Vertex Form Calculator

Convert the formula of a parabola from standard to vertex form or vertex to standard form using the vertex formula calculator below.

Standard Form:
y = ax² + bx + c

Vertex Form:
y = a(x - h)² + k

Results:

Vertex Form

 

Standard Form

 

Vertex

 

Y-interept

 


How to Convert Standard Form to Vertex Form

Every parabola has either a minimum (upwards opening parabola) or a maximum point (downwards opening parabola), which is known as the vertex. More formally, the vertex is the extremal value of the quadratic curve.

A parabola is the graph of a quadratic equation, which is often referred to as the standard form equation, or just standard form. Don’t confuse this with standard form in the context of scientific notation, which is a very different thing.

So you might be wondering how to find vertex from standard form? Recall the standard form quadratic equation is:

y = ax² + bx + c

You can find the vertex P(h, k) given the standard form using the following formulas.

h = -b2a
k = c – 4a

Thus, h is equal to negative b divided by 2 times a, and k is equal to c minus b squared divided by 4 times a.

Using the results from these formulas, you can write the standard form in vertex form using the vertex formula:

y = a(x – h)² + k

For example, let’s convert the equation y = 4x² + 3x + 1 from standard form to vertex form.

Start by finding h.

h = -32 × 4
h = -38
h = -0.375

Then find k.

k = 1 – 4 × 4
k = 1 – 916
k = 1 – 0.5625
k = 0.4375

Then substitute these values in the vertex formula and the answer in vertex form is:

y = 4(x + 0.375)² + 0.4375

How to Convert Vertex Form to Standard Form

If you’re following along so far, you might ask yourself how to find standard form from vertex form? Recall from above that vertex form for a parabola looks like this.

y = ax² + bx + c

You can reverse the formulas above that were used to convert standard form to vertex form to find the values b and c given the vertex form of a parabola.

b = -2ah
c = ah² + k

So, value b is equal to -2 times a times h, and c is equal to a times h squared plus k.

Then, using these values, write the equation in standard form.

y = ax² + bx + c

For example, let’s convert the equation y = 3(x + 2)² + 1 from vertex form to standard form.

Start by finding b.

b = -2 × 3 × 2
b = -12

Then find c.

c = 3 × 2² + 1
c = 3 × 4 + 1
c = 13

Then substitute these values in the standard form equation:

y = 3x² – 12x + 13