
Graph, Domain and Range of Absolute Value Functions; Examples with Detailed SolutionsExample 1: f is a function given by
f (x) = x  2
 Find the x and y intercepts of the graph of f.
 Find the domain and range of f.
 Sketch the graph of f.
Solution to Example 1

a  The y intercept is given by
(0 , f(0)) = (0 ,2) = (0 , 2)

The x coordinate of the x intercepts is equal to the solution of the equation
x  2 = 0
which is
x = 2

The x intercepts is at the point (2 , 0)

b  The domain of f is the set of all real numbers
Since x  2 is either positive or zero for x = 2; the range of f is given by the interval [0 , +infinity).

c  To sketch the graph of f(x) = x  2, we first sketch the graph of y = x  2 and then take the absolute value of y.
The graph of y = x  2 is a line with x intercept (2 , 0) and y intercept (0 , 2). (see graph below)
 We next use the definition of the absolute value to graph f(x) = x  2 =  y .
If y >= 0 then  y  = y , if y <0 then  y  = y.

For values of x for which y is positive, the graph of  y  is the same as that of y = x  2. For values of x for which y is negative, the graph of  y  is a reflection on the x axis of the graph of y. The graph of y = x  2 above has y negative on the interval (infinity , 2) and it is this part of the graph that has to be reflected on the x axis. (see graph below).
 Check that the range is given by the interval [0 , +infinity), the domain is the set of all real numbers, the y intercept is at (0 , 2) and the x intercept at (2, 0).
Example 2: f is a function given by
f (x) = (x  2)^{2}  4
 Find the x and y intercepts of the graph of f.
 Find the domain and range of f.
 Sketch the graph of f.
Solution to Example 2

a  The y intercept is given by
(0 , f(0)) = (0 ,(2)^{2}  4) = (0 , 0)

The x coordinates of the x intercepts are equal to the solutions of the equation
(x  2)^{2}  4 = 0
which is solved
(x  2)^{2} = 4
Which gives the solutions
x = 0 and x = 4

The x intercepts is at the point (0 , 0) and (4 , 0)

b  The domain of f is the set of all real numbers
Since (x  2)^{2}  4 is either positive or zero for x = 4 and x = 0; the range of f is given by the interval [0 , +infinity).

c  To sketch the graph of f(x) = (x  2)^{2}  4, we first sketch the graph of y = (x  2)^{2}  4 and then take the absolute value of y.
The graph of y = (x  2)^{2}  4 is a parabola with vertex at (2,4), x intercepts (0 , 0) and (4 , 0) and a y intercept (0 , 0). (see graph below)

The graph of f is given by reflecting on the x axis part of the graph of y = (x  2)^{2}  4 for which y is negative. (see graph below).
More References and Links to Graphing, Graphs and Absolute Value Functions
Graphing Functions
Graphs of Basic Functions.
Absolute Value Functions.
Definition of the Absolute Value.
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