**Domain and Range of Quadratic Function Worksheet : **

Worksheet given in this section will be much useful for the students who would like to practice problems on finding domain and range of a quadratic function.

Before look at the worksheet, if you would like to learn, how to find domain and range of a quadratic function,

**Problem 1 : **

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

y = x^{2} + 5x + 6

**Problem 2 : **

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

y = -2x^{2} + 5x - 7

**Problem 1 : **

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

y = x^{2} + 5x + 6

**Solution : **

**Domain : **

In the quadratic function, y = x^{2} + 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 : **

Comparing the given quadratic function y = x^{2} + 5x + 6 with

y = ax^{2} + bx + c

we get

a = 1

b = 5

c = 6

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

Find the x-coordinate at the vertecx.

x = -b / 2a

Substitute 1 for a and 5 for b.

x = -5/2(1)

x = -5/2

x = -2.5

Substitute -2.5 for x in the given quadratic function to find y-coordinate at the vertex.

y = (-2.5)^{2} + 5(-2.5) + 6

y = 6.25 - 12.5 + 6

y = - 0.25

So, y-coordinate of the vertex is -0.25

Because the parabola is open upward, range is all the real values greater than or equal to -0.25

**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 = x^{2} + 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

**Problem 2 : **

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

y = -2x^{2} + 5x - 7

**Solution : **

**Domain :**

In the quadratic function, y = -2x^{2} + 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 : **

Comparing the given quadratic function y = -2x^{2} + 5x - 7 with

y = ax^{2} + bx + c

we get

a = -2

b = 5

c = -7

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

x = -b / 2a

Substitute -2 for a and 5 for b.

x = -5/2(-2)

x = -5/(-4)

x = 5/4

x = 1.25

Substitute 1.25 for x in the given quadratic function to find y-coordinate at the vertex.

y = -2(1.25)^{2} + 5(1.25) - 7

y = -3.125 + 6.25 - 7

y = -3.875

So, y-coordinate of the vertex is -3.875.

Because the parabola is open downward, range is all the real values greater 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 = -2x^{2} + 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, how to find the domain and range of a quadratic function.

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