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 equationwhere and are real numbers and
Graph the absolute value function using a table of values.
To draw the graph, we can plot these points, then connect them with a V-shaped curve.
The graph shows the function
Describe the function's key features including intercepts, the intervals for which it increases and decreases, its minimum and maximum values, and its end behavior. Then, show the features on the graph.
To begin, we'll describe each of the features. A graph's - and -intercepts are the points where the graph intersects with the -axis and -axis, respectively. It can be seen that intersects the -axis at two different points, and The function intersects the -axis at Therefore, the intercepts are Since is an absolute value function, it has a V-shaped graph. This means the function will have both an increasing and a decreasing interval. Looking from left to right on the graph, it can be seen that from the left side of the graph until decreases. Additionally, from to the right side of the graph, increases. Thus, we can express the increasing and decreasing intervals of as follows. Looking at the graph, we can see that both the left end and the right end extend upward. Thus, the end behavior of can be written as follows. Since the function is decreasing in one interval and increasing in another, the function has a minimum. From the graph, we can tell that the minimum is the point Let us now show these features on the graph.