# Domain and Range of a Parabola

**Table of Content (8 min. reading time)**

1 – Step by Step Solver: Find the Range of a Parabola

2 – What is the Domain of a Parabola?

3 – How do I write the Range in Interval Notation?

4 – Example: Domain and Range of a Parabola

5 – Is there a Domain and Range Parabola Calculator?

### Domain and Range of a Parabola 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 Domain and Range of a Parabola with Steps.

### What is the Domain of a Parabola?

Great news: The Domain of ANY Parabola always is all real numbers. In other words, we can plug any real number into quadratic equation in standard form y=ax^2+bx+c or in vertex form y=a(x-h)^2+kThe reason for that is quadratic equations fall in the category of polynomials and thus don’t contain fractions, roots or radicals nor logarithms. Those 3 categories of functions have restrictions with regard to their domain.

#### Thus, the Domain of any Parabola in Interval Notation is simply (-\infty , \infty) .

### How do I write the Range in Interval Notation?

In the above example the Range is y>=3 . The corresponding Interval Notation is [3 , \infty)Notice that the 3 is included in the Range so we use bracket [ in the Interval Notation. Since there is no upper bound for the Range we denote that with the Infinity Symbol which is always followed by the Open Interval symbol “)”. Think of Infinity as NOT being a concrete endpoint.

**Example2:**If the Range is y<=5 then the corresponding Interval Notation is (-oo , 5] .

Again, the 5 is included in the Range so we use the bracket ] in the Interval Notation.

### Example: Domain and Range of a Parabola?

We are given the Standard Form

y=3x^2- 6x-2 .

Since ANY x value can be plugged into this equation (we have no fractions, radicals nor logarithms) we can conclude that the Domain is ‘all real numbers’. In Interval Notation: (-\infty , \infty)

To find the Range we 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) .

The Range is simply all real numbers greater or equal to -5 or simply y>=-5.

Using Interval Notation: [-5 , \infty )

Explanation: This parabola opens to the top since the leading coefficient 3 is greater than 0. This implies that the vertex is a minimum and therefore the parabola takes on any value greater than 5. Since the Range contains all possible y- values the Range is [-5 , \infty)

Easy, wasn’t it?

Tip: When using the above Range of a Parabola Calculator for 3x^2-6x-2

we must enter the 3 coefficients a,b,c as a=3, b=-6 and c=-2.

Get it now? Try the above Range of a Parabola Calculator again if needed.