# Nakafa Framework: LLM

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

Output docs content for large language models.

---

export const metadata = {
  title: "Horizontal Reflection",
  description: "Master horizontal reflection in functions with clear examples. Learn how to reflect graphs across the y-axis and apply this transformation effectively.",
  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 Reflection

Horizontal reflection is a geometric transformation that reflects the graph of a function across the y-axis, like seeing the reflection of an object in a vertical mirror. Imagine standing in front of a mirror, your right hand will appear as your left hand in the mirror, similarly with function graphs that are reflected horizontally.

If we have a function <InlineMath math="f(x)" />, then horizontal reflection produces a new function <InlineMath math="g(x) = f(-x)" /> which is the reflection of the original function across the y-axis.

### Rules of Horizontal Reflection

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

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

This transformation changes every point <InlineMath math="(x, y)" /> on the original graph to <InlineMath math="(-x, y)" /> on the reflected graph.

## Visualization of Horizontal Reflection

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

<LineEquation
  title={<>Horizontal Reflection of Quadratic Function <InlineMath math="f(x) = (x - 2)^2" /></>}
  description="Notice how the graph reflects across the y-axis, forming a symmetric reflection."
  showZAxis={false}
  cameraPosition={[12, 8, 12]}
  data={[
    {
      points: Array.from({ length: 41 }, (_, i) => {
        const x = (i - 20) * 0.25;
        return { x, y: (x - 2) * (x - 2), z: 0 };
      }),
      color: getColor("PURPLE"),
      labels: [{ text: "f(x) = (x - 2)²", offset: [1, 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("ORANGE"),
      labels: [{ text: "g(x) = (-x - 2)²", offset: [-1, -1, 0] }],
      showPoints: false,
    },
  ]}
/>

From the visualization above, we can observe:
- The original function <InlineMath math="f(x) = (x - 2)^2" /> has its vertex at <InlineMath math="(2, 0)" />
- The reflected function <InlineMath math="g(x) = (-x - 2)^2" /> has its vertex at <InlineMath math="(-2, 0)" />
- Both graphs are symmetric across the y-axis

## Horizontal Reflection on Linear Functions

Now let's apply the same concept to the linear function <InlineMath math="f(x) = 2x + 3" />.

<LineEquation
  title={<>Horizontal Reflection of Linear Function <InlineMath math="f(x) = 2x + 3" /></>}
  description="The reflected line has opposite slope but the same y-intercept."
  showZAxis={false}
  cameraPosition={[10, 6, 10]}
  data={[
    {
      points: Array.from({ length: 21 }, (_, i) => {
        const x = (i - 10) * 0.5;
        return { x, y: 2 * x + 3, z: 0 };
      }),
      color: getColor("VIOLET"),
      labels: [{ text: "f(x) = 2x + 3", offset: [2.5, 0.5, 0] }],
      showPoints: false,
    },
    {
      points: Array.from({ length: 21 }, (_, i) => {
        const x = (i - 10) * 0.5;
        return { x, y: 2 * (-x) + 3, z: 0 };
      }),
      color: getColor("AMBER"),
      labels: [{ text: "g(x) = -2x + 3", offset: [-2.5, -0.5, 0] }],
      showPoints: false,
    },
  ]}
/>

Notice that:
- The original function <InlineMath math="f(x) = 2x + 3" /> has a positive slope of 2
- The reflected function <InlineMath math="g(x) = -2x + 3" /> has a negative slope of 2
- Both lines intersect the y-axis at the same point <InlineMath math="(0, 3)" />

## Important Properties of Horizontal Reflection

### Y-axis as Axis of Symmetry

Horizontal reflection uses the y-axis as the mirror line. Every point on the original graph has the same distance to the y-axis as the corresponding point on the reflected graph.

