The Modulus (Absolute Value) Function

Modulus Function: The modulus function, also written as the absolute value function, is defined as:

$∣ x ∣ = {x - x if x \geq 0 if x < 0$

It returns the non-negative magnitude of any real number.

In plain terms, the modulus function strips away any negative sign. So $∣ - 5 ∣ = 5$ , $∣3∣ = 3$ , and $∣0∣ = 0$ .

When applied to functions, the modulus transforms graphs in predictable and examinable ways. There are two distinct cases you need to master:

$y = ∣ f (x) ∣$ , modulus applied to the output
$y = f (∣ x ∣)$ , modulus applied to the input

These two cases produce very different graph shapes, so it is critical not to mix them up.

Warning

$∣ f (x) ∣$ and $f (∣ x ∣)$ are not the same transformation. Confusing these two is one of the most common errors in this topic.

Expressing modulus functions as piecewise functions

Before graphing, it is often helpful to write $∣ f (x) ∣$ explicitly as a piecewise function. This is standard IB HL practice and makes the case structure visible:

$∣ f (x) ∣ = {f (x) - f (x) if f (x) \geq 0 if f (x) < 0$

For example, $∣2 x - 1∣$ can be written as:
$∣2 x - 1∣ = {2 x - 1 - (2 x - 1) = 1 - 2 x if x \geq \frac{1}{2} if x < \frac{1}{2}$

This piecewise form directly tells you the gradient on each branch and where corner points occur , essential information for both sketching and solving.

Graphing y = |f(x)|

When the modulus is applied to the output of a function, any part of the graph that lies below the x-axis is reflected upward into the region $y \geq 0$ . The parts of the graph already above the x-axis remain unchanged.

Method for sketching $y = ∣ f (x) ∣$ :

Sketch $y = f (x)$ as normal.
Identify all portions where $f (x) < 0$ (i.e., below the x-axis).
Reflect those portions in the x-axis (flip them upward).
The x-intercepts of $f (x)$ become corner points (sharp, non-differentiable points) on $y = ∣ f (x) ∣$ .

Note

The graph of $y = ∣ f (x) ∣$ always satisfies $y \geq 0$ for all $x$ in the domain. It can never go below the x-axis by definition.

Note on terminology: The points at x-intercepts are called corner points or sharp points , they are not differentiable because the gradient changes sign abruptly. Do not call them cusps; a cusp has a specific different meaning in mathematics.

Example

Sketch $y = ∣ x^{2} - 4∣$ .

First, write as a piecewise function:
$∣ x^{2} - 4∣ = {x^{2} - 4 - (x^{2} - 4) = 4 - x^{2} if x \leq - 2 or x \geq 2 if - 2 < x < 2$

Step 1: Sketch $y = x^{2} - 4$ . This is an upward-opening parabola with vertex at $(0, - 4)$ and x-intercepts at $x = \pm 2$ .

Step 2: Identify the region below the x-axis: this occurs for $- 2 < x < 2$ , where $x^{2} - 4 < 0$ .

Step 3: Reflect that central portion upward. The section of the parabola between $x = - 2$ and $x = 2$ flips to become an upward-opening arch above the x-axis (the downward-pointing section $- (x^{2} - 4) = 4 - x^{2}$ is a downward parabola, but since we are reflecting the negative portion upward, the central arch peaks at $(0, 4)$ and opens downward , creating the characteristic W-shape overall).

Key features of $y = ∣ x^{2} - 4∣$ :

Touches the x-axis at $x = \pm 2$ (corner points , not differentiable here)
Local maximum at $(0, 4)$ (from the reflected vertex)
The outer arms of the parabola ( $x < - 2$ and $x > 2$ ) are unchanged, rising without bound
Overall shape: a W-shape with two outer rising arms and a central peak at $(0, 4)$

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9 more sections in this topic

The Modulus (Absolute Value) Function

Modulus Function: The modulus function, also written as the absolute value function, is defined as:

$∣ x ∣ = {x - x if x \geq 0 if x < 0$

It returns the non-negative magnitude of any real number.

In plain terms, the modulus function strips away any negative sign. So $∣ - 5 ∣ = 5$ , $∣3∣ = 3$ , and $∣0∣ = 0$ .

When applied to functions, the modulus transforms graphs in predictable and examinable ways. There are two distinct cases you need to master:

$y = ∣ f (x) ∣$ , modulus applied to the output
$y = f (∣ x ∣)$ , modulus applied to the input

These two cases produce very different graph shapes, so it is critical not to mix them up.

Warning

$∣ f (x) ∣$ and $f (∣ x ∣)$ are not the same transformation. Confusing these two is one of the most common errors in this topic.

Expressing modulus functions as piecewise functions

Before graphing, it is often helpful to write $∣ f (x) ∣$ explicitly as a piecewise function. This is standard IB HL practice and makes the case structure visible:

$∣ f (x) ∣ = {f (x) - f (x) if f (x) \geq 0 if f (x) < 0$

For example, $∣2 x - 1∣$ can be written as:
$∣2 x - 1∣ = {2 x - 1 - (2 x - 1) = 1 - 2 x if x \geq \frac{1}{2} if x < \frac{1}{2}$

This piecewise form directly tells you the gradient on each branch and where corner points occur , essential information for both sketching and solving.

Graphing y = |f(x)|

Method for sketching $y = ∣ f (x) ∣$ :

Sketch $y = f (x)$ as normal.
Identify all portions where $f (x) < 0$ (i.e., below the x-axis).
Reflect those portions in the x-axis (flip them upward).
The x-intercepts of $f (x)$ become corner points (sharp, non-differentiable points) on $y = ∣ f (x) ∣$ .

Note

The graph of $y = ∣ f (x) ∣$ always satisfies $y \geq 0$ for all $x$ in the domain. It can never go below the x-axis by definition.

Example

Sketch $y = ∣ x^{2} - 4∣$ .

First, write as a piecewise function:
$∣ x^{2} - 4∣ = {x^{2} - 4 - (x^{2} - 4) = 4 - x^{2} if x \leq - 2 or x \geq 2 if - 2 < x < 2$

Step 1: Sketch $y = x^{2} - 4$ . This is an upward-opening parabola with vertex at $(0, - 4)$ and x-intercepts at $x = \pm 2$ .

Step 2: Identify the region below the x-axis: this occurs for $- 2 < x < 2$ , where $x^{2} - 4 < 0$ .

Key features of $y = ∣ x^{2} - 4∣$ :

Touches the x-axis at $x = \pm 2$ (corner points , not differentiable here)
Local maximum at $(0, 4)$ (from the reflected vertex)
The outer arms of the parabola ( $x < - 2$ and $x > 2$ ) are unchanged, rising without bound
Overall shape: a W-shape with two outer rising arms and a central peak at $(0, 4)$

Free preview

AHL 2.16—Graphing modulus equations and inequalities

The Modulus (Absolute Value) Function

Graphing y = |f(x)|

9 more sections in this topic

AHL 2.16—Graphing modulus equations and inequalities

The Modulus (Absolute Value) Function

Graphing y = |f(x)|

9 more sections in this topic