Unit 3: Week 3 - Converting a Quadratic Function 2

Converting a Quadratic Function 2

Converting a Quadratic Function from Standard Form to Vertex Form (by Completing the Square)

We obtained the vertex form of a quadratic function from its standard form by first deriving a formula for the vertex. But we can also obtain the vertex form from the standard form “algebraically” by using the method of completing the square. We use the following examples to demonstrate the method.

Example:

Write the quadratic function  in vertex form using the method of completing the square. Then find the vertex.

Solution:

Note that the coefficient of the x-term is 4. So the square of half of the coefficient of the x-term is .

To complete the square we need to add and subtract 4 to the expression which defines the function:

.

Now observe that  and  . Thus f can be written in vertex form as

.

The vertex of f  is .

Example

Write the quadratic function   in vertex form using the method of completing the square. Then find the vertex.

Solution:

First, divide both sides of   by the coefficient of the x2-term:

If we simplify the first two terms on the right-hand side, we get

Note that the coefficient of the x-term is -2 and so the square of half of the coefficient of the x-term is 1. Now complete the square by adding and subtract 1 from the expression on the right-hand side:

Since   and  , we can rewrite the right-hand side like this:

By multiplying both sides of the equation by 3, we get the vertex form of f :

This time the vertex of f is (1,2).

Using the navigation on the left, please proceed to the next page.

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