# Nakafa Framework: LLM

URL: https://nakafa.com/en/subject/high-school/12/mathematics/function-transformation/horizontal-dilation
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/subject/high-school/12/mathematics/function-transformation/horizontal-dilation/en.mdx

Output docs content for large language models.

---

export const metadata = {
  title: "Horizontal Dilation",
  description: "Understand horizontal dilation in functions with clear examples. Learn how to stretch or compress graphs horizontally and apply these concepts to various functions.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "05/26/2025",
  subject: "Function Transformation",
};

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

## Basic Concepts of Horizontal Dilation

Horizontal dilation is a geometric transformation that changes the size of a function graph horizontally, like pulling or compressing a rubber band left and right. Imagine holding a photo with both hands on the left and right sides, then stretching or compressing it horizontally without changing the height of the photo.

If we have a function <InlineMath math="f(x)" />, then horizontal dilation produces a new function <InlineMath math="g(x) = f(kx)" /> where <InlineMath math="k" /> is the scale factor that determines how much the horizontal size changes.

### Rules of Horizontal Dilation

For any function <InlineMath math="f(x)" />, horizontal dilation is defined as:

<BlockMath math="g(x) = f(kx)" />

Where <InlineMath math="k" /> is the scale factor that affects the transformation:
- If <InlineMath math="k > 1" />, the graph is compressed horizontally (reduced)
- If <InlineMath math="0 < k < 1" />, the graph is stretched horizontally (enlarged)
- If <InlineMath math="k = 1" />, the graph does not change
- If <InlineMath math="k < 0" />, the graph undergoes reflection as well as dilation

Note that the effect of horizontal dilation is counterintuitive: a larger scale factor actually compresses the graph.

## Visualization of Horizontal Dilation

Let's see how horizontal dilation works on the quadratic function <InlineMath math="f(x) = x^2" /> with various scale factors.

<LineEquation
  title={<>Horizontal Dilation of Quadratic Function <InlineMath math="f(x) = x^2" /></>}
  description="Notice how the graph is compressed or stretched horizontally with different scale factors."
  showZAxis={false}
  data={[
    {
      points: Array.from({ length: 41 }, (_, i) => {
        const x = (i - 20) * 0.25;
        return { x, y: x * x, z: 0 };
      }),
      color: getColor("PURPLE"),
      labels: [{ text: "f(x) = x²", offset: [4, 2, 0] }],
      showPoints: false,
    },
    {
      points: Array.from({ length: 41 }, (_, i) => {
        const x = (i - 20) * 0.25;
        return { x, y: (2 * x) * (2 * x), z: 0 };
      }),
      color: getColor("ORANGE"),
      labels: [{ text: "g(x) = (2x)²", offset: [4, 3, 0] }],
      showPoints: false,
    },
    {
      points: Array.from({ length: 41 }, (_, i) => {
        const x = (i - 20) * 0.25;
        return { x, y: (0.5 * x) * (0.5 * x), z: 0 };
      }),
      color: getColor("SKY"),
      labels: [{ text: "h(x) = (0.5x)²", offset: [4, 1, 0] }],
      showPoints: false,
    },
  ]}
/>

From the visualization above, we can observe:
- The original function <InlineMath math="f(x) = x^2" /> as reference
- Function <InlineMath math="g(x) = (2x)^2" /> is horizontally compressed by factor 2
- Function <InlineMath math="h(x) = (0.5x)^2" /> is horizontally stretched by factor 0.5
- All graphs have the same vertex at <InlineMath math="(0, 0)" />

## Horizontal Dilation on Linear Functions

Now let's apply the same concept to the linear function <InlineMath math="f(x) = x + 2" />.

<LineEquation
  title={<>Horizontal Dilation of Linear Function <InlineMath math="f(x) = x + 2" /></>}
  description="The dilated line has slope that changes according to the scale factor."
  showZAxis={false}
  data={[
    {
      points: Array.from({ length: 21 }, (_, i) => {
        const x = (i - 10) * 0.5;
        return { x, y: x + 2, z: 0 };
      }),
      color: getColor("AMBER"),
      labels: [{ text: "f(x) = x + 2", offset: [2, 0.5, 0] }],
      showPoints: false,
    },
    {
      points: Array.from({ length: 21 }, (_, i) => {
        const x = (i - 10) * 0.5;
        return { x, y: (2 * x) + 2, z: 0 };
      }),
      color: getColor("TEAL"),
      labels: [{ text: "g(x) = 2x + 2", offset: [1, 1.5, 0] }],
      showPoints: false,
    },
    {
      points: Array.from({ length: 21 }, (_, i) => {
        const x = (i - 10) * 0.5;
        return { x, y: (0.5 * x) + 2, z: 0 };
      }),
      color: getColor("ROSE"),
      labels: [{ text: "h(x) = 0.5x + 2", offset: [3, -0.5, 0] }],
      showPoints: false,
    },
  ]}
/>

