How to convert from Vertex to Standard Form?

The Vertex Form of a Parabola is
where (h,k) are the Vertex Coordinates.

The Standard form of a Parabola is

Let’s do an easy example first

Let y=2(x-1)²-5
we first apply the binomial formula to expand and get
Next, we distribute the 2 to get
y= 2x²-4x+2-5
With 2-5=-3 we finally arrive at the Standard Form:

In general, we obtain the Standard Form from the Vertex Form by using these 2 steps:
Given: y=a(x-h)²+k
Step1: (Use Binomial Formula) y=a(x²-2hx+h²)+k
Step2: (Distribute and Combine 2 like terms ah² and k) y=ax²-(2ah)x+(ah²+k)

How do you find the Vertex of a Quadratic Equation?

Every Parabola has either a..
..Minimum (when opened to the top due to leading coefficient a>0) or
..Maximum (when opened to the bottom due to leading coefficient a<0).
The Vertex is just that particular point on the Graph of a Parabola.
See the illustration of the two possible vertex locations below:


Example: What if the leading coefficient a is negative?

Standard and Vertex form of a Quadratic Equation
Let’s do another example with a negative leading coefficient a=-2.

We are given the quadratic equation in vertex format

First, apply the binomial formula
(x+3)² = x²+6x+9

Thus we have
y= -2(x²+6x+9)-7

Next, distribute the 2 to get
y= -2x²-12x-18-7

Since -18-7=-25 we finally get the standard form
y= -2x²-12x-25

Here, a=-2, b=-12 and c=-25 are the coefficients in the Standard Form
y= ax²+bx+c

Get it now? Try our Vertex to Standard Form Calculator a few more times.