Formulas for Quadratic Functions

Let's talk about the quadratic function or parabolas. We will first look at a quadratic function in vertex form, and then we will look at a quadratic function in standard form. We find vertex and intercept formulas for both.

Here is a parabola in vertex form:

(h,k) would be your vertex. The intercepts can setting either x or the y to zero. So to find the y-intercept, you set the x equal to 0.

So the y-intercept is (0, ah^2+k). To find the x-intercept, you set y equal to 0.

This is an interesting formula, because if your a and k are the same sign, you'll end with two complex but not real roots.

Now, let's look at a parabola in standard form.

It does not give you the vertex right away.  If I use a little calculus, take the first derivative with respect to x, and set equal to 0, I will get the minimum or the maximum, which is the vertex. 

So 

Now, let's find k. To find k (the y-coordinate of the vertex), we would just have to plug what h is (x-coordinate) into the function.

We found the vertex for standard. Let's find formulas for the intercepts.

For the y-intercept,

y-intercept is (0,c). The formula for the x-intercepts I found in a previous post (http://mathtalkwithjd.blogspot.com/2015/09/deriving-quadratic-formula.html).
It's the quadratic formula.

If I wanted to, I could make D=b^2-4ac. It would simplify the formulas for standard form, but first let's review the formulas we found for vertex form.

Formulas for Vertex Form of a Parabola

Formulas for Standard Form of a Parabola

Hopefully, you have found this helpful. Please e-mail me if you have questions. Let's talk math.