How to Graph Absolute Value Function?

Graphing absolute value equations produces a distinctive V-shaped graph. Every absolute value function has a vertex — the tip of the V — and opens either upward or downward depending on the sign of the leading coefficient. Once you identify the vertex and direction, you can sketch the entire graph by plotting a few additional points. This guide walks you through the rules and transformations, with worked examples, two video lessons, and practice problems.

What Does an Absolute Value Graph Look Like?

The parent function f(x) = |x| produces a V-shape with its vertex at the origin (0, 0). The two sides of the V are straight lines with slopes of 1 (right side) \(\color{blue}{\text{ and } -1}\) (left side). Transformations shift, stretch, compress, or flip this base graph.

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How to Graph an Absolute Value Function

Identifying the Vertex

For a function in the form \(\color{blue}{f(x) = a}\)|\(\color{blue}{x – h}\)| + k:

• The vertex is at (h, k).
• If a > 0, the V opens upward; if a < 0, the V opens downward.
• |a| stretches (|a| > 1) or compresses (0 < |a| < 1) the graph vertically.

Quick example: \(\color{blue}{f(x) = 2}\)|\(\color{blue}{x + 1}\)| − 4 → vertex at (−1, −4); opens upward; steeper than the parent.

Plotting Additional Points

Choose x-values on each side of the vertex, evaluate f(x), and plot the points. Connect them in a V-shape.

Quick example: For f(x) = |\(\color{blue}{x – 2}\)| + 3 (vertex at (2, 3)):
\(\color{blue}{x = 0}\): f(0) = |\(\color{blue}{0-2}\)| + \(\color{blue}{3 = 5}\), giving (0, 5)
\(\color{blue}{x = 4}\): f(4) = |\(\color{blue}{4-2}\)| + \(\color{blue}{3 = 5}\), giving (4, 5)

Domain and Range

• Domain: All real numbers (x can be any value).
• Range (opens up): y ≥ k (all values at or above the vertex y-coordinate).
• Range (opens down): y ≤ k (all values at or below the vertex y-coordinate).

Step-by-Step Summary

1. Write the function in vertex form: \(\color{blue}{f(x) = a}\)|\(\color{blue}{x – h}\)| + k.
2. Identify the vertex (h, k).
3. Determine the direction: a > 0 opens up; a < 0 opens down.
4. Note the slope of each arm: the right arm has slope a; the left arm has \(\color{blue}{\text{ slope } -a}\).
5. Plot the vertex and at least two points on each side, then connect with straight lines.
6. State the domain (all reals) and range (y ≥ k or y ≤ k).

Watch: Graphs of Absolute Value Functions (Video Lesson)

Khan Academy explains the vertex form and how each parameter transforms the graph:

Graphing Absolute Value Equations – Worked Examples

Example 1: Graph f(x) = |x|. Identify the vertex, direction, domain, and range.

Vertex: (0, 0)  Direction: opens up  Domain: all real numbers  Range: \(\color{blue}{y \ge 0}\)
Points: (−2, 2), (−1, 1), (0, 0), (1, 1), (2, 2)

Example 2: Graph f(x) = |\(\color{blue}{x – 2}\)| + 3. Find the vertex and two more points.

Vertex: (2, 3)  Direction: opens up
\(\color{blue}{x = 0}\): |\(\color{blue}{0-2}\)|+\(\color{blue}{3 = 2+3 = 5}\) → (0, 5)
\(\color{blue}{x = 4}\): |\(\color{blue}{4-2}\)|+\(\color{blue}{3 = 2+3 = 5}\) → (4, 5)
Range: \(\color{blue}{y \ge 3}\)

Example 3: Graph \(\color{blue}{f(x) = 2}\)|\(\color{blue}{x + 1}\)| − 4. Find the vertex, direction, and range.

Vertex: (−1, −4)  Direction: opens up (\(\color{blue}{a = 2}\) > 0); steeper than parent
x = −3: 2|−\(\color{blue}{3+1}\)|−\(\color{blue}{4 = 2(2)-4 = 0}\) → (−3, 0)
\(\color{blue}{x = 1}\): 2|\(\color{blue}{1+1}\)|−\(\color{blue}{4 = 2(2)-4 = 0}\) → (1, 0)
Range: y ≥ −4

Example 4: Graph f(x) = −|x| + 5. Find the vertex, direction, and range.

