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

Learn horizontal reflection in functions with clear examples. Learn how to reflect graphs across the vertical axis and apply this transformation.

---

## 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.

Visible text: Horizontal reflection is a geometric transformation that reflects the graph of a function across the -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 $$f(x)$$, then horizontal reflection produces a new function $$g(x) = f(-x)$$ which is the reflection of the original function across the $$y$$-axis.

Visible text: If we have a function , then horizontal reflection produces a new function which is the reflection of the original function across the -axis.

### Rules of Horizontal Reflection

For any function $$f(x)$$, horizontal reflection is defined as:

Visible text: For any function , horizontal reflection is defined as:

```math
g(x) = f(-x)
```

This transformation changes every point $$(x, y)$$ on the original graph to $$(-x, y)$$ on the reflected graph.

Visible text: This transformation changes every point on the original graph to on the reflected graph.

## Visualization of Horizontal Reflection

Let's see how horizontal reflection works on the quadratic function $$f(x) = (x - 2)^2$$.

Visible text: Let's see how horizontal reflection works on the quadratic function .

Component: LineEquation
Props:
- title: Horizontal Reflection of Quadratic Function $$f(x) = (x - 2)^2$$
  Visible text: Horizontal Reflection of Quadratic Function
- description: Notice how the graph reflects across the $$y$$-axis, forming a symmetric reflection.
  Visible text: Notice how the graph reflects across the -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 $$f(x) = (x - 2)^2$$ has its vertex at $$(2, 0)$$
- The reflected function $$g(x) = (-x - 2)^2$$ has its vertex at $$(-