Standard Form to Vertex Form Calculator


Solution with Steps

Enter the 3 coefficients a,b,c of the Quadratic Equation in the above 3 boxes.
Next, press the button to find the Vertex and Vertex Form with Steps.

How do you convert from Standard Form to Vertex Form?

The Quadratic Equation in Standard Form is
\boxed{ y=ax^2+bx+c }

Then, the Vertex (h,k) can be found from the above Standard Form using
\boxed{ h= {-b \over 2a} , k=f( {-b \over 2a }) }

Once computed, the vertex coordinates are plugged into the Vertex Form of a Parabola, see below.

vertex form

How do you locate the Vertex on the Graph of a Parabola?

Every Parabola has either a minimum (when opened to the top) or a maximum (when opened to the bottom).
The Vertex is just that particular point on the Graph of a Parabola.
See the illustration of the two possible vertex locations below:


Example: How do you convert from Standard Form to Vertex Form?

We are given the Standard Form
y=3x^2- 6x-2 .
First, compute the x-coordinate of the vertex
h={ - b \over 2a}= { -(-6)\over (2*3)} = 1 .
Next, compute the y-coordinate of the vertex by plugging h=1 into the given equation:
k= 3*(1)^2-6(1)-2=-5 .

Therefore, the vertex is
(h,k)=(1,-5) .

Thus, we transformed the above Standard Form into the Vertex Form
y=(x-1)^2-5 .

Easy, wasn’t it?

Tip: When using the above Standard Form to Vertex Form Calculator to solve
3x^2-6x-2=0 we must enter the 3 coefficients a,b,c as
a=3, b=-6 and c=-2.

Then, the calculator will find the Vertex (h,k)=(1,-5) step by step.

Finally, the Vertex Form of the above Quadratic Equation is
y=(x-1)^2-5 .

Get it now? Try the above Standard Form to Vertex Standard Calculator again.

How do I find h and k in Vertex Form?

There are two ways to find h and k, the vertex x- and y- coordinates.

1) The fast way: Given y=ax^2+bx+c we first compute h= {-b \over 2a} and next k=f(h) .
Example: y=3x^2+6x+4 thus h= {-6 \over 2*3} = -1 and k=f(-1)=3(-1)^2+6(-1)+4=3-6+4=1
Thus, Vertex Coordinates are (k,h)=(-1,1)

2) The long way: We do the Complete-the-Square procedure to convert
y=ax^2+bx+c into
y=a(x-h)^2+k .
Please click HERE to do this procedure.

What are h and k in Vertex Form?

h and k are the Vertex x- and y- coordinates of the Graph of a Quadratic Equation. They give the Location of a Minimum (when a>0) or Maximum (when a<0).

You may also think of h and k as shifts/transformations:
Shifting the Standard Parabola
h units right yields
y=(x-h)^2 .
Shifting it k units up yields
y=(x-h)^2+k .
By performing those 2 shifts we moved the Vertex from (0,0) to the new location (h,k) .