# Nakafa Framework: LLM

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

Output docs content for large language models.

---

export const metadata = {
  title: "Horizontal Translation",
  description: "Learn how horizontal translation shifts a function's graph left or right without altering its shape, enhancing your understanding of function transformations.",
  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 Translation

Horizontal translation is a geometric transformation that shifts the graph of a function left or right along the x-axis without changing the shape of the graph. Imagine sliding an object horizontally on a table, its shape remains the same, only its position changes.

If we have a function <InlineMath math="f(x)" />, then horizontal translation produces a new function <InlineMath math="g(x) = f(x - h)" /> where <InlineMath math="h" /> is the translation constant.

### Rules of Horizontal Translation

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

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

Where:
- If <InlineMath math="h > 0" />, the graph shifts to the **right** by <InlineMath math="h" /> units
- If <InlineMath math="h < 0" />, the graph shifts to the **left** by <InlineMath math="|h|" /> units
- If <InlineMath math="h = 0" />, there is no translation (graph remains the same)

## Visualization of Horizontal Translation

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

<LineEquation
  title={<>Horizontal Translation of Quadratic Function <InlineMath math="f(x) = x^2" /></>}
  description="Notice how the graph shifts horizontally without changing the parabola shape."
  showZAxis={false}
  cameraPosition={[12, 8, 12]}
  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: [0.5, 1, 0] }],
      showPoints: false,
    },
    {
      points: Array.from({ length: 41 }, (_, i) => {
        const x = (i - 20) * 0.25;
        return { x, y: (x - 3) * (x - 3), z: 0 };
      }),
      color: getColor("ORANGE"),
      labels: [{ text: "g(x) = (x - 3)²", offset: [0.5, 1, 0] }],
      showPoints: false,
    },
    {
      points: Array.from({ length: 41 }, (_, i) => {
        const x = (i - 20) * 0.25;
        return { x, y: (x + 2) * (x + 2), z: 0 };
      }),
      color: getColor("TEAL"),
      labels: [{ text: "h(x) = (x + 2)²", offset: [0.5, 1, 0] }],
      showPoints: false,
    },
  ]}
/>

From the visualization above, we can observe:
- The original function <InlineMath math="f(x) = x^2" /> has its vertex at <InlineMath math="(0, 0)" />
- Function <InlineMath math="g(x) = (x - 3)^2" /> shifts right by 3 units with vertex at <InlineMath math="(3, 0)" />
- Function <InlineMath math="h(x) = (x + 2)^2" /> shifts left by 2 units with vertex at <InlineMath math="(-2, 0)" />

## Horizontal Translation on Linear Functions

Now let's apply the same concept to the linear function <InlineMath math="f(x) = 2x + 1" />.

<LineEquation
  title={<>Horizontal Translation of Linear Function <InlineMath math="f(x) = 2x + 1" /></>}
  description="The line maintains the same slope, only its horizontal position changes."
  showZAxis={false}
  cameraPosition={[10, 6, 10]}
  data={[
    {
      points: Array.from({ length: 21 }, (_, i) => {
        const x = (i - 10) * 0.5;
        return { x, y: 2 * x + 1, z: 0 };
      }),
      color: getColor("VIOLET"),
      labels: [{ text: "f(x) = 2x + 1", offset: [1, 0.5, 0] }],
      showPoints: false,
    },
    {
      points: Array.from({ length: 21 }, (_, i) => {
        const x = (i - 10) * 0.5;
        return { x, y: 2 * (x - 4) + 1, z: 0 };
      }),
      color: getColor("AMBER"),
      labels: [{ text: "g(x) = 2(x - 4) + 1", offset: [1, 0.5, 0] }],
      showPoints: false,
    },
    {
      points: Array.from({ length: 21 }, (_, i) => {
        const x = (i - 10) * 0.5;
        return { x, y: 2 * (x + 3) + 1, z: 0 };
      }),
      color: getColor("CYAN"),
      labels: [{ text: "h(x) = 2(x + 3) + 1", offset: [1, 0.5, 0] }],
      showPoints: false,
    },
  ]}
/>

Notice that:
- All lines have the same slope of 2
- Function <InlineMath math="g(x) = 2(x - 4) + 1" /> shifts right by 4 units
- Function <InlineMath math="h(x) = 2(x + 3) + 1" /> shifts left by 3 units

## Important Properties of Horizontal Translation

### Graph Shape Remains Unchanged

Horizontal translation preserves the original shape of the graph. The vertical distance between points on the graph remains the same, only the horizontal position changes.

### Effect on Coordinate Points

If point <InlineMath math="(a, b)" /> is on the graph of <InlineMath math="f(x)" />, then after horizontal translation by <InlineMath math="h" />, that point becomes <InlineMath math="(a + h, b)" /> on the graph of <InlineMath math="f(x - h)" />.

### Domain and Range

- **Domain**: Shifts by <InlineMath math="h" /> units
- **Range**: Does not change after horizontal translation

If the domain of the original function is <InlineMath math="[c, d]" />, then the domain after horizontal translation <InlineMath math="h" /> becomes <InlineMath math="[c + h, d + h]" />.