### 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(-x)" /> is <InlineMath math="(-a, b)" />.

### Domain and Range

- **Domain**: Changes to the opposite of the original domain
- **Range**: Does not change after horizontal reflection

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

## Application Examples

### Exponential Function Example

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

<LineEquation
  title={<>Horizontal Reflection of Exponential Function <InlineMath math="f(x) = 2^x" /></>}
  description="The reflected exponential curve produces a decreasing curve with different characteristics."
  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: [3, 1, 0] }],
      showPoints: false,
    },
    {
      points: Array.from({ length: 31 }, (_, i) => {
        const x = (i - 15) * 0.3;
        return { x, y: Math.pow(2, -x), z: 0 };
      }),
      color: getColor("EMERALD"),
      labels: [{ text: "g(x) = 2^(-x)", offset: [-3, 1, 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 function that was originally increasing becomes decreasing after reflection

## Horizontal Reflection on Square Root Functions

Let's see how horizontal reflection affects the square root function.

<LineEquation
  title={<>Horizontal Reflection of Square Root Function <InlineMath math="f(x) = \sqrt{x}" /></>}
  description="The reflected square root curve produces a curve that opens in the opposite direction."
  showZAxis={false}
  cameraPosition={[10, 6, 10]}
  data={[
    {
      points: Array.from({ length: 21 }, (_, i) => {
        const x = i * 0.25;
        return { x, y: Math.sqrt(x), z: 0 };
      }),
      color: getColor("CYAN"),
      labels: [{ text: "f(x) = √x", offset: [1, 1.5, 0] }],
      showPoints: false,
    },
    {
      points: Array.from({ length: 21 }, (_, i) => {
        const x = -i * 0.25;
        if (x <= 0) {
          return { x, y: Math.sqrt(-x), z: 0 };
        }
        return null;
      }).filter(Boolean),
      color: getColor("ROSE"),
      labels: [{ text: "g(x) = √(-x)", offset: [-1, 1.5, 0] }],
      showPoints: false,
    },
  ]}
/>

Notice that:
- The domain of the original function <InlineMath math="f(x) = \sqrt{x}" /> is <InlineMath math="[0, \infty)" />
- The domain of the reflected function <InlineMath math="g(x) = \sqrt{-x}" /> is <InlineMath math="(-\infty, 0]" />
- Both curves meet at the origin <InlineMath math="(0, 0)" />

## Exercises

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

2. If the graph of function <InlineMath math="g(x) = 3^x + 1" /> is reflected across the y-axis, determine:
   - The equation of the resulting reflected function
   - The domain of the function after reflection

3. Function <InlineMath math="h(x) = |x + 2|" /> undergoes horizontal reflection. Determine the vertex of the resulting reflected function.

### Answer Key

1. Horizontal reflection: <InlineMath math="f'(x) = f(-x) = (-x)^2 + 3(-x) + 2 = x^2 - 3x + 2" />

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

2. Equation of the resulting reflected function:
    
   - Horizontal reflection: <InlineMath math="g'(x) = g(-x) = 3^{-x} + 1" />
   - Domain after reflection: Remains <InlineMath math="(-\infty, \infty)" /> because exponential functions are defined for all real numbers

   Visualization:

   <LineEquation
     title={<>Function <InlineMath math="g(x) = 3^x + 1" /> and Its Reflection Result</>}
     description="The increasing exponential curve is reflected to become a decreasing curve with the same asymptote."
     showZAxis={false}
     cameraPosition={[8, 6, 8]}
     data={[
       {
         points: Array.from({ length: 31 }, (_, i) => {
           const x = (i - 15) * 0.2;
           return { x, y: Math.pow(3, x) + 1, z: 0 };
         }),
         color: getColor("VIOLET"),
         labels: [{ text: "g(x) = 3^x + 1", offset: [3, 1, 0] }],
         showPoints: false,
       },
       {
         points: Array.from({ length: 31 }, (_, i) => {
           const x = (i - 15) * 0.2;
           return { x, y: Math.pow(3, -x) + 1, z: 0 };
         }),
         color: getColor("TEAL"),
         labels: [{ text: "g'(x) = 3^(-x) + 1", offset: [-3, 1, 0] }],
         showPoints: false,
       },
     ]}
   />

3. The original function <InlineMath math="h(x) = |x + 2|" /> has its vertex at <InlineMath math="(-2, 0)" />.
   After horizontal reflection: <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 Reflection Result</>}
     description="The absolute value function is reflected across the y-axis producing a function with opposite vertex position."
     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: [-1, 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: [1, -1, 0] }],
         showPoints: false,
       },
     ]}
   />