# Nakafa Framework: LLM

URL: https://nakafa.com/en/subject/high-school/11/mathematics/function-modeling/absolute-value-function
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/subject/high-school/11/mathematics/function-modeling/absolute-value-function/en.mdx

Output docs content for large language models.

---

import { getColor } from "@repo/design-system/lib/color";
import { LineEquation } from "@repo/design-system/components/contents/line-equation";

export const metadata = {
  title: "Absolute Value Function",
  description: "Master absolute value functions with interactive graphs, transformations, and step-by-step solutions. Learn properties, equations, and real applications.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "05/18/2025",
  subject: "Functions and Their Modeling",
};

## Understanding Absolute Value Functions

An absolute value function is a function that produces positive or zero values from any input, regardless of the original sign of the input. Geometrically, absolute value can be understood as the distance of a number from the zero point on the number line.

### Mathematical Definition

For any real number <InlineMath math="x" />, the absolute value function is defined as:

<BlockMath math="f(x) = |x| = \begin{cases} x, & \text{if } x \geq 0 \\ -x, & \text{if } x < 0 \end{cases}" />

**Components of absolute value functions:**

- The symbol <InlineMath math="|x|" /> is read as "absolute value of x"
- The function result is always non-negative (<InlineMath math="|x| \geq 0" />)
- This function is even: <InlineMath math="|-x| = |x|" />

## Properties of Absolute Value Functions

Absolute value functions have several important properties that need to be understood:

**Basic properties:**

<div className="flex flex-col gap-4">
  <BlockMath math="|x| \geq 0 \text{ for all } x \in \mathbb{R}" />
  <BlockMath math="|-x| = |x|" />
  <BlockMath math="|x|^2 = x^2" />
  <BlockMath math="|xy| = |x| \cdot |y|" />
  <BlockMath math="\left|\frac{x}{y}\right| = \frac{|x|}{|y|}, \text{ with } y \neq 0" />
</div>

**Triangle inequality properties:**

<div className="flex flex-col gap-4">
  <BlockMath math="|x + y| \leq |x| + |y|" />
  <BlockMath math="||x| - |y|| \leq |x - y|" />
</div>

## Graphs of Absolute Value Functions

The following is a visualization of the basic absolute value function:

<LineEquation
  title={
    <>
      Graph of Function <InlineMath math="f(x) = |x|" />
    </>
  }
  description={
    <>
      The graph shows the characteristic shape of an absolute value function
      that forms the letter V.
    </>
  }
  data={[
    {
      points: Array.from({ length: 21 }, (_, i) => {
        const x = i - 10;
        return { x, y: Math.abs(x), z: 0 };
      }),
      color: getColor("PURPLE"),
      showPoints: false,
      smooth: false,
      labels: [{ text: "f(x) = |x|", at: 15, offset: [1, -1, 0] }],
    },
  ]}
  cameraPosition={[0, 0, 15]}
  showZAxis={false}
/>

**Value table for function <InlineMath math="f(x) = |x|" />:**

| <InlineMath math="x" /> | <InlineMath math="-4" /> | <InlineMath math="-3" /> | <InlineMath math="-2" /> | <InlineMath math="-1" /> | <InlineMath math="0" /> | <InlineMath math="1" /> | <InlineMath math="2" /> | <InlineMath math="3" /> | <InlineMath math="4" /> |
| ---------------------- | ---------------------- | ---------------------- | ---------------------- | ---------------------- | ---------------------- | ---------------------- | ---------------------- | ---------------------- | ---------------------- |
| <InlineMath math="f(x) = x " /> | <InlineMath math="x" /> | <InlineMath math="x" /> | <InlineMath math="x" /> | <InlineMath math="x" /> | <InlineMath math="x" /> | <InlineMath math="x" /> | <InlineMath math="x" /> | <InlineMath math="x" /> | <InlineMath math="x" /> |

## Transformations of Absolute Value Functions

Absolute value functions can be transformed in various ways:

### Vertical Translation

The function <InlineMath math="f(x) = |x| + k" /> shifts the graph upward (if <InlineMath math="k > 0" />) or downward (if <InlineMath math="k < 0" />).

