Graphing Square Root Functions

Example 1: Graph

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.

Example 2: Graph

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

Example 3: Graph

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

Example 4: Graph

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.

Example 5: Graph

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

Example 6: Graph

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.

Example 7: Graph

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.