Graphs of Absolute Value Functions

OverviewThis 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.

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.

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.

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.

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.

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

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).

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).

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.