<LineEquation
  title={<>Vertical Translation</>}
  description={
    <>
      Comparison of <InlineMath math="f(x) = |x|" /> with{" "}
      <InlineMath math="g(x) = |x| + 2" /> and{" "}
      <InlineMath math="h(x) = |x| - 2" />.
    </>
  }
  data={[
    {
      points: Array.from({ length: 21 }, (_, i) => {
        const x = i - 10;
        return { x, y: Math.abs(x), z: 0 };
      }),
      color: getColor("PURPLE"),
      showPoints: false,
      smooth: false,
      labels: [{ text: "f(x) = |x|", at: 12, offset: [1, 0.5, 0] }],
    },
    {
      points: Array.from({ length: 21 }, (_, i) => {
        const x = i - 10;
        return { x, y: Math.abs(x) + 2, z: 0 };
      }),
      color: getColor("ORANGE"),
      showPoints: false,
      smooth: false,
      labels: [{ text: "g(x) = |x| + 2", at: 12, offset: [1, 0.5, 0] }],
    },
    {
      points: Array.from({ length: 21 }, (_, i) => {
        const x = i - 10;
        return { x, y: Math.abs(x) - 2, z: 0 };
      }),
      color: getColor("CYAN"),
      showPoints: false,
      smooth: false,
      labels: [{ text: "h(x) = |x| - 2", at: 12, offset: [1, 0.5, 0] }],
    },
  ]}
  cameraPosition={[0, 0, 15]}
  showZAxis={false}
/>

### Horizontal Translation

The function <InlineMath math="f(x) = |x - h|" /> shifts the graph to the right (if <InlineMath math="h > 0" />) or to the left (if <InlineMath math="h < 0" />).

<LineEquation
  title={<>Horizontal Translation</>}
  description={
    <>
      Comparison of <InlineMath math="f(x) = |x|" /> with{" "}
      <InlineMath math="g(x) = |x - 3|" /> and{" "}
      <InlineMath math="h(x) = |x + 3|" />.
    </>
  }
  data={[
    {
      points: Array.from({ length: 21 }, (_, i) => {
        const x = i - 10;
        return { x, y: Math.abs(x), z: 0 };
      }),
      color: getColor("PURPLE"),
      showPoints: false,
      smooth: false,
      labels: [{ text: "f(x) = |x|", at: 12, offset: [1, 0.5, 0] }],
    },
    {
      points: Array.from({ length: 21 }, (_, i) => {
        const x = i - 10;
        return { x, y: Math.abs(x - 3), z: 0 };
      }),
      color: getColor("PINK"),
      showPoints: false,
      smooth: false,
      labels: [{ text: "g(x) = |x - 3|", at: 12, offset: [3, 0.5, 0] }],
    },
    {
      points: Array.from({ length: 21 }, (_, i) => {
        const x = i - 10;
        return { x, y: Math.abs(x + 3), z: 0 };
      }),
      color: getColor("AMBER"),
      showPoints: false,
      smooth: false,
      labels: [{ text: "h(x) = |x + 3|", at: 12, offset: [1, 0.5, 0] }],
    },
  ]}
  cameraPosition={[0, 0, 15]}
  showZAxis={false}
/>

### Stretching and Compression

The function <InlineMath math="f(x) = a|x|" /> changes the slope of the graph:

- If <InlineMath math="a > 1" />: the graph becomes steeper
- If <InlineMath math="0 < a < 1" />: the graph becomes gentler
- If <InlineMath math="a < 0" />: the graph is inverted (reflection across the x-axis)

To make it easier to understand, let's look at the following example:

<LineEquation
  title={<>Stretching and Compression</>}
  description={
    <>
      Comparison of <InlineMath math="f(x) = |x|" /> with{" "}
      <InlineMath math="g(x) = 2|x|" /> and <InlineMath math="h(x) = 0.5|x|" />.
    </>
  }
  data={[
    {
      points: Array.from({ length: 21 }, (_, i) => {
        const x = i - 10;
        return { x, y: Math.abs(x), z: 0 };
      }),
      color: getColor("PURPLE"),
      showPoints: false,
      smooth: false,
      labels: [{ text: "f(x) = |x|", at: 12, offset: [1, 0.5, 0] }],
    },
    {
      points: Array.from({ length: 21 }, (_, i) => {
        const x = i - 10;
        return { x, y: 2 * Math.abs(x), z: 0 };
      }),
      color: getColor("TEAL"),
      showPoints: false,
      smooth: false,
      labels: [{ text: "g(x) = 2|x|", at: 12, offset: [1, 1, 0] }],
    },
    {
      points: Array.from({ length: 21 }, (_, i) => {
        const x = i - 10;
        return { x, y: 0.5 * Math.abs(x), z: 0 };
      }),
      color: getColor("ROSE"),
      showPoints: false,
      smooth: false,
      labels: [{ text: "h(x) = 0.5|x|", at: 12, offset: [3, 0.2, 0] }],
    },
  ]}
  cameraPosition={[0, 0, 15]}
  showZAxis={false}
/>

## General Form of Absolute Value Functions

The general form of an absolute value function is:

<BlockMath math="f(x) = a|x - h| + k" />

where:

- <InlineMath math="a" />: stretching/compression factor and reflection
- <InlineMath math="h" />: horizontal translation
- <InlineMath math="k" />: vertical translation
- The vertex is located at <InlineMath math="(h, k)" />

