Analyzing Graphs of Absolute Value Functions
Absolute Value Function
An absolute value function is any function that contains the absolute value of a variable expression. These functions can also be described as any function that is a transformation of the absolute value parent function, Since the absolute value of an expression is never negative, the graph of the function will always lie on or above the -axis. Note that This means that the points and both lie on As it turns out, these points lie directly across from each other. In fact, this symmetry exists for all inverse input values. Thus, absolute value graphs have a distinct V-shape.
Any function belonging to the absolute value function family can be written using the equation
where and are real numbers andTo draw the graph, we can plot these points, then connect them with a V-shaped curve.
Solving Absolute Value Equations Graphically
An absolute value equation is an equation that contains the absolute value of a variable expression. An example of this kind of equation is
As is the case with most equations, these can be solved graphically. This is done by moving all terms except the constant term to one side. The function, which is the non-constant side of the equation, is then graphed and the solution(s) to the equation are found as the point(s) on the graph having the -coordinate that equals the constant.When solving an equation graphically, the first step is to rearrange the equation so that the constant term is alone on one side. The left-hand side of the equation can now be expressed as a function, We'll draw the graph of
The solutions to the equation are the -coordinates of the points on the graph that have the -coordinate
Since makes a true statement, it is also a solution. Thus, the equation has the solutions