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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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Practice Vertex Form of a Quadratic Equation
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Estimated11 minsto complete
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Vertex Form of a Quadratic Equation

What if you had a quadratic function like ? How could you rewrite it in vertex form to find its vertex and intercepts? After completing this Concept, you'll be able to rewrite and graph quadratic equations like this one in vertex form.

Guidance

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.

Example A

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

a)

b)

Solution

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 )

Example B

Graph the parabola given by the function .

Solution

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:

Example C

Graph the parabola given by the function .

Solution:

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:

Watch this video for help with the Examples above.

Vocabulary

• 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 .

• 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.

Guided Practice

Graph the parabola given by the function .

Solution:

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:

Practice

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.

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