## Contents:

This article focuses on the graph of the quadratic function called “Parabola” and Vertex of a parabola.

Following are the key features of this article.

- Introduction about the parabola, Vertex of a parabola.
- Types of parabolas and their Orientations explained in detail.
- Finding the Domain and Range of a parabola using the orientation and vertex of a parabola.
- A summary table giving domain and range of
**ANY**quadratic function. - Worked out examples

# 1.The Vertex of a Parabola:-

## I). What is a parabola?

A parabola is a curve which belongs to the family of conic sections and resembles a “U” or an Arch shape with an apex point which is called the vertex of a Parabola

The graph of a quadratic function is called as a parabola.

For example, When a ball is thrown horizontally, it takes a parabolic path in air and touches the ground.

The parabolic curves have different orientations and they can appear stretched, compressed, flipped.

There are 3 key transformations which can be applied to the quadratics functions namely *Dilations, Translations, and reflections, which are discussed in detail*

Let us classify the parabolic graphs based on their orientation.

### Types of parabolic graphs:

Based on the orientation, Parabolic graphs are broadly classified into 2 types :

**1)Vertical/Regular parabola**:- These parabolas are named as a vertical parabola since the axis of symmetry will be a vertical line.

The axis of symmetry of the vertical parabola will be * parallel to the “y”* axis always. There can be two types of the vertical parabola which are widely seen in the real-world application and so the reason are called as a Regular parabola.

**i)** **Up-Parabola**

** ii)** **Down-Parabola**

**2)** **Horizontal/sideways parabola**: These parabolas earned their name as “Horizontal parabola” since the axis of symmetry is a horizontal line and is parallel to the “x” axis.

Since these parabolas open to the left or right side visually, they are called as the **sideways parabola** in general. Following are the two types of the sideways parabola.

**i)** **Right -Parabola**

**ii)Left-Parabola**

## II). What is the vertex of a parabola or Quadratic function?

Whether it is a vertical /horizontal parabola, the “U” shaped graph gets tapered towards a point.

At this point. the graph starts moving in the opposite direction.

The point on the graph, where the graph takes a “U”, turn to change its direction, is called the “vertex of the parabola”.

The vertex is an extreme point of a parabola and it can be the highest / lowest of a quadratic graph or the furthest Left/Right point.

For a regular parabola, vertex is also known as maximum or minimum of a parabola.

The vertex of a parabola can also be defined as a point, where the graph of the quadratic function meets the axis of symmetry.

Visual inspection of the parabolic graph can give the vertex coordinates.

However, there is a direct formula to * find the vertex of the parabola*, using the coefficients a,b,c of the quadratic function.

# 2.Domain and Range of a Parabola:

It is easy to find the Domain and range of a parabola when the graph of the same is given unlike when the equation is given.

The domain of a graph is the set of “x” values that a function can take. Here “x” is the independent variable.

The range of a graph is the set of values that the dependent variable “y “takes up.

The vertex of a parabola or a quadratic function helps in finding the domain and range of a parabola.

So, in the next few steps using the coordinates of the vertex of a parabola, we are going to arrive at a table, which can be referred to find the domain and range of any Quadratic graph.

Hence, all that needed to inspect is the orientation of the quadratic graph, and the vertex point, which has the information of maximum or minimum.

In a Vertical/General parabola, for any real value of “x” (-∞,∞) ( which means the value “x” can be any value from – infinity to +infinity ) the graph can have

- Minimum “y” for up-parabola

OR

- Maximum “y” for down-parabola

On similar lines, for a horizontal/sideways parabola, for any value of “y” taken, the graph can have

- Minimum “X” for Right-parabola

OR

- Maximum “X”for Left-parabola

Therefore, in conclusion, a Quadratic graph will either have a maximum, or a minimum value but not both at a time.

Sounds confusing??? Let us look at an example for the same.

### Domain and Range of a Parabola opening up:-

Let us consider the vertical parabola, which is opening up(the one in RED ). The graph extends infinitely along positive “y” axis.

The general form of this equation would be **Y=ax^2 +bx+c.**

The vertex of the up-parabola is at (10,10).

Since the graph does not extend down, beyond the point (10,10) the minimum of parabola does not fall below 10 for any real value of “x”.

Therefore the **DOMAIN is x ∈(-∞,∞)**

The **range is y∈ [10,∞).** we use the ” [ ” closed interval for 10 because Y =10 is included in the range.

### Domain and Range of a Parabola opening down:-

The down-parabola ( the one in Black) opens down and moves infinitely along the negative Y-axis, and has the vertex at (-10,-10).

The general form of this equation would be** Y = – ax^2 +bx +c**.We observe that the sign of the leading coefficient “a” is negative. This is the only difference between up and down facing parabola.

Since the graph does not move up beyond y = -10, the maximum value that the parabola can take is -10 for any real value of “x”.