Vertex: (0, 5)  Direction: opens down (a = −1 < 0)
x = −2: −|−2|+5 = −\(\color{blue}{2+5 = 3}\) → (−2, 3)
\(\color{blue}{x = 2}\): −|2|+5 = −\(\color{blue}{2+5 = 3}\) → (2, 3)
Range: \(\color{blue}{y \le 5}\)

Domain, Range, and Transformations (Video Lesson)

The Organic Chemistry Tutor covers graphing absolute value functions including domain and range in detail:

Exercises: Graphing Absolute Value Equations

1. Identify the vertex and direction for f(x) = |\(\color{blue}{x + 4}\)| − 1.
2. Identify the vertex and direction for f(x) = −3|\(\color{blue}{x – 5}\)| + 2.
3. Graph f(x) = |\(\color{blue}{x – 1}\)| and find the range.
4. Write the vertex form of an absolute value function with vertex (3, −2) that opens upward.
5. What is the range of f(x) = −|\(\color{blue}{x + 2}\)| + 6?
6. How does the graph of \(\color{blue}{f(x) = 4}\)|x| differ from the parent function f(x) = |x|?

Answers

1. Vertex: (−4, −1); opens upward.
2. Vertex: (5, 2); opens downward (a = −3 < 0).
3. Vertex (1, 0), opens up; range: \(\color{blue}{y \ge 0}\).
4. f(x) = |\(\color{blue}{x – 3}\)| − 2 (or any \(\color{blue}{f(x) = a}\)|\(\color{blue}{x-3}\)|−2 with a > 0).
5. \(\color{blue}{y \le 6}\) (opens down, vertex at (−2, 6)).
6. The graph is narrower (steeper) than the parent by a factor of 4; both open up with the same vertex (0, 0).

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Free Graphing Absolute Value Equations Worksheet

Ready to practice on your own? Download our free Graphing Absolute Value Equations worksheet below, work through each problem at your own pace, and then check your answers. If a few give you trouble, scroll back up to the worked examples and try again, steady practice is the surest way to master Graphing Absolute Value Equations before a quiz or test.

Download Absolute Value Equations Worksheet

Frequently Asked Questions

What is the formula for graphing absolute value functions?

The general form is y = a|x − h| + k. The vertex, or tip of the V, is at (h, k). If a is positive the V opens up; if a is negative it opens down. The size of a sets the steepness: each side has slope a or −a. For y = |x − 3| + 1, the vertex is (3, 1) and the V opens upward with slopes 1 and −1.

What is the graph of an absolute value function called?

The graph is usually called a V-shaped graph, or simply a V, because of its two straight rays meeting at a sharp point. That point is called the vertex. The parent function y = |x| has its vertex at the origin, and every other absolute value graph is a shifted, stretched or flipped version of that V.

How do you graph absolute value functions with transformations?

Start from y = |x| and apply the changes one at a time. For y = −2|x + 1| + 3, the vertex moves to (−1, 3), the negative sign flips the V to open downward, and the 2 makes each side twice as steep. Plot the vertex, then step 1 unit left and right and drop 2 units each way to (0, 1) and (−2, 1), and draw the rays.

How do you graph absolute value functions with a table?

Choose x-values on both sides of the vertex, compute y, and plot the points. For y = |x − 2|, the vertex is at x = 2, so try x = 0, 1, 2, 3 and 4. The y-values are 2, 1, 0, 1 and 2. Plot them and connect with two straight lines meeting at (2, 0). Picking values evenly around the vertex shows the symmetry clearly.

How do you find the domain and range of an absolute value function?

The domain of y = a|x − h| + k is all real numbers, since you can take the absolute value of any number. The range depends on the vertex and direction. If the V opens up, the range is y ≥ k; if it opens down, the range is y ≤ k. For y = −|x| + 5, the range is y ≤ 5.

What does absolute value do to a graph?

Taking the absolute value of a function, y = |f(x)|, reflects any part of the graph that lies below the x-axis up above it, because absolute value turns negative outputs positive. That is why y = |x| looks like the line y = x with its left half flipped upward. Parts of the graph already above the x-axis stay exactly where they are.

Related Topics

• Solving Absolute Value Equations
• Solving Absolute Value Inequalities
• Graphing Absolute Value Inequalities
• Piecewise Functions
• Graphing Linear Functions