# Nakafa Framework: LLM

URL: https://nakafa.com/en/subject/high-school/11/mathematics/geometric-transformation/reflection-over-y-axis
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/subject/high-school/11/mathematics/geometric-transformation/reflection-over-y-axis/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: "Reflection over Y Axis",
  description: "Learn y-axis reflection with complete examples for points, triangles, and lines. Master the P'(-x, y) rule with interactive visualizations.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "05/10/2025",
  subject: "Geometric Transformation",
};

## Understanding Reflection over the Y-axis

Reflection over the Y-axis is a type of geometric transformation that moves every point on an object to a new position. Imagine the Y-axis as a mirror. Every point will have an image on the opposite side of the Y-axis at the same distance from the Y-axis.

### Rule for Reflection over the Y-axis

If a point <InlineMath math="P(x, y)" /> is reflected over the Y-axis, its image's coordinates, <InlineMath math="P'(x', y')" />, will follow the rule:

<div className="flex flex-col gap-4">
  <BlockMath math="x' = -x" />
  <BlockMath math="y' = y" />
</div>

Thus, the image of point <InlineMath math="P(x, y)" /> is <InlineMath math="P'(-x, y)" />.

Note that the value of the y-coordinate does not change, while the value of the x-coordinate becomes its opposite (negative if positive, positive if negative).

## Reflecting a Point

Suppose we have point <InlineMath math="A(3, 4)" />. If point A is reflected over the Y-axis, its image, <InlineMath math="A'" />, can be determined as follows:

The original x-coordinate is 3, so <InlineMath math="x' = -3" />.

The original y-coordinate is 4, so <InlineMath math="y' = 4" />.

Thus, the image of point A is <InlineMath math="A'(-3, 4)" />.

Let's visualize this:

<LineEquation
  title={
    <>
      Point <InlineMath math="A(3,4)" /> and its Image{" "}
      <InlineMath math="A'(-3,4)" />
    </>
  }
  description="Visualization of the reflection of point A over the Y-axis."
  data={[
    {
      points: [{ x: 3, y: 4, z: 0 }],
      color: getColor("EMERALD"),
      showPoints: true,
      labels: [{ text: "A(3,4)", at: 0, offset: [0.3, 0.3, 0] }],
    },
    {
      points: [{ x: -3, y: 4, z: 0 }],
      color: getColor("SKY"),
      showPoints: true,
      labels: [{ text: "A'(-3,4)", at: 0, offset: [0.3, 0.3, 0] }],
    },
  ]}
  showZAxis={false}
  cameraPosition={[0, 0, 12]}
/>

## Reflecting a Triangle

Now, let's reflect a triangle <InlineMath math="PQR" /> with vertices <InlineMath math="P(1, 2)" />, <InlineMath math="Q(4, 4)" />, and <InlineMath math="R(2, 0)" /> over the Y-axis.

To reflect a triangle, we need to reflect each of its vertices.

1.  Point <InlineMath math="P(1, 2)" />: Its image is <InlineMath math="P'(-1, 2)" />.
2.  Point <InlineMath math="Q(4, 4)" />: Its image is <InlineMath math="Q'(-4, 4)" />.
3.  Point <InlineMath math="R(2, 0)" />: Its image is <InlineMath math="R'(-2, 0)" />.

By connecting the image points <InlineMath math="P'Q'R'" />, we obtain the reflected triangle.

<LineEquation
  title={
    <>
      Triangle <InlineMath math="PQR" /> and its Image{" "}
      <InlineMath math="P'Q'R'" />
    </>
  }
  description="Visualization of the reflection of triangle PQR over the Y-axis."
  data={[
    ...[
      {
        from: { x: 1, y: 2, z: 0, label: "P(1,2)" },
        to: { x: 4, y: 4, z: 0, label: "Q(4,4)" },
      },
      {
        from: { x: 4, y: 4, z: 0, label: "Q(4,4)" },
        to: { x: 2, y: 0, z: 0, label: "R(2,0)" },
      },
      {
        from: { x: 2, y: 0, z: 0, label: "R(2,0)" },
        to: { x: 1, y: 2, z: 0, label: "P(1,2)" },
      },
    ].map((segment) => ({
      points: [segment.from, segment.to],
      color: getColor("AMBER"),
      showPoints: true,
      labels: [
        { text: segment.from.label, at: 0, offset: [0.5, 0.5, 0] },
        { text: segment.to.label, at: 1, offset: [0.5, 0.5, 0] },
      ],
    })),
    ...[
      {
        from: { x: -1, y: 2, z: 0, label: "P'(-1,2)" },
        to: { x: -4, y: 4, z: 0, label: "Q'(-4,4)" },
      },
      {
        from: { x: -4, y: 4, z: 0, label: "Q'(-4,4)" },
        to: { x: -2, y: 0, z: 0, label: "R'(-2,0)" },
      },
      {
        from: { x: -2, y: 0, z: 0, label: "R'(-2,0)" },
        to: { x: -1, y: 2, z: 0, label: "P'(-1,2)" },
      },
    ].map((segment) => ({
      points: [segment.from, segment.to],
      color: getColor("TEAL"),
      showPoints: true,
      labels: [
        { text: segment.from.label, at: 0, offset: [-0.5, 0.5, 0] },
        { text: segment.to.label, at: 1, offset: [-0.5, 0.5, 0] },
      ],
    })),
  ]}
  showZAxis={false}
  cameraPosition={[0, 0, 12]}
/>

