# 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.

 Example 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.

Step by Step Math Worksheets SolversNew !
Linear ProgrammingNew ! Online Step by Step Calculus Calculators and SolversNew ! Factor Quadratic Expressions - Step by Step CalculatorNew ! Step by Step Calculator to Find Domain of a Function New !
Free Trigonometry Questions with Answers -- Interactive HTML5 Math Web Apps for Mobile LearningNew ! -- Free Online Graph Plotter for All Devices
Home Page -- HTML5 Math Applets for Mobile Learning -- Math Formulas for Mobile Learning -- Algebra Questions -- Math Worksheets -- Free Compass Math tests Practice
Free Practice for SAT, ACT Math tests -- GRE practice -- GMAT practice Precalculus Tutorials -- Precalculus Questions and Problems -- Precalculus Applets -- Equations, Systems and Inequalities -- Online Calculators -- Graphing -- Trigonometry -- Trigonometry Worsheets -- Geometry Tutorials -- Geometry Calculators -- Geometry Worksheets -- Calculus Tutorials -- Calculus Questions -- Calculus Worksheets -- Applied Math -- Antennas -- Math Software -- Elementary Statistics High School Math -- Middle School Math -- Primary Math
Math Videos From Analyzemath
Author - e-mail

Updated: February 2015