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URL: https://nakafa.com/en/subjects/mathematics/geometric-transformation/reflection-over-x-axis
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/material/lesson/mathematics/geometric-transformation/reflection-over-x-axis/en.mdx

Learn horizontal-axis reflection transformations. Learn how coordinates change with triangles, lines, and interactive visualizations.

---

## Understanding Reflection across the Horizontal Axis

Reflection across the $$x$$-axis is a type of geometric transformation that moves every point of an object to a new position symmetrical to the $$x$$-axis. Imagine the $$x$$-axis as a flat mirror.

Visible text: Reflection across the -axis is a type of geometric transformation that moves every point of an object to a new position symmetrical to the -axis. Imagine the -axis as a flat mirror.

If a point $$P$$ has coordinates $$(x,y)$$, then its reflection, which we'll call <InlineMath math="P'" />, will have the same $$x$$-coordinate, but its $$y$$-coordinate will be the negative of the original $$y$$ value.

Visible text: If a point has coordinates , then its reflection, which we'll call <InlineMath math="P'" />, will have the same -coordinate, but its -coordinate will be the negative of the original value.

Mathematically, if the initial point is $$P(x,y)$$, then after reflection across the $$x$$-axis, its image is <InlineMath math="P'(x,-y)" />.

Visible text: Mathematically, if the initial point is , then after reflection across the -axis, its image is <InlineMath math="P'(x,-y)" />.

### Visualizing Points and Their Reflections

Let's observe some points and their reflections after being mirrored across the $$x$$-axis.

Visible text: Let's observe some points and their reflections after being mirrored across the -axis.

Notice how the $$y$$-coordinate changes sign, while the $$x$$-coordinate remains the same.

Visible text: Notice how the -coordinate changes sign, while the -coordinate remains the same.

Component: LineEquation
Props:
- title: Point Reflections Across the X-Axis
- description: Visualization of points $$A, B, C, D$$ and their
reflections <InlineMath math="A', B', C', D'" /> after mirroring across
the $$x$$-axis.
  Visible text: Visualization of points and their
reflections <InlineMath math="A', B', C', D'" /> after mirroring across
the -axis.
- cameraPosition: [0, 0, 15]
- showZAxis: false
- data: [
...[
{ x: -5, y: 2, label: "A" },
{ x: -3, y: 1, label: "B" },
{ x: 1, y: 2, label: "C" },
{ x: 4, y: -2, label: "D" },
].map((p, i) => ({
points: [
{ x: p.x, y: p.y, z: 0 },
{ x: p.x, y: p.y, z: 0 }, // Single point
],
color: getColor("AMBER"),
labels: [{ text: p.label, at: 0, offset: [0.3, 0.3, 0] }],
showPoints: true,
})),
...[
{ x: -5, y: -2, label: "A'" },
{ x: -3, y: -1, label: "B'" },
{ x: 1, y: -2, label: "C'" },
{ x: 4, y: 2, label: "D'" },
].map((p, i) => ({
points: [
{ x: p.x, y: p.y, z: 0 },
{ x: p.x, y: p.y, z: 0 }, // Single point
],
color: getColor("CYAN"),
labels: [{ text: p.label, at: 0, offset: [0.3, 0.3, 0] }],
showPoints: true,
})),
]

Based on the interactive visualization above, we can observe the relationship between the original points (pre-image) and their reflections (image) as follows:

- Point $$A(-5,2)$$ becomes <InlineMath math="A'(-5,-2)" />
- Point $$B(-3,1)$$ becomes <InlineMath math="B'(-3,-1)" />
- Point $$C(1,2)$$ becomes <InlineMath math="C'(1,-2)" />
- Point $$D(4,-2)$$ becomes <InlineMath math="D'(4,2)" />

Visible text: - Point becomes <InlineMath math="A'(-5,-2)" />
- Point becomes <InlineMath math="B'(-3,-1)" />
- Point becomes <InlineMath math="C'(1,-2)" />
- Point becomes <InlineMath math="D'(4,2)" />

The visible pattern is that the $$x$$ value remains constant, and the $$y$$ value changes sign (becomes its opposite).

Visible text: The visible pattern is that the value remains constant, and the value changes sign (becomes its opposite).

## Property of Reflection across the Horizontal Axis

Based on the observations above, we can formulate the property of reflection across the $$x$$-axis:

Visible text: Based on the observations above, we can formulate the property of reflection across the -axis:

```math
P(x,y) \xrightarrow{\text{X-axis}} P'(x,-y)
```

This means the image of point $$P(x,y)$$ reflected across the $$x$$-axis is <InlineMath math="P'(x,-y)" />. The $$x$$-axis in this case acts as the line $$y=0$$.

Visible text: This means the image of point reflected across the -axis is <InlineMath math="P'(x,-y)" />. The -axis in this case acts as the line .

## Application Examples

### Reflecting a Triangle

Determine the image of triangle $$ABC$$ with vertices $$A(-1,4)$$, $$B(2,1)$$, and $$C(-2,-1)$$ reflected across the $$x$$-axis.

Visible text: Determine the image of triangle with vertices , , and reflected across the -axis.

To determine the image of triangle $$ABC$$, we apply the reflection property to each of its vertices:

Visible text: To determine the image of triangle , we apply the reflection property to each of its vertices:

Component: MathContainer
Children:

```math
A(-1,4) \xrightarrow{\text{X-axis}} A'(-1,-4)
```

```math
B(2,1) \xrightarrow{\text{X-axis}} B'(2,-1)
```

```math
C(-2,-1) \xrightarrow{\text{X-axis}} C'(-2,1)
```

Consequently, the image of triangle $$ABC$$ is triangle <InlineMath math="A'B'C'" /> with vertices <InlineMath math="A'(-1,-4)" />, <InlineMath math="B'(2,-1)" />, and <InlineMath math="C'(-2,1)" />.