**Transformation table:**

| Parameter                       | Value        | Effect on Graph       |
| ------------------------------- | ------------ | --------------------- |
| <InlineMath math="a > 1" />     | Positive > 1 | Graph becomes steeper |
| <InlineMath math="0 < a < 1" /> | Positive < 1 | Graph becomes gentler |
| <InlineMath math="a < 0" />     | Negative     | Graph is inverted     |
| <InlineMath math="h > 0" />     | Positive     | Shift to the right    |
| <InlineMath math="h < 0" />     | Negative     | Shift to the left     |
| <InlineMath math="k > 0" />     | Positive     | Shift upward          |
| <InlineMath math="k < 0" />     | Negative     | Shift downward        |

## Absolute Value Equations and Inequalities

**Solving absolute value equations:**

To solve <InlineMath math="|x| = a" /> with <InlineMath math="a \geq 0" />:

<div className="flex flex-col gap-4">
  <BlockMath math="|x| = a \Rightarrow x = a \text{ or } x = -a" />
</div>

**Example:** Solve <InlineMath math="|x - 3| = 5" />

<div className="flex flex-col gap-4">
  <BlockMath math="|x - 3| = 5" />
  <BlockMath math="x - 3 = 5 \text{ or } x - 3 = -5" />
  <BlockMath math="x = 8 \text{ or } x = -2" />
</div>

**Solving absolute value inequalities:**

For <InlineMath math="|x| < a" /> with <InlineMath math="a > 0" />:

<BlockMath math="|x| < a \Rightarrow -a < x < a" />

For <InlineMath math="|x| > a" /> with <InlineMath math="a > 0" />:

<BlockMath math="|x| > a \Rightarrow x < -a \text{ or } x > a" />

## Exercises

1. Determine the value of <InlineMath math="f(x) = |2x - 6|" /> for <InlineMath math="x = -1, 0, 3, 5" />

2. Solve the equation <InlineMath math="|3x + 1| = 7" />

3. Solve the inequality <InlineMath math="|x - 2| < 4" />

4. Determine the vertex of the function <InlineMath math="f(x) = 2|x - 3| + 1" />

5. The distance between two cities is 150 km. If city A is located at coordinate -50 km, where is city B located?

### Answer Key

1. **Calculating function values for various inputs:**

   Substitute each value of x into the function <InlineMath math="f(x) = |2x - 6|" />:

   <div className="flex flex-col gap-4">
     <BlockMath math="f(-1) = |2(-1) - 6| = |-2 - 6| = |-8| = 8" />
     <BlockMath math="f(0) = |2(0) - 6| = |0 - 6| = |-6| = 6" />
     <BlockMath math="f(3) = |2(3) - 6| = |6 - 6| = |0| = 0" />
     <BlockMath math="f(5) = |2(5) - 6| = |10 - 6| = |4| = 4" />
   </div>

2. **Solving absolute value equations:**

   For the equation <InlineMath math="|3x + 1| = 7" />, we use the definition of absolute value which produces two possibilities:

   <div className="flex flex-col gap-4">
     <BlockMath math="3x + 1 = 7 \quad \text{or} \quad 3x + 1 = -7" />
     <BlockMath math="3x = 6 \quad \text{or} \quad 3x = -8" />
     <BlockMath math="x = 2 \quad \text{or} \quad x = -\frac{8}{3}" />
   </div>

3. **Solving absolute value inequalities:**

   For <InlineMath math="|x - 2| < 4" />, we use the property that <InlineMath math="|a| < b" /> is equivalent to <InlineMath math="-b < a < b" />:

   <div className="flex flex-col gap-4">
     <BlockMath math="-4 < x - 2 < 4" />
     <BlockMath math="-4 + 2 < x < 4 + 2" />
     <BlockMath math="-2 < x < 6" />
   </div>

   So the solution set is <InlineMath math="x \in (-2, 6)" />.

4. **Determining the vertex:**

   From the function <InlineMath math="f(x) = 2|x - 3| + 1" />, we can identify the parameters:

   - <InlineMath math="a = 2" /> (stretching factor)
   - <InlineMath math="h = 3" /> (horizontal translation)
   - <InlineMath math="k = 1" /> (vertical translation)

   The vertex is located at <InlineMath math="(h, k) = (3, 1)" />.

5. **Calculating position based on distance:**

   Given that the distance between cities A and B is 150 km, with city A at coordinate -50 km. Let city B be at coordinate <InlineMath math="x_B" />:

   <div className="flex flex-col gap-4">
     <BlockMath math="|-50 - x_B| = 150" />
     <BlockMath math="-50 - x_B = 150 \quad \text{or} \quad -50 - x_B = -150" />
     <BlockMath math="x_B = -50 - 150 = -200 \quad \text{or} \quad x_B = -50 + 150 = 100" />
   </div>

   So city B can be located at coordinate 100 km or -200 km.