Notice that:
- The original function <InlineMath math="f(x) = x + 2" /> has slope 1
- Function <InlineMath math="g(x) = f(2x) = 2x + 2" /> has slope 2 (horizontally compressed)
- Function <InlineMath math="h(x) = f(0.5x) = 0.5x + 2" /> has slope 0.5 (horizontally stretched)
- All lines intersect the y-axis at the same point <InlineMath math="(0, 2)" />

## Important Properties of Horizontal Dilation

### Effect on Coordinate Points

If point <InlineMath math="(a, b)" /> is on the graph of <InlineMath math="f(x)" />, then the corresponding point on the graph of <InlineMath math="f(kx)" /> is <InlineMath math="(\frac{a}{k}, b)" />.

### Domain and Range

- **Domain**: Changes according to the scale factor <InlineMath math="k" />
- **Range**: Does not change after horizontal dilation

If the domain of the original function is <InlineMath math="[c, d]" />, then the domain after horizontal dilation with factor <InlineMath math="k > 0" /> becomes <InlineMath math="[\frac{c}{k}, \frac{d}{k}]" />.

### Axis Intercepts

- **x-intercept**: Changes according to the scale factor
- **y-intercept**: Does not change

## Application Examples

### Exponential Function

Let's look at horizontal dilation on the exponential function <InlineMath math="f(x) = 2^x" />.

<LineEquation
  title={<>Horizontal Dilation of Exponential Function <InlineMath math="f(x) = 2^x" /></>}
  description="The exponential curve undergoes width changes according to the scale factor."
  showZAxis={false}
  cameraPosition={[8, 6, 8]}
  data={[
    {
      points: Array.from({ length: 31 }, (_, i) => {
        const x = (i - 15) * 0.3;
        return { x, y: Math.pow(2, x), z: 0 };
      }),
      color: getColor("INDIGO"),
      labels: [{ text: "f(x) = 2^x", offset: [2, 1, 0] }],
      showPoints: false,
    },
    {
      points: Array.from({ length: 31 }, (_, i) => {
        const x = (i - 15) * 0.3;
        return { x, y: Math.pow(2, 2 * x), z: 0 };
      }),
      color: getColor("EMERALD"),
      labels: [{ text: "g(x) = 2^(2x)", offset: [1, 2, 0] }],
      showPoints: false,
    },
  ]}
/>

For exponential functions:
- The horizontal asymptote remains at <InlineMath math="y = 0" /> for both functions
- The y-intercept remains the same at <InlineMath math="(0, 1)" />
- The growth rate of the function changes according to the scale factor

## Horizontal Dilation with Negative Factor

Let's see what happens when the scale factor is negative.

<LineEquation
  title={<>Horizontal Dilation with Negative Factor <InlineMath math="f(x) = x^2 + 1" /></>}
  description="Negative scale factor causes reflection as well as dilation."
  showZAxis={false}
  cameraPosition={[10, 6, 10]}
  data={[
    {
      points: Array.from({ length: 41 }, (_, i) => {
        const x = (i - 20) * 0.25;
        return { x, y: x * x + 1, z: 0 };
      }),
      color: getColor("CYAN"),
      labels: [{ text: "f(x) = x² + 1", offset: [2, 1, 0] }],
      showPoints: false,
    },
    {
      points: Array.from({ length: 41 }, (_, i) => {
        const x = (i - 20) * 0.25;
        return { x, y: (-2 * x) * (-2 * x) + 1, z: 0 };
      }),
      color: getColor("PINK"),
      labels: [{ text: "g(x) = (-2x)² + 1", offset: [3, 2, 0] }],
      showPoints: false,
    },
  ]}
/>

When the scale factor is negative:
- The graph undergoes reflection across the y-axis
- Simultaneously undergoes dilation according to the absolute value of the scale factor
- The graph shape remains the same because the quadratic function is symmetric

## Exercises

1. Given the function <InlineMath math="f(x) = x^2 - 2x + 1" />. Determine the equation of the function resulting from horizontal dilation with scale factor 3.

