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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.

points and graph of SQRT(x)


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




points and graph of SQRT(x - 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




points and graph of - SQRT(-2x + 4) + 1


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.

points and graph of SQRT (- x<sup> 2</sup> + 4)
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




points and graph of SQRT (x<sup> 2</sup> - 9)


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.

points and graph of SQRT (x<sup> 2</sup> - 6x + 9)


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.

points and graph of x<sup> 2</sup> + 4x + 6
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


points and graph of SQRT (x<sup> 2</sup> - 6x + 9)


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


More references and links on graphing.
Graphing Functions