**Domain and range of a quadratic function :**

The general form of a quadratic function is

y = ax² + bx + c

Domain is nothing but the values of "x" for which the given quadratic function is defined.

Range is nothing but the values of "y" that we get for the the given domain (values of "x").

The domain of any quadratic function is all real values.

Because, in any quadratic function which is in the form

y = ax² + bx + c,

we can plug any real value for "x"

Because, "y" will be defined for all real values of "x".

Therefore, the domain of the quadratic function in the form y = ax² + bx + c is all real values.

That is,

**Domain = { x | x ∈ R }**

To know the range of a quadratic function in the form y = ax² + bx + c, we have to know the following two stuff.

They are,

(i) Parabola is open upward or downward

(ii) "y" - coordinate at the vertex. ** **

Let us see, how to know whether the graph (parabola) of the quadratic function is open upward or downward.

**(i) Parabola is open upward or downward :**

y = ax² + bx + c

If the leading coefficient or the sign of "a" is positive, the parabola is open upward and "a" is negative, the parabola is open downward.

Let us see, how to know whether the graph (parabola) of the quadratic function is open upward or downward.

**(ii) ****"y" - coordinate at the vertex :**

To know y - coordinate of the vertex, we have to find the value "x" using the formula given below.

**x = -b / 2a**

Now, we have to plug x = -b/2a in the given quadratic function.

So, y - coordinate of the quadratic function is

**y = f(-b/2a)**

**How to find range from the above two stuff :**

(i) If the parabola is open upward, the range is all the real values greater than or equal to y = f(-b/2a)

(i) If the parabola is open downward, the range is all the real values less than or equal to y = f(-b/2a)

**Example 1 : **

Find the domain and range of the quadratic function given below.

y = x² + 5x + 6

**Solution : **

**Domain : **

In the quadratic function, y = x² + 5x + 6, we can plug any real value for "x".

Because, "y" is defined for all real values of "x"

Therefore, the domain of the given quadratic function is all real values.

That is,

**Domain = { x | x ∈ R }**

**Range : **

When we compare the given quadratic function with y = x² + 5x + 6, we get

a = 1

b = 5

c = 6

Since the leading coefficient "a" is positive, the parabola is open upward.

"x" coordinate of the vertex = -b / 2a

"x" coordinate of the vertex = -5 / 2(1)

"x" coordinate of the vertex = -5 / 2

"x" coordinate of the vertex = -2.5

"y" coordinate of the vertex = y(-2.5)

"y" coordinate of the vertex = (-2.5)² + 5(-2.5) + 6

"y" coordinate of the vertex = 6.25 - 12.5 + 6

"y" coordinate of the vertex = -0.25 or -1/4

**Vertex ( -2.5, -0.25 )**

Since the parabola is open upward, range is all the real values greater than or equal to -1/4

**Range = { y | y ≥ -0.25 }**

To have better understanding on domain and range of a quadratic function, let us look at the graph of the quadratic function y = x² + 5x + 6.

When we look at the graph, it is clear that "x" (Domain) can take any real value and "y" (Range) can take all real values greater than or equal to -0.25

**Example 2 : **

Find the domain and range of the quadratic function given below.

y = -2x² + 5x - 7

**Solution : **

**Domain :**

In the quadratic function, y = -2x² + 5x - 7, we can plug any real value for "x".

Because, "y" is defined for all real values of "x"

Therefore, the domain of the given quadratic function is all real values.

That is,

**Domain = { x | x ∈ R }**

**Range : **

When we compare the given quadratic function with y = -2x² + 5x - 7, we get

a = -2

b = 5

c = -7

Since the leading coefficient "a" is negative, the parabola is open downward.

"x" coordinate of the vertex = -b / 2a

"x" coordinate of the vertex = -5 / 2(-2)

"x" coordinate of the vertex = -5 / (-4)

"x" coordinate of the vertex = 1.25

"y" coordinate of the vertex = y(1.25)

"y" coordinate of the vertex = -2(1.25)² + 5(1.25) - 7

"y" coordinate of the vertex = -3.125 + 6.25 - 7

"y" coordinate of the vertex = -3.875

**Vertex ( 1.25, -3.875 )**

Since the parabola is open downward, range is all the real values less than or equal to -3.875

**Range = { y | y ****≤**** -3.875 }**

To have better understanding on domain and range of a quadratic function, let us look at the graph of the quadratic function y = -2x² + 5x - 7.

When we look at the graph, it is clear that "x" (Domain) can take any real value and "y" (Range) can take all real values less than or equal to -3.875

After having gone through the stuff given above, we hope that the students would have understood "Domain and range of a quadratic function".

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