2.1 Quadratic Functions

Polynomial Functions: can be written in the form

Polynomial functions are classified by degree of the leading term, as follows:

    Degree                Name

0                             Constant

1                             Linear

2                             Quadratic

3                             Cubic

4                             Quartic

5                             5th degree

Today, we are focusing on QUADRATIC FUNCTIONS.

Quadratic Function: Let a, b, and c be real numbers and a cannot equal zero. The function

is called a quadratic function. For such functions, the graph is a parabola. 

An especially convenient form of quadratics for graphing is called vertex form, expressed as:

where the vertex is represented as (h, k). If a > 0, then the parabola opens upward. If a < 0, then the parabola opens downward.

In order to change an equation from standard form to vertex form, it is sometimes necessary to complete the process known as "Completing the Square". The procedure is as follows:

1. Write original function, in standard form.

2. Divide all terms by the leading coefficient of the first term. 

3. Divide the value of b by 2, then square it.

4. Move the value c to the very right of the equation. 

5. Add/subtract the value from step 3 to the first two terms, the do the opposite to the value of c, all the way on the right.

6. Factor the perfect square and regroup terms.

Example:   

       

The vertex of the above function would be (-5, 25), and the parabola would open downward as a result of the negative value of a.