The Vertex Form of the quadratic model is : , where a is a constant and (h, k) is the coordinate of the vertex. With this form of the quadratic formula one can easily visualize the shape of the graph.

First the coefficient of the first term of the formula, a tell us whether the function is has a maximum or a minimum value. . When  we have a a function with a maximum, therefore the graph opens downward, (see Figure 5.5.2) and when  we have a function with a minimum, therefore the graph opens upward, (see Figure 5.5.1).

One can readily find the formula for a quadratic model if the vertex and one other point is given. This will be covered later. Also given any other form of the quadratic formula it is often useful to rewrite it into the Vertex Form, since the vertex give us the very useful x-y coordinate of the optima (maximum or minimum) of the function.

There are several techniques for finding the vertex or Vertex Form of the quadratic function:

Complete the square method of finding the Vertex Form of the quadratic

Steps: Expand the quadratic into the Standard Form 

Then Isolate the x terms: 

To make  a perfect square, add , the square of half the coefficient of x.

This gives the perfect square 

Example 5.5.1 Complete the Square and Write in Vertex Form for the Quadratic:

Table 5.5.3 Complex Example for Completing the Square to convert to Vertex Form of Quadratic Functions
 

Figure 5.5.4

For the Quadratic  Write in vertex form: So the vertex is (1, 4)

Association to vertex

The Standard Form and the Vertex Form of the Quadratic is related to each other by this association:

and if the right side of the equation is expanded we get:

So 

Example 5.4.5 Write  into the vertex Form of the Quadratic

That is rewrite  in this form 

Table 5.5.4 Association of Vertex Form with Standard Form of Quadratic Functions
 

to Vertex Form  and  and  , So  ,  So

Figure 5.5.5

Parabola and the Quadratic Model

The Quadratic Function is also another version of the parabola and from the symmetric of the parabola we know that given the Standard Form of the quadratic / parabola: 

Then the Vertex is given by the coordinate: 

Example 5.5.6 Sketch the graph of the quadratic function without the use of a graphing Calculator. 

Table 5.5.5 Parabola and the Quadratic Model
 

First find the Vertex using knowledge of the symmetry of the parabola model: Vertex => : h =  and k =  So the vertex is

Figure 5.5.6

Sketching the Quadratic Using the Vertex and another point:

Knowing the vertex of a quadratic model is useful information for sketching the function given another point. For since the vertex is the optimal value of the function then any other point will suggest whether the graphs opens downward (maximum) or opens upward (minimum).

For example a quadratic function with a Vertex at (2, 3) and another point on the graph at (-1, 16) is a function with a minimum at (2, 3) and opens upward.

Precalculus: Contemporary Models
by Pin D. Ling