2. If the graph of function <InlineMath math="g(x) = \sqrt{x}" /> undergoes horizontal dilation with factor <InlineMath math="\frac{1}{2}" />, determine:
   - The equation of the resulting dilated function
   - The domain of the function after dilation

3. Function <InlineMath math="h(x) = |x - 2|" /> undergoes horizontal dilation with factor -1. Determine the vertex of the resulting dilated function.

### Answer Key

1. Horizontal dilation with factor 3: <InlineMath math="f'(x) = f(3x) = (3x)^2 - 2(3x) + 1 = 9x^2 - 6x + 1" />

   <LineEquation
     title={<>Function <InlineMath math="f(x) = x^2 - 2x + 1" /> and Its Dilation Result</>}
     description="The original parabola is horizontally compressed by factor 3 producing a narrower parabola."
     showZAxis={false}
     cameraPosition={[12, 8, 12]}
     data={[
       {
         points: Array.from({ length: 41 }, (_, i) => {
           const x = (i - 20) * 0.25;
           return { x, y: x * x - 2 * x + 1, z: 0 };
         }),
         color: getColor("PURPLE"),
         labels: [{ text: "f(x) = x² - 2x + 1", offset: [4, 1, 0] }],
         showPoints: false,
       },
       {
         points: Array.from({ length: 41 }, (_, i) => {
           const x = (i - 20) * 0.25;
           return { x, y: 9 * x * x - 6 * x + 1, z: 0 };
         }),
         color: getColor("ORANGE"),
         labels: [{ text: "f'(x) = 9x² - 6x + 1", offset: [3, 2, 0] }],
         showPoints: false,
       },
     ]}
   />

2. Equation of the resulting dilated function:
    
   - Horizontal dilation: <InlineMath math="g'(x) = g(\frac{1}{2}x) = \sqrt{\frac{1}{2}x} = \sqrt{\frac{x}{2}}" />
   - Domain after dilation: <InlineMath math="[0, \infty)" /> becomes <InlineMath math="[0, \infty)" /> (unchanged because the scale factor is positive)

   Visualization:

   <LineEquation
     title={<>Function <InlineMath math="g(x) = \sqrt{x}" /> and Its Dilation Result</>}
     description="The square root curve is horizontally stretched by factor 0.5 producing a wider curve."
     showZAxis={false}
     data={[
       {
         points: Array.from({ length: 21 }, (_, i) => {
           const x = i * 0.25;
           return { x, y: Math.sqrt(x), z: 0 };
         }),
         color: getColor("VIOLET"),
         labels: [{ text: "g(x) = √x", offset: [2, 1, 0] }],
         showPoints: false,
       },
       {
         points: Array.from({ length: 21 }, (_, i) => {
           const x = i * 0.25;
           return { x, y: Math.sqrt(0.5 * x), z: 0 };
         }),
         color: getColor("TEAL"),
         labels: [{ text: "g'(x) = √(x/2)", offset: [3, 0.5, 0] }],
         showPoints: false,
       },
     ]}
   />

3. The original function <InlineMath math="h(x) = |x - 2|" /> has its vertex at <InlineMath math="(2, 0)" />.
   After horizontal dilation with factor -1: <InlineMath math="h'(x) = |-x - 2| = |-(x + 2)| = |x + 2|" />, the vertex becomes <InlineMath math="(-2, 0)" />.

   <LineEquation
     title={<>Function <InlineMath math="h(x) = |x - 2|" /> and Its Dilation Result</>}
     description="The absolute value function undergoes reflection and dilation with factor -1."
     showZAxis={false}
     cameraPosition={[10, 6, 10]}
     data={[
       {
         points: Array.from({ length: 41 }, (_, i) => {
           const x = (i - 20) * 0.25;
           return { x, y: Math.abs(x - 2), z: 0 };
         }),
         color: getColor("INDIGO"),
         labels: [{ text: "h(x) = |x - 2|", offset: [2.5, 1, 0] }],
         showPoints: false,
       },
       {
         points: Array.from({ length: 41 }, (_, i) => {
           const x = (i - 20) * 0.25;
           return { x, y: Math.abs(-x - 2), z: 0 };
         }),
         color: getColor("EMERALD"),
         labels: [{ text: "h'(x) = |-x - 2|", offset: [2.5, 2, 0] }],
         showPoints: false,
       },
     ]}
   />