This document provides 21 examples of how to graph absolute value functions by making variations to the basic absolute value function f(x) = |x|. These variations include changing the width of the graph using f(a*x), translating the graph along the x-axis using f(x-a), translating along the y-axis using f(x)+a, and combining multiple transformations. All examples are explained to show how each variation affects the vertex and orientation of the absolute value graph.

1. Graphs of Absolute Value Functions

2. Overview
This set of tutorials provides 21 examples of graphs of
absolute value functions, taking into account variations in the
location of the vertex and orientation of the graph.
The basic absolute value function is shown below. The
worked out examples look at these variations: a•f(x), f(x – a),
f(a•x), f(x) – a, among others.

3. This is a graph of the base function f(x) = |x|. Students should note
that the vertex of the graph is at the origin and that all values of f(x)
are non-negative.

4. In this variation we look at f(a•x). In this case a > 1, and as a result
narrows the graph of the function, when compared to the base
function in Example 1.

5. In this variation we also look at f(a•x). Although in this case a < 1, the absolute
value symbol ensures that a remains positive. Note that increasing values of a
result in a narrower graph of the function, when compared to the base function
in Example 1.

6. In this variation we look at a•f(x). In this case a > 1, and as a result
narrows the graph of the function, when compared to the base
function in Example 1.

7. In this variation we also look at a•f(x). However, note that a < 1,
and since the value of a is outside the absolute value sign, it
causes the graph to flip its orientation relative to the x-axis

8. In this variation we also look at f(x – a), to see that this results in a
horizontal shift of the base graph along the x-axis. When the function
is written in the form f(x – a), the coordinates of the vertex are (a, 0).

9. In this variation we also look at f(x – a), to see that this results in a
horizontal shift of the base graph along the x-axis. When the function is
written in the form f(x – a), the coordinates of the vertex are (a, 0).

10. In this variation we look at f(x) + a. This results in a vertical shift of
the base function. In this example a > 0, which results in an upward
shift of the graph.

11. In this variation we look at f(x) + a. This results in a vertical shift of
the base function. In this example a < 0, which results in an
downward shift of the graph.

12. In this variation we look at f(a•x – b), which combines two previous variations:
changing the width of graph and translating the graph along the x-axis. Since a
> 1, the graph is narrower and when written in a•x – b format, the horizontal
shift is toward negative x.

13. In this variation we also look at f(a•x – b), which combines two previous
variations: changing the width of graph and translating the graph along the x-
axis. Since a > 1, the graph is narrower and when written in a•x – b format, the
horizontal shift is toward positive x.

14. In this variation we also look at f(a•x – b), which combines two previous variations: changing the
width of graph and translating the graph along the x-axis. Although a < 1, the absolute value sign
ensures the orientation is above the x-axis. When written in a•x – b format, the horizontal shift is
toward positive x.

15. In this variation we look at f(–x – a). This maintains the width of the base function
but when written in the form x – a, the horizontal shift is toward negative x. So,
although the negative sign doesn’t change the vertical orientation of the graph, it
does change the horizontal orientation.

16. In this variation we also look at f(a•x – b) + c, which combines three previous variations:
changing the width of graph, translating the graph along the x-axis, and translating the graph
along the y-axis. This example results in the vertex in the Quadrant II. For this and the remaining
examples, the x coordinate of the vertex is the ratio of b and a.

17. In this variation we also look at f(a•x – b) + c, which combines three previous variations:
changing the width of graph, translating the graph along the x-axis, and translating the graph
along the y-axis. This example results in the vertex in the Quadrant I. For this and the remaining
examples, the x coordinate of the vertex is the ratio of b and a.

18. In this variation we also look at f(a•x – b) + c, which combines three previous variations:
changing the width of graph, translating the graph along the x-axis, and translating the graph
along the y-axis. This example results in the vertex in the Quadrant III. For this and the
remaining examples, the x coordinate of the vertex is the ratio of b and a.

19. In this variation we also look at f(a•x – b) + c, which combines three previous variations:
changing the width of graph, translating the graph along the x-axis, and translating the graph
along the y-axis. This example results in the vertex in the Quadrant IV. For this and the
remaining examples, the x coordinate of the vertex is the ratio of b and a.

20. In this variation we also look at f(a•x – b) + c, which combines three previous variations:
changing the width of graph, translating the graph along the x-axis, and translating the graph
along the y-axis. This example results in the vertex in the Quadrant I. For this and the remaining
examples, the x coordinate of the vertex is the ratio of b and a.

21. In this variation we also look at f(a•x – b) + c, which combines three previous variations:
changing the width of graph, translating the graph along the x-axis, and translating the graph
along the y-axis. This example results in the vertex in the Quadrant II. For this and the remaining
examples, the x coordinate of the vertex is the ratio of b and a.

22. In this variation we also look at f(a•x – b) + c, which combines three previous variations:
changing the width of graph, translating the graph along the x-axis, and translating the graph
along the y-axis. This example results in the vertex in the Quadrant IV. For this and the
remaining examples, the x coordinate of the vertex is the ratio of b and a.

23. In this variation we also look at f(a•x – b) + c, which combines three previous
variations: changing the width of graph, translating the graph along the x-axis, and
translating the graph along the y-axis. This example results in the vertex in the
Quadrant III. The x coordinate of the vertex is the ratio of b and a.