Graphing Square Root Functions

A step by step tutorial on graphing and sketching square root functions . The graph, domain, range of these functions and other properties are discussed.

In what follows, SQRT means square root.

The domain of function f defined by f(x) = SQRT ( x ) is the set of all real positive numbers and zero because the square root of negative numbers are not real numbers (think of SQRT (- 4), is it real?). In inequality form, the domain of f(x) = SQRT ( x ) is written as

x ≥ 0

in interval form the domain is given by

[ 0 , + infinity)

Example 1: Graph

f( x ) = SQRT (x)

and find the range of f.

Solution to Example 1:

Because the domain of f is the set of all positive real numbers and zero, we might construct a table of values as follows:

 x 0 1 4 9 16 SQRT (x) 0 1 2 3 4

The values of x were selected so that the square root of these values are whole numbers which make it easy to plot the points shown in the table.

The range of f is given by the interval [0 , +infinity).

Example 2: Graph

f( x ) = SQRT (x - 3)

and find the range of f.

Solution to Example 2:

First find the domain of the square root function given above by stating that the expression under the square root must be positive or equal to zero

x - 3 ≥ 0

Solve the above inequality to obtain the domain of f as the set of all real values such that

x ≥ 3

We now select values of x in the domain to construct a table of values.

 x 3 4 7 12 SQRT (x - 3) 0 1 2 3

The interval [0 , +infinity) represents the range of f.

Example 3: Graph

f( x ) = - SQRT (- 2x + 4) + 1

and find the range of f.

Solution to Example 3:

The domain of the function given above is found by setting

- 2x + 4 ≥ 0

Solve the above inequality to obtain the domain of f as the set of all real values such that

x ≤ 2

We now select values of x in the domain of f to construct a table of values. These values are selected so that the square root term is a whole number and give points that are easy to plot.

 x 2 3/2 0 -5/2 -6 - SQRT (-2 x + 4 ) + 1 1 0 -1 -2 -3

The range of f is given by the interval (-infinity , 1].

Example 4: Graph

f( x ) = SQRT (- x 2 + 4)

and find the range of f.

Solution to Example 4:

The domain of function given above is found by solving the polynomial inequality

- x 2 + 4 ≥ 0

The solution set of the above inequality is given by the interval

[-2 , 2]

which is also the domain of the above function.

Let us write the given function as an equation as follows

y = SQRT (- x 2 + 4)

Square both sides and arrange to obtain.

x 2 + y 2 = 2 2

The equation obtained is that of a circle. Hence the graph of f(x) = SQRT (- x 2 + 4) is the upper half of a circle sinsce SQRT (- x 2 + 4) is positive. Hence the graph below.

The interval [0 , 2] represents the range of f.

Example 5: Graph

f( x ) = SQRT (x 2 - 9)

and find the range of f.

Solution to Example 5:

The domain of the function given above is found by solving

x
2 - 9 ≥ 0

Which gives a domain reprsented by

(-infinity , -3] U [3 , + infinity)

We now select values of x in the domain of f to construct a table of values, noting f(x) = f(-x) hence a symmetry of the graph with respect to the y axis.

 x 3 5 8 SQRT (x 2 - 9) 0 4 7.4

The range of f is given by the interval [0 , + infinity).

Example 6: Graph

f( x ) = SQRT (x 2 - 6x + 9)

and find the range of f.

Solution to Example 6:

Let us use write the expression under the square root as a square as follows

x 2 - 6x + 9 = (x - 3) 2

Hence

f( x ) = SQRT (x 2 - 6x + 9)

= SQRT ( (x - 3) 2 ) = | x - 3 |

The given function has been rewitten as an absolute value function. Function f may be written as a piecewise function and graphed as follows.

The range of f is given by the interval [0 , + infinity).

Example 7: Graph

f( x ) = SQRT (x 2 + 4x + 6)

and find the range of f.

Solution to Example 7:

Use completing the square to rewtite the expression under the square root as follows

x 2 + 4x + 6 = (x + 2) 2 + 2

The expression under the square root is always positive hence the domain of f is the set of all real numbers. Let us first look at the graph of (x + 2) 2 + 2. It is a parabola.

We would expect the graph of f to have the same axis of symmetry, the vertical line, x = -2 as the above graph. The table of values may constructed as follows.

 x -2 0 2 4 SQRT ( (x + 2) 2 + 2 ) 1.4 2.4 4.2 6.2

The range of f is given by the interval [SQRT(2) , + infinity).

More references and links on graphing.
Graphing Functions