Finding the Vertex (completing the square)

The process of completing the square can be used to rewrite a quadratic function of the form into the form . As we saw earlier this last form makes it easy to find the vertex of the parabola.

Example 1

Put into vertex form.

Solution .

The general form that we need involves the square of a binomial plus or minus some constant. Let's review some squares of binomials to recall what they have in common.

, ,

Note that the square of one-half of the coefficient of the middle term gives the last term in this type of perfect square trinomial.

Now let's enter the equation.

Group the first two terms together.

Because the square of half of -4 is 4, we add 4 and subtract 4 from the right side of the equation.

Factoring the trinomial and simplifying, we have

From this vetex form we see that the vertex is (2, -1)

Example 2

Put into vertex form.

Solution .

Group the first two terms together and factor out the -3.

Half of -5/3 is -5/6. Put 25/36 , the square of this, inside the parenthesis and add 3 times 25/36 or 25/12 to the -2.

Factoring the trinomial and simplifying, we have

The vertex of the curve is (5/6, 1/12)