How to Graph Absolute Value Functions
What Is an Absolute Value Function?
The absolute value of a number is its distance from zero on the number line, always a non-negative result. The function y = |x| assigns to each input x the corresponding non-negative distance. For positive x, |x| = x, so the right half of the graph follows the line y = x. For negative x, |x| = -x, so the left half follows y = -x. These two linear pieces meet at the origin, forming the characteristic V-shape. Unlike a smooth curve, the graph has a sharp corner at the vertex where the two lines join.
Shape and Symmetry of y = |x|
The parent absolute value function y = |x| is an even function, meaning f(-x) = f(x) for all x. Its graph is perfectly symmetric about the y-axis. The vertex sits at the origin (0, 0), which is the lowest point of the V. The domain is all real numbers, and the range is [0, infinity) because the function never produces negative outputs. The two arms of the V each have a slope of magnitude 1: the right arm has slope +1 and the left arm has slope -1. Understanding this parent shape is the foundation for graphing any transformed absolute value function.
Vertex Form: y = a|x - h| + k
Transformed absolute value functions are most clearly written in vertex form: y = a|x - h| + k. In this form, the vertex of the V is at the point (h, k). The parameter h shifts the graph horizontally: positive h moves the vertex right, negative h moves it left. The parameter k shifts the graph vertically: positive k moves the vertex up, negative k moves it down. The coefficient a controls both the width and the orientation of the V, making this form very easy to interpret at a glance once you recognize the roles of each parameter.
Effect of the Coefficient a
The coefficient a in y = a|x - h| + k determines the steepness and direction of the V. When a > 0, the V opens upward with vertex as a minimum point. When a < 0, the V opens downward (an inverted V) with vertex as a maximum point. The magnitude |a| controls the steepness: |a| > 1 makes the arms steeper than the parent function, while 0 < |a| < 1 makes the arms shallower. For example, y = 2|x| has arms with slopes of +2 and -2, producing a narrower V. The function y = (1/2)|x| has arms with slopes of +0.5 and -0.5, a wider and flatter V.
Finding Key Points and Plotting
To graph y = a|x - h| + k, start by identifying and plotting the vertex (h, k). Next, choose x-values to the right of h and compute the corresponding y-values. For each such point (x, y), the symmetric point at (2h - x, y) also lies on the graph. Plot both and connect them to the vertex with straight lines. For accuracy, choose at least two x-values on each side. For example, to graph y = 2|x - 3| + 1: the vertex is at (3, 1). At x = 4, y = 2(1) + 1 = 3, giving (4, 3). By symmetry, (2, 3) is also on the graph. At x = 5, y = 2(2) + 1 = 5, so (5, 5) and (1, 5) are both on the graph.
X-Intercepts and Domain Considerations
To find the x-intercepts of y = a|x - h| + k, set y = 0 and solve |x - h| = -k/a. If -k/a is negative, there are no x-intercepts (the V does not reach the x-axis). If -k/a = 0, the vertex is on the x-axis and that is the only x-intercept. If -k/a is positive, there are two x-intercepts at x = h + (-k/a) and x = h - (-k/a). For instance, y = |x - 2| - 3 has vertex at (2, -3). Setting y = 0 gives |x - 2| = 3, so x - 2 = 3 or x - 2 = -3, yielding x = 5 and x = -1 as the two x-intercepts.
Common Applications and Mistakes
Absolute value functions model situations where only the magnitude of a difference matters, such as the deviation from a target, the distance between two values, or the error in a measurement. A common mistake is forgetting that the graph has a sharp corner at the vertex, not a smooth curve. Another error is applying the horizontal shift in the wrong direction: in y = |x - 3|, the vertex moves to x = +3 (right), not to x = -3. Always extract h directly from the expression inside the absolute value by setting the argument equal to zero.