# How To Find Domain And Range Of A Parabola ? (Quadratic Equation With Graphs)

Domain and range of a parabola shaped graph is the first step towards finding the minimum and maximum values of any parabolic function . If you are still unaware of the basics of domain and range , you can check my previous detailed posts for sure. In this post , we’ll mainly focus on finding the domain and range of a parabola (quadratic equations) . So just sit back and read further .

**How To Find Domain And Range Of A Parabola ?**

Let’s understand the steps to find the domain and range of a parabola :

**Plot the graph of f(x) i.e. y = f(x) , for this you need to have the knowledge of graphs of basic maths functions****In any graph, we can have Domain as all the x – coordinate values (along x-axis) of the graph****And Range is all y – coordinate values (along y-axis) of the graph****Finally you have to include/exclude the endpoints in the interval carefully by looking at the graph (for which f(x) is a valid function).**

The above steps are the same for finding the domain and range of any function graphically . Moreover, there is also a readymade formula to find the minima and maxima of a quadratic function (parabola) .

You can even find the **domain and range of a parabola** algebraically , but for everything a basic understanding of parabolic function is necessary .

**What exactly is a Parabola Function ?**

The parabolic function is simply a specific type in the **family of curves** aong others like the ellipse , hyperbola , etc . For any parabolic function , we’ll see a common shape as the ‘**U**‘ shaped pattern .

The parabolic functions are related to the quadratic equations normally . You may also encounter a sideways parabola sometimes . In this post we’ll see the method to find domain and range of any parabola just by inspection quickly .

Always remember that the **degree** of a parabolic function **will be always ‘2’.**

**Types Of Parabola And Its Graph !!**

There are normally four types of parabola , i.e. a **vertical and horizontal** parabola indeed . In a vertical parabola , we’ll have either an **Up or Down facing** parabola like below :

In a vertical parabola , the upwards or a downwards opening is basically based upon the sign of the **coefficient of x²** in the quadratic equation .

In the equation **f(x) = ax² + bx +c **

for a > 0 value :

for a < 0 value :

Now for horizontal parabola , we basically have a function defined in terms of ‘y’ variable i.e. :

f(y) = ay² + by + c

Again depending upon the sign of ‘a’ , the graph opening will be different :

for a > 0 value :

for a < 0 value :

Now , hopefully the types of parabola is pretty clear to you . Lets now understand the process to solve fprdomain and range in parabolic function .

Now clearly, the **minimum and maximum** value of any parabola will depend upon the shape of the graph firstly . For example in an **upward-opening** vertical parabola, we’ll have the minimum point at the vertex of the parabola and the maximum value of the function should be always infinity as shown :

**Domain And Range Of A Square Function Using Graph !!**

A parabola is formed whenever we have two values of ‘x’ , that will give same value for the f(x) . Basically we are talking about the square function here .

A parabolic function will always resemble a symmetric nature along the **axis of symmetry** . This **U** shaped graph is thus due to the possible same value of ‘f(x)’ for **two ‘x’ values** .

First, let us understand the concept of a square function along with its graph.

A square function will always result in a positive value , but it can accept any values in R . The square function is given by :

**f(x) = x² ** , is a square function .

Let’s plot the graph of f(x) as given below :

Clearly the domain of f(x) is R i.e. (-∞ , ∞) and the range of f(x) is [0,∞) , the graph of a square function is a parabola .

Now let’s try to find the domain and range of functions other than the standard function graphically . Always just keep in mind that in order to find the domain and range of a graph , you have to follow the below steps :

1. For finding domain graphically, just start to move in the direction from the L**eft to Right** along the x-axis .

2. Now for finding the range of a graph , just move from **Down to Up** direction along the y-axis ,

The above two points will be clear from solving some examples to find domain and range graphically .

**How To Find Domain And Range Of A Parabola (Quadratic Function) ?**

Now this is interesting. Firstly you have to plot the graph of the given quadratic function i.e. a parabola shaped plot precisely . Also, the opening of the parabola will depend on the coefficient of x² in the quadratic equation .

Let say **f(x) = ax² + bx + c **

Now the above equation is the standard quadratic equation , we can have different shaped parabola lying anywhere in the cartesian plane depending upon the roots and sign of the coefficient of ‘x²’

If **a > 0 **, then the graph of f(x) will be :

If **a < 0 ** , then the graph of f(x) will be :

Also remember that the **domain** of any vertical parabolic equation will always be **R i.e. (-∞,∞) ** , which means that it can take any value for ‘x’ , and for the **range** we need to care for the start and endpoint of the quadratic equation along the y-axis . Let’s try an example now to make this concept clear :

Now let us try to find the domain and range of the below parabolic function graphically :

**f(x) = x² + 5x + 6 **

Now first we need to plot the graph , for this let’s find out the roots :

by using the **middle term split up method** , we can write the above function as follows :

f(x) = x² + 5x + 6 = (x+2)(x+3) = 0 ;

The roots of the above equ is the point where f(x) = 0 ;

Hence the roots will be given as **x = -2 , -3 ;**

Now plotting the graph for the same :

**f(x) = x² + 5x + 6** ; the graph will look like as follows :

Clearly the domain is **R i.e. (-∞,∞) …..**(since there is no endpoint along the x-axis if we go from left to right)

Now range will be given as the points along the y-axis :

As seen in the graph , there is no endpoint of the graph in the upward direction along the y-axis. Now the start point of the graph is given as ‘**vertex**‘

where , the **axis of symmetry** = **midpoint** of both the roots on the number line

Also , for the y-coordinate of the **Vertex of f(x)** = Put the value of ‘**x**‘ if equ of f(x) …… (x is the axis of symmetry)

Now the graph will cut the y-axis at some point where , x = 0

**f(x) = x² + 5x + 6 ;**

On putting x =0 , f(x) = 6 ;

Now from the graph it is clear that the range of f(x) is given as [y coordinate of vertex,∞)

**Shortcut tip :** the vertex for any quadratic function can be given as the formula :

**Vertex = (-b/2a , -D/4a)** , where **D = b² – 4ac for the equation f(x) = ax² + bx + c ; **

On applying the formula , in the equation : f(x) = x² + 5x + 6 ;

a = 1 , b = 5 and c = 6 , Also D = 1

**Vertex of f(x) = (-5/2 , -1/4) **

The **range of f(x) is [-1/4 , ∞) **

For the above parabolic function , since it is vertical (upward-opening) parabola and hence a thumb rule will be that the domain of a vertical parabola will always be same i.e. (-∞ , ∞) in the number line .

Similarly, for a horizontal parabola the range will be **R** , as the endpoint will be **∞** in both the direction along the y – axis .

Hope you really enjoyed this post regarding the basics of parabola . Stay tuned for more interesting stuff in this series .

Aric is a tech enthusiast , who love to write about the tech related products and ‘How To’ blogs . IT Engineer by profession , right now working in the Automation field in a Software product company . The other hobbies includes singing , trekking and writing blogs .