intercepts of a curve are the locations where the curve intersects the x and y axes. An x intercept is a point at which the curve intersects the x-axis. A y intercept is a point at which the curve intersects the y-axis.

Parabola

A parabola is the characteristic shape of a quadratic function graph, resembling a "U".

Vertex

A vertex is a corner of a three-dimensional object. It is the point where three or more faces meet.

Image Attributions

  1. [1]^ License: CC BY-NC 3.0
  2. [2]^ License: CC BY-NC 3.0
  3. [3]^ License: CC BY-NC 3.0

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