Factored form, when it can be done, is a very useful form of a quadratic function.

Let’s imagine you are asked the following question:

For the quadratic function y=x^2 – 6x -16 find the

a) vertex

b) domain and range

c) axis of symmetry

d) max or min value (and tell whether it is a max or a min)

e) y intercept

f) x intercept(s), if any

Solution: Let’s assume we are going to use factored form. (An alternative is vertex form; I’ll tackle that method in another post.)

Our first step is to factor x^2 – 6x – 16 (you might want to review my article here on factoring easy trinomials).

We seek the numbers that multiply to make -16, but add to make -6. Realizing they are -8 and 2, we write

y=x^2 – 6x -16 ⇒ y=(x-8)(x+2)

Looking at the factored form, we notice that when x=8 or x=-2, y must be 0. Therefore, the points (8,0) and (-2,0) must be on the graph.

In the original function y=x^2 – 6x -16, note that the coefficient of x^2 is positive. Therefore, this graph will have a minimum value; from the vertex it will rise. Knowing this, and that it includes (8,0) and (-2,0), we can make a quick sketch of the graph:

Midway between the two x-intercepts, aka zeros, (-2,0) and (8,0), lies the vertex. The middle between -2 and 8 is found thus:

middle=(-2+8)/2 = 6/2 = 3

Therefore, the vertex occurs at x=3. To find the y coordinate, we plug 3 in for x in the original equation:

y=3^2 – 6(3) -16 = 9-18-16=-25

Therefore, the vertex is at (3,-25).

The domain of a quadratic function is all real numbers unless it’s a word problem with real-life limitations. In this case, the domain is all real numbers.

The range is y≥-25, which you can see from the graph when you place (3,-25) in for the vertex.

The axis of symmetry is x=3; it’s the vertical line that cuts the graph down the middle. For a quadratic function with vertex (p,q), the axis of symmetry is x=p.

The graph has a min; its min value is -25. The max or min value of a quadratic function with vertex (p,q) is q.

To find the y intercept, set x to 0. Doing that in the original function, we see y=-16.

We’ve already found the x intercepts: they’re at x=-2 and x=8.

While this article is a good primer, there’s still more to mention. In fact, I’ll be saying much more about quadratic functions in future posts.

HTH:)

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.