# Graph, Domain and Range of Absolute Value Functions

This is a step by step tutorial on how to graph functions with absolute value. Properties of the graph of these functions such as domain, range, x and y intercepts are also discussed. Free graph paper is available.

## Graph, Domain and Range of Absolute Value Functions; Examples with Detailed Solutions

Example 1: f is a function given by

f (x) = |x - 2|

1. Find the x and y intercepts of the graph of f.
2. Find the domain and range of f.
3. 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|

1. Find the x and y intercepts of the graph of f.
2. Find the domain and range of f.
3. 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.
{"ft_lang_label":"__label__en","ft_lang_prob":0.8530403,"math_prob":0.9997272,"size":2830,"snap":"2023-40-2023-50","text_gpt3_token_len":915,"char_repetition_ratio":0.17232838,"word_repetition_ratio":0.36283186,"special_character_ratio":0.36501768,"punctuation_ratio":0.0927357,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9998746,"pos_list":[0,1,2,3,4,5,6,7,8],"im_url_duplicate_count":[null,5,null,5,null,5,null,5,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-09-26T06:26:36Z\",\"WARC-Record-ID\":\"<urn:uuid:44b9c199-2d08-45cb-934b-edf6a8f1b7f4>\",\"Content-Length\":\"23740\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:342e89f1-e288-4e31-9074-1f8eb917ac2d>\",\"WARC-Concurrent-To\":\"<urn:uuid:02e55313-b0c4-4c40-a868-0aec7efbffd9>\",\"WARC-IP-Address\":\"3.210.81.252\",\"WARC-Target-URI\":\"https://www.analyzemath.com/Graphing/Graph_Abs_Val_Func.html\",\"WARC-Payload-Digest\":\"sha1:TXKC3J5NEC4BHXOHIFIFT3U3HEPZTA2W\",\"WARC-Block-Digest\":\"sha1:JDBBXCWXLZTTQY5Z422333EO3PFIYSEU\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-40/CC-MAIN-2023-40_segments_1695233510149.21_warc_CC-MAIN-20230926043538-20230926073538-00461.warc.gz\"}"}