Finding a Formula for the Vertex ("short cut")

Let's graph the two quadratic functions used in the previous section on finding the vertex by completing the square.

Note that in each case the vertex is on the axis of symmetry and that the x-intercepts are equidistant from the axis of symmetry. In the picture on the left, the axis of symmetry is x = 2 with the x-intercepts at 1 and 3. Each intercept is 1 unit from the axis of symmetry. In the graph of the right, the axis of symmetry is x = 5/6. While it seems that one intercept is 1, let's use the quadratic formula to find the intercepts.

We are solving Letting a = 3, b = 5, and c = -2 in

or gives us

or . These yield

or which give us or .

Note that these two x-intercepts are 1/6 of a unit from the axis of symmetry, x=5/6.

Let's take a closer look at one of the last steps in getting these two intercepts. If we rewrite

or as or , we can see that

or . Note that each answer involves the axis of symmetry plus or minus the same amount. If you trace this back, you will see that the 5/6 came from the while the 1/6 came from

Finding the Vertex

by a formula

The x value of the vertex of is given by .

The y value at the vertex is found by evaluating . This value will be the largest value for F if a<0 or will be the smallest value for F if a>0.

Example 1

Find the vertex to .

Solution .

Using that the x value at the vertex is , we have that or . The y value at the vertex is . This gives us

Below is a sketch of the graph of this function whose vertex is at ( 0.3, 12.45)

Example 2

Find the minimum value of

Solution .

First find the x value at the vertex. Thus x = 70. The minimum value of g is

or 23.

This minimum value of g is consistent with the following graph.