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

The general form of a quadratic function is

y = ax^{2} + bx + c

Domain is all real values of x for which the given quadratic function is defined.

Range is all real values of y for the given domain (real values values of x).

The general form a quadratic function is

y = ax^{2} + bx + c

The domain of any quadratic function in the above form is all real values.

Because, in the above quadratic function, y is defined for all real values of x.

Therefore, the domain of the quadratic function in the form y = ax^{2} + bx + c is all real values.

That is,

**Domain = {x | x ∈ R}**

To know the range of a quadratic function in the form

y = ax^{2} + 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 of the Parabola . ** **

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 = ax^{2} + 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.

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

To know y - coordinate of the vertex, first 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 = 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

**Example 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.

Apart from the stuff given in this section, if you need any other stuff in math, please use our google custom search here.

Widget is loading comments...

You can also visit our following web pages on different stuff in math.

**WORD PROBLEMS**

**Word problems on simple equations **

**Word problems on linear equations **

**Word problems on quadratic equations**

**Area and perimeter word problems**

**Word problems on direct variation and inverse variation **

**Word problems on comparing rates**

**Converting customary units word problems **

**Converting metric units word problems**

**Word problems on simple interest**

**Word problems on compound interest**

**Word problems on types of angles **

**Complementary and supplementary angles word problems**

**Trigonometry word problems**

**Markup and markdown word problems **

**Word problems on mixed fractrions**

**One step equation word problems**

**Linear inequalities word problems**

**Ratio and proportion word problems**

**Word problems on sets and venn diagrams**

**Pythagorean theorem word problems**

**Percent of a number word problems**

**Word problems on constant speed**

**Word problems on average speed **

**Word problems on sum of the angles of a triangle is 180 degree**

**OTHER TOPICS **

**Time, speed and distance shortcuts**

**Ratio and proportion shortcuts**

**Domain and range of rational functions**

**Domain and range of rational functions with holes**

**Graphing rational functions with holes**

**Converting repeating decimals in to fractions**

**Decimal representation of rational numbers**

**Finding square root using long division**

**L.C.M method to solve time and work problems**

**Translating the word problems in to algebraic expressions**

**Remainder when 2 power 256 is divided by 17**

**Remainder when 17 power 23 is divided by 16**

**Sum of all three digit numbers divisible by 6**

**Sum of all three digit numbers divisible by 7**

**Sum of all three digit numbers divisible by 8**

**Sum of all three digit numbers formed using 1, 3, 4**

**Sum of all three four digit numbers formed with non zero digits**