## Reflecting a Line Equation

Suppose we have a line with the equation <InlineMath math="y = x + 2" />. To find the equation of its image after reflection over the Y-axis, we substitute <InlineMath math="x" /> with <InlineMath math="-x" /> (because <InlineMath math="x' = -x" />) and <InlineMath math="y" /> with <InlineMath math="y" /> (because <InlineMath math="y' = y" />) into the original equation.

Original equation:

<BlockMath math="y = x + 2" />

Substitute <InlineMath math="x \rightarrow -x" />:

<BlockMath math="y = (-x) + 2" />

The equation of the image is:

<BlockMath math="y = -x + 2" />

Let's visualize these two lines:

<LineEquation
  title={
    <>
      Line <InlineMath math="y = x + 2" /> and its Image{" "}
      <InlineMath math="y = -x + 2" />
    </>
  }
  description="Reflection of a line over the Y-axis."
  data={[
    {
      // Original Line y = x + 2
      points: Array.from({ length: 11 }, (_, i) => {
        const x = (i - 5) * 0.8; // x from -4 to 4 for better fit
        return { x, y: x + 2, z: 0 };
      }),
      color: getColor("PURPLE"),
      smooth: true,
      labels: [{ text: "y=x+2", at: 9, offset: [0.7, -0.5, 0] }],
    },
    {
      // Reflected Line y = -x + 2
      points: Array.from({ length: 11 }, (_, i) => {
        const x = (i - 5) * 0.8; // x from -4 to 4
        return { x, y: -x + 2, z: 0 };
      }),
      color: getColor("PINK"),
      smooth: true,
      labels: [{ text: "y=-x+2", at: 1, offset: [-0.7, -0.5, 0] }],
    },
  ]}
  showZAxis={false}
  cameraPosition={[0, 0, 15]}
/>

## Exercises

1.  Determine the coordinates of the image of point <InlineMath math="K(-5, 8)" /> if it is reflected over the Y-axis!
2.  A triangle <InlineMath math="ABC" /> has vertices <InlineMath math="A(2, 1)" />, <InlineMath math="B(5, 3)" />, and <InlineMath math="C(3, 6)" />. Determine the coordinates of the image triangle <InlineMath math="A'B'C'" /> after reflection over the Y-axis!
3.  Determine the equation of the image of the line <InlineMath math="3x - 2y + 6 = 0" /> if it is reflected over the Y-axis!
4.  A line has the equation <InlineMath math="y = -3x - 4" />. Determine the equation of its image after reflection over the Y-axis.

### Key Answers

1.  The image of point <InlineMath math="K(-5, 8)" /> is <InlineMath math="K'(5, 8)" />.

    **Explanation:** <InlineMath math="x' = -(-5) = 5" />, <InlineMath math="y' = 8" />.

2.  The coordinates of the image triangle <InlineMath math="A'B'C'" /> are:

    - <InlineMath math="A'(-2, 1)" />
    - <InlineMath math="B'(-5, 3)" />
    - <InlineMath math="C'(-3, 6)" />

3.  The equation of the image of the line <InlineMath math="3x - 2y + 6 = 0" /> is <InlineMath math="-3x - 2y + 6 = 0" />.

    **Explanation:** Substitute <InlineMath math="x" /> with <InlineMath math="-x" /> into the original equation:

    <div className="flex flex-col gap-4">
      <BlockMath math="3(-x) - 2y + 6 = 0" />
      <BlockMath math="-3x - 2y + 6 = 0" />
    </div>

4.  The equation of the image of the line <InlineMath math="y = -3x - 4" /> is <InlineMath math="y = 3x - 4" />.

    **Explanation:** Substitute <InlineMath math="x" /> with <InlineMath math="-x" />:

    <div className="flex flex-col gap-4">
      <BlockMath math="y = -3(-x) - 4" />
      <BlockMath math="y = 3x - 4" />
    </div>