Write a quadratic function in vertex form calculator

Q: How do you write a quadratic function in vertex form?

The vertex form of a quadratic function is f(x) = a(x – h)2 + k, where a, h, and k are constants. of the parabola is at (h, k).

Question

This Standard form to vertex form calculator is a free tool to assist you in parabolic equations. It tells:

Table of Contents Show

How to use this calculator?
What is vertex form?
General vertex form
How to convert the standard form into vertex form?
How to find vertex from the quadratic equations?
Find vertex from the standard form:
Find vertex from the vertex form:
How do you convert from Standard Form to Vertex Form?
How do you locate the Vertex on the Graph of a Parabola?
Example: How do you convert from Standard Form to Vertex Form?
How do I find h and k in Vertex Form?
What are h and k in Vertex Form?
How do you write a quadratic function in vertex form?

Vertex form equation
Standard form equation
Vertex
Y-intercept

You can find the vertex of a parabola from a vertex equation. But if you have a standard form equation then you can convert it to vertex form using this calculator.

How to use this calculator?

To operate this calculator, you have to:

Choose what you want to calculate; vertex form or standard form to vertex form.
Enter the values.
Hit the calculate button.

What is vertex form?

Quadratic equations are represented in two types of equations; Standard form and Vertex form. Both of these equations have their uses.

The vertex form is the one that gives you information about the vertex, maximum or minimum point, of a parabola. It can be derived from the standard form.

General vertex form

A general vertex form is represented as:

How to convert the standard form into vertex form?

As mentioned before, you can convert the standard form into vertex form. A standard form is written as:

Y = ax2+bx+c

You will have to:

Make the coefficient of x2, 1.
Find the value to complete the square from the coefficient of x.
Close the square.
The equation is in standard form.

Example:

Convert y= 5x2 + 10x + 2 into vertex form.

Solution:

Step 1: Make the coefficient of x2 to be 1.

For this purpose, extract 5 from the whole equation, such as;

Y = 5 (x2 + 2x + ⅖)

Step 2: Divide the coefficient of x by 2 and take its square.

Coefficient of x = 2, so:

(2/2)2 = 12 = 1

Step 3: Add and subtract this value in the parenthesis.

Y = 5 (x2 + 2x + 1 - 1 + ⅖)

Step 4: Complete the square.

Observe that x2 + 2x + 1 = (x + 1)2 therefore,

Y = 5 (x + 1)2 - 5(1 + ⅖)

Y = 5 (x + 1)2 - 5(⅗)

Y = 5 (x + 1)2 - 3

This is the vertex form. In the last step, when we completed the square, we had to take the constant terms (i.e 1 + ⅖) aside. But since 5 was extracted from them as well, we multiplied it separately.

How to find vertex from the quadratic equations?

You can simply find the vertex from the quadratic equations. To know how to? keep reading.

Find vertex from the standard form:

If you don’t want to convert the standard form into the vertex form, find the vertex point using these formulas.

h = -b / (2a)

k = c - b2 / (4a)

Example:

Find the vertex of a parabola from the equation y = x2 - 3x + 1.

Solution:

The equation is in standard form so:

Step 1: Identify the elements.

a = 1

b = -3

c = 1

Step 2: Put the values in the formulas.

For h:

h = -b / (2a)

h = -(-3) / 2(1)

h = 3/2 or 1.5

For k:

k = c - b2 / (4a)

k = 1 - (-3)2 /4(1)

k = - 5/4 or -1.25

So, the vertex is (3/2, -5/4). You can use these values to find the vertex form as well. Just put these values in the places of h and k while a will remain the same i.e y = 1(x + 3/2)2 - 5/4.

Find vertex from the vertex form:

There is no rocket science in it. Simply identify the values of h and k and put them in the general vertex form equation.

Example:

What is the vertex for equation y = 9(x + 3) + 2.

Solution:

Identify the values.

h = 3

k = 2

So, the vertex is (3, 2).

How do you convert from Standard Form to Vertex Form?

The Quadratic Equation in Standard Form is

 y=ax²+bx+c

Then, the Vertex (h,k) can be found from the above Standard Form using

  h= -b/2a , k=f(h)

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

Example: Convert from Standard Form to Vertex Form

Let’s convert

y=2x²+8x+3

into Vertex Form.
Then,
h = -b/(2a) = -(8)/(2*2) = -2 .
Next, compute k, the vertex y-coordinate, by plugging h = -2
into
k = 3*(-2)²+8(-2)+3 = -1 .
Thus, the vertex is (h,k)=(-2,-1) .

Since -(-2)=2 we converted to the Vertex Form

 y=(x+2)²-1

.

Watch the video below for a great explanation of how to convert from Standard to Vertex Form.

Learn how to find the Vertex Form using this helpful Video on YouTube. She is truly the expert!

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=3x²- 6x-2

.
First, compute the x-coordinate of the vertex
h = – b/2a = -(-6) / (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
3x²-6x-2=0 we must enter the 3 coefficients a,b,c as
a=3, b=-6, 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 our 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. There is a fast way and a long way.

1) The fast way: Given y = ax²+bx+c we first compute h = -b / 2a and next k=f(h) .
Example: y=3x²+6x+4 thus h =-6/2*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=ax²+bx+c into
y=a(x-h)²+k .
We create a separate page to learn this method. Please click HEREto 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
y=x²
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
the origin (0,0) to the new location (h,k) .