Therefore the **DOMAIN is x ∈ (-∞,∞)**

The **range is y∈ (-∞,-10]**

### Domain and Range of a Parabola opening Right:-

The Right- parabola ( the one in Purple ) has the vertex at(5,-5) and .X=5 is the “MINIMUM VALUE of X” that the graph can have.

The general form of this equation would be **X = ay^2+ by +c**

The graph starts at x =5 and moves infinitely along the positive “x” axis for any value of “Y” chosen. Therefore, the “Y” coordinate can extend from -infinity to +infinity.

Therefore the **DOMAIN is x ∈ [5,∞)**

The **range is y∈ (- ∞, ∞)**

### Domain and Range of a Parabola opening left:-

And lastly, the Left- parabola ( the one in Green ) has the vertex at(-5,5) and the “MAXIMUM VALUE OF “x” is -5.

The general form of this equation would be **X = – ay^2+ by +c** and the sign of the leading coefficient is negative.

The graph starts at x =-5 and moves infinitely along the negative “x” axis for any value of “Y” chosen. Therefore, the “Y” coordinate can extend from -infinity to +infinity.

Therefore the **DOMAIN is x ∈ (-∞,-5]**

The **range is y∈ (-∞,∞)**

Here is the summary chart, a cheat sheet, for finding the Domain and range of any parabolic graph.

The table below gives the overview of the Domain and Range of a parabola with different orientations.

As a pre-requisite, we first need to identify the orientation and then use this table to pick up the correct Domain and range of parabola.

Let us take a generic vertex (h,k) for the parabola of any orientation and thereby arrive at the Domain and range of a parabola of our interest.

Type of Parabola | Minimum "Y" Co-ordinte value of the Graph. | Maximum"Y" Co-ordinte Value of the Graph | Minimum "X" Co-ordinte Value of the Graph | Maximum "X" Co-ordinte Value of the Graph | Domain of the parabola | Range of the parabola |
---|---|---|---|---|---|---|

Up-Parabola (Parabola opening up) | k | INF | INF | INF | (-INF, INF) | [k,INF) |

Down-Parabola (Parabola opening down) | -INF | k | -INF | INF | (-INF, INF) | (-INF, k] |

Right-Parabola (Parabola opening Right) | -INF | INF | h | INF | [h, INF) | (-INF, INF) |

Left-Parabola (Parabola opening Left) | -INF | INF | -INF | h | (-INF, h] | (-INF, INF) |

Now we arrived at a table, which works to find the domain and range of a parabola of any orientation.

# 4. Summary:-

- A Parabola belongs to the family of conic sections and it is “U” shaped curve.
- The vertex of a parabola is the maximum or minimum point of a parabola.
- The graph of a quadratic function gives as a parabola.
- The vertex of a parabola passes through the axis of symmetry of the graph.
- The orientation of a quadratic graph along with the vertex point is sufficient to arrive at the Domain and Range of parabola.

# 5.Worked out Examples:

**Example-1**. **Identify the orientation and vertex of the parabola. Find the Domain & Range of given parabola thereby.**

**Strategy Step-by-Step:-**

**Step1: **Identify the orientation of the given graph.

Since the two ends of the graph move up towards positive”y” indefinitely, we can conclude that the given parabola is an Up-Parabola. Therefore, there would be a minimum value for the parabola at the vertex.

**Step2: **Identify the vertex of the given quadratic graph.

It is evident that the vertex is located in the Fourth Quadrant, and the coordinates of the vertex **(h,k) are(3,-1) ,** which gives h=3 and k=-1 on a comparison. The minimum value of graph is -1.

**Step3: **Use the chart to identify the generic domain and range of a parabola, for a given orientation.

As per the summary chart, We observe that the for the up-Parabola Domain is x ∈(- ∞,∞) and the range is y∈ [k,∞). We already have k=-1**.** Therefore **The Domain is x ∈(- ∞,∞) and the range is y∈ [-1,∞)** for the given parabola**.**

**Example-2. Identify the orientation and vertex of the parabola. Find the Domain & Range of given parabola thereby.**

**Step1: **Identify the orientation of the given graph.

Since the two ends of the graph move right towards positive “X” infinitely,we can conclude that the given parabola is a Right-Parabola . Therefore, there would be a minimum value of “X” for the parabola.

**Step2: **Identify the vertex of the given quadratic graph.

Since the vertex is located on the “y” axis, The “x” coordinate would be “0”.Therefore the coordinates of the vertex **(h,k) are (0,2),**which gives h=0 and y =2 .The graph takes a minimum value of “x” as 0 at the vertex.

**Step3: **Use the chart to identify the domain and range of a parabola, for a given orientation.

As per the summary chart, we observe that the for the Right-Parabola, Domain is x ∈ [ h,∞) and the range is y∈ (-∞,∞). We already have h =0. Therefore, for the Example-2 given, **The Domain is x ∈ [0,∞) and the range is y∈ (-∞,∞).**

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The graph of a quadratic function with vertex (-1,-2) is shown in the figure below.

Find the range and the domain.

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