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|?

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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 vertex of an absolute value function?

The vertex is the lowest point of a V that opens up (minimum) or the highest point of a V that opens down (maximum). For \(\color{blue}{f(x) = a}\)|\(\color{blue}{x – h}\)| + k, the vertex is always (h, k).

How does a negative leading coefficient affect the graph?

A negative leading coefficient (a < 0) flips the V upside down, so the graph opens downward. The vertex becomes a maximum point instead of a minimum.

Can an absolute value function have a horizontal V?

The standard absolute value function always produces lines with non-zero slopes on each side of the vertex. The arms can be made steeper (|a| > 1) or shallower (0 < |a| < 1), but they can never be perfectly horizontal unless you are working with transformations of other functions.

Related Topics

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