Visible text: Consequently, the image of triangle is triangle <InlineMath math="A'B'C'" /> with vertices <InlineMath math="A'(-1,-4)" />, <InlineMath math="B'(2,-1)" />, and <InlineMath math="C'(-2,1)" />.

Component: LineEquation
Props:
- title: Reflection of Triangle $$ABC$$ across the X-Axis
  Visible text: Reflection of Triangle across the X-Axis
- description: Visualization of triangle $$ABC$$ and its
reflection <InlineMath math="A'B'C'" /> after mirroring across
the $$x$$-axis.
  Visible text: Visualization of triangle and its
reflection <InlineMath math="A'B'C'" /> after mirroring across
the -axis.
- cameraPosition: [0, 0, 15]
- showZAxis: false
- data: [
// Triangle ABC (Original)
...[
{
from: { x: -1, y: 4, z: 0, label: "A" },
to: { x: 2, y: 1, z: 0, label: "B" },
},
{
from: { x: 2, y: 1, z: 0, label: "B" },
to: { x: -2, y: -1, z: 0, label: "C" },
},
{
from: { x: -2, y: -1, z: 0, label: "C" },
to: { x: -1, y: 4, z: 0, label: "A" },
},
].map((segment) => ({
points: [segment.from, segment.to],
color: getColor("ORANGE"),
showPoints: true,
labels: [
{
text: segment.from.label,
at: 0,
offset: [0.3, 0.3, 0],
},
{ text: segment.to.label, at: 1, offset: [0.3, 0.3, 0] },
],
})),
// Triangle A'B'C' (Reflected)
...[
{
from: { x: -1, y: -4, z: 0, label: "A'" },
to: { x: 2, y: -1, z: 0, label: "B'" },
},
{
from: { x: 2, y: -1, z: 0, label: "B'" },
to: { x: -2, y: 1, z: 0, label: "C'" },
},
{
from: { x: -2, y: 1, z: 0, label: "C'" },
to: { x: -1, y: -4, z: 0, label: "A'" },
},
].map((segment) => ({
points: [segment.from, segment.to],
color: getColor("PURPLE"),
showPoints: true,
labels: [
{
text: segment.from.label,
at: 0,
offset: [0.3, 0.3, 0],
},
{ text: segment.to.label, at: 1, offset: [0.3, 0.3, 0] },
],
})),
]

### Reflecting a Line

If a line has the equation $$2x - 3y = 0$$ and is reflected across the $$x$$-axis, determine the equation of its reflected line.

Visible text: If a line has the equation and is reflected across the -axis, determine the equation of its reflected line.

**Alternative Solution:**

Let an arbitrary point $$P(a,b)$$ lie on the line $$2x - 3y = 0$$. Then, the following holds:

Visible text: Let an arbitrary point lie on the line . Then, the following holds:

```math
2a - 3b = 0
```

The point $$P(a,b)$$ reflected across the $$x$$-axis produces the image <InlineMath math="P'(a,-b)" />.

Visible text: The point reflected across the -axis produces the image <InlineMath math="P'(a,-b)" />.

To obtain the equation of the reflected line, we substitute the coordinates of the image into new variables. Let <InlineMath math="x' = a" /> and <InlineMath math="y' = -b" />.

From this, we get <InlineMath math="a = x'" /> and <InlineMath math="b = -y'" />.

Substitute <InlineMath math="a = x'" /> and <InlineMath math="b = -y'" /> into the original equation $$2a - 3b = 0$$:

Visible text: Substitute <InlineMath math="a = x'" /> and <InlineMath math="b = -y'" /> into the original equation :

Component: MathContainer
Children:

```math
2(x') - 3(-y') = 0
```

```math
2x' + 3y' = 0
```

Since <InlineMath math="x'" /> and <InlineMath math="y'" /> are arbitrary variables representing the coordinates on the reflected line, we can rewrite them as $$x$$ and $$y$$.

Visible text: Since <InlineMath math="x'" /> and <InlineMath math="y'" /> are arbitrary variables representing the coordinates on the reflected line, we can rewrite them as and .

Thus, the equation of the reflected line is:

Component: ContentStack
Children:

```math
2x + 3y = 0
```

Component: LineEquation
Props:
- title: Reflection of Line $$2x - 3y = 0$$ across the X-Axis
  Visible text: Reflection of Line across the X-Axis
- description: The original line $$2x - 3y = 0$$ (lime green) and its
reflection $$2x + 3y = 0$$ (magenta) after mirroring.
  Visible text: The original line (lime green) and its
reflection (magenta) after mirroring.
- cameraPosition: [0, 0, 15]
- showZAxis: false
- data: [
{
// Original Line: 2x - 3y = 0 => y = (2/3)x
points: [
{ x: -6, y: (2 / 3) * -6, z: 0 },
{ x: 6, y: (2 / 3) * 6, z: 0 },
],
color: getColor("LIME"),
labels: [{ text: "2x - 3y = 0", at: 1, offset: [0.5, 0.5, 0] }],
},
{
// Reflected Line: 2x + 3y = 0 => y = -(2/3)x
points: [
{ x: -6, y: -(2 / 3) * -6, z: 0 },
{ x: 6, y: -(2 / 3) * 6, z: 0 },
],
color: getColor("MAGENTA"),
labels: [{ text: "2x + 3y = 0", at: 1, offset: [0.5, -0.5, 0] }],
},
]

This shows how the equation of a line changes after being reflected across the $$x$$-axis.

Visible text: This shows how the equation of a line changes after being reflected across the -axis.