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, f(x)=∣x∣. Since the absolute value of an expression is never negative, the graph of the function f(x)=∣x∣ will always lie on or above the x-axis. Note that ∣-5∣=5and∣5∣=5. This means that the points (-5,5) and (5,5) both lie on f(x)=∣x∣. 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 y=a∣x−h∣+k
where a, h, and k are real numbers and a=0.Graph the absolute value function f(x)=∣x−2∣−3 using a table of values.
x | ∣x−2∣−3 | f(x) |
---|---|---|
-2 | ∣-2−2∣−3 | 1 |
-1 | ∣-1−2∣−3 | 0 |
0 | ∣0−2∣−3 | -1 |
1 | ∣1−2∣−3 | -2 |
2 | ∣2−2∣−3 | -3 |
3 | ∣3−2∣−3 | -2 |
4 | ∣4−2∣−3 | -1 |
5 | ∣5−2∣−3 | 0 |
6 | ∣6−2∣−3 | 1 |
7 | ∣7−2∣−3 | 2 |
To draw the graph, we can plot these points, then connect them with a V-shaped curve.
An absolute value equation is an equation that contains the absolute value of a variable expression. An example of this kind of equation is ∣∣∣2x2−8∣∣∣=5.
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 y-coordinate that equals the constant.Solve the equation graphically. ∣2x−5∣−3=0
When solving an equation graphically, the first step is to rearrange the equation so that the constant term is alone on one side. ∣2x−5∣−3=0⇔∣2x−5∣=3 The left-hand side of the equation can now be expressed as a function, f(x)=∣2x−5∣. We'll draw the graph of f.
The solutions to the equation y=∣2x−5∣=3 are the x-coordinates of the points on the graph that have the y-coordinate 3.
Since x=4 makes a true statement, it is also a solution. Thus, the equation has the solutions x=1andx=4.