## Application Examples

### Exponential Function Example

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

<LineEquation
  title={<>Horizontal Translation of Exponential Function <InlineMath math="f(x) = 2^x" /></>}
  description="The exponential curve maintains its characteristics after horizontal translation."
  showZAxis={false}
  cameraPosition={[8, 5, 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: [0.5, 1, 0] }],
      showPoints: false,
    },
    {
      points: Array.from({ length: 31 }, (_, i) => {
        const x = (i - 15) * 0.3;
        return { x, y: Math.pow(2, x - 2), z: 0 };
      }),
      color: getColor("EMERALD"),
      labels: [{ text: "g(x) = 2^(x-2)", offset: [0.5, 1, 0] }],
      showPoints: false,
    },
    {
      points: Array.from({ length: 31 }, (_, i) => {
        const x = (i - 15) * 0.3;
        return { x, y: Math.pow(2, x + 1), z: 0 };
      }),
      color: getColor("ROSE"),
      labels: [{ text: "h(x) = 2^(x+1)", offset: [0.5, 1, 0] }],
      showPoints: false,
    },
  ]}
/>

For exponential functions:
- The horizontal asymptote remains at <InlineMath math="y = 0" /> for all functions
- The y-intercept changes due to horizontal shift
- Function <InlineMath math="g(x) = 2^{x-2}" /> shifts right by 2 units
- Function <InlineMath math="h(x) = 2^{x+1}" /> shifts left by 1 unit

## Difference from Vertical Translation

It's important to understand the difference between horizontal and vertical translation:

### Horizontal Translation
- Changes the function input: <InlineMath math="f(x - h)" />
- Affects the x position of each point
- Domain changes, range remains

### Vertical Translation
- Changes the function output: <InlineMath math="f(x) + k" />
- Affects the y position of each point
- Domain remains, range changes

## Exercises

1. Given the function <InlineMath math="f(x) = x^2 + 2x + 1" />. Determine the equation of the function resulting from horizontal translation to the right by 3 units.

2. If the graph of function <InlineMath math="g(x) = \sqrt{x}" /> is translated horizontally to the left by 4 units, determine:
   - The equation of the resulting translated function
   - The domain of the function after translation

3. Function <InlineMath math="h(x) = 3^x" /> undergoes horizontal translation such that point <InlineMath math="(0, 1)" /> becomes <InlineMath math="(2, 1)" />. Determine the translation constant value and the equation of the resulting translated function.

### Answer Key

1. Horizontal translation to the right by 3 units: <InlineMath math="f'(x) = f(x - 3) = (x - 3)^2 + 2(x - 3) + 1 = x^2 - 4x + 4" />

   <LineEquation
     title={<>Function <InlineMath math="f(x) = x^2 + 2x + 1" /> and Its Translation Result</>}
     description="Original quadratic function and the result of horizontal translation to the right by 3 units."
     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: [1, 1, 0] }],
         showPoints: false,
       },
       {
         points: Array.from({ length: 41 }, (_, i) => {
           const x = (i - 20) * 0.25;
           return { x, y: x * x - 4 * x + 4, z: 0 };
         }),
         color: getColor("ORANGE"),
         labels: [{ text: "f'(x) = x² - 4x + 4", offset: [1, 1, 0] }],
         showPoints: false,
       },
     ]}
   />

2. Equation of the resulting translated function:
    
   - Translation to the left by 4 units: <InlineMath math="g'(x) = g(x + 4) = \sqrt{x + 4}" />
   - Domain after translation: <InlineMath math="x + 4 \geq 0" />, so <InlineMath math="x \geq -4" /> or <InlineMath math="[-4, \infty)" />

   Visualization:

   <LineEquation
     title={<>Function <InlineMath math="g(x) = \sqrt{x}" /> and Its Translation Result</>}
     description="Original square root function and the result of horizontal translation to the left by 4 units."
     showZAxis={false}
     cameraPosition={[8, 6, 8]}
     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: [1, 0.5, 0] }],
         showPoints: false,
       },
       {
         points: Array.from({ length: 21 }, (_, i) => {
           const x = i * 0.25 - 4;
           if (x + 4 >= 0) {
             return { x, y: Math.sqrt(x + 4), z: 0 };
           }
           return null;
         }).filter(Boolean),
         color: getColor("TEAL"),
         labels: [{ text: "g'(x) = √(x + 4)", offset: [1, 0.5, 0] }],
         showPoints: false,
       },
     ]}
   />

3. Point <InlineMath math="(0, 1)" /> on <InlineMath math="h(x) = 3^x" /> becomes <InlineMath math="(2, 1)" />, meaning horizontal translation by <InlineMath math="h = 2" /> units to the right.
   Equation of the translation result: <InlineMath math="h'(x) = 3^{x-2}" />

   <LineEquation
     title={<>Function <InlineMath math="h(x) = 3^x" /> and Its Translation Result</>}
     description="Original exponential function and the result of horizontal translation to the right by 2 units."
     showZAxis={false}
     cameraPosition={[10, 6, 10]}
     data={[
       {
         points: Array.from({ length: 31 }, (_, i) => {
           const x = (i - 15) * 0.2;
           return { x, y: Math.pow(3, x), z: 0 };
         }),
         color: getColor("INDIGO"),
         labels: [{ text: "h(x) = 3^x", offset: [0.5, 1, 0] }],
         showPoints: false,
       },
       {
         points: Array.from({ length: 31 }, (_, i) => {
           const x = (i - 15) * 0.2;
           return { x, y: Math.pow(3, x - 2), z: 0 };
         }),
         color: getColor("EMERALD"),
         labels: [{ text: "h'(x) = 3^(x-2)", offset: [0.5, 1, 0] }],
         showPoints: false,
       },
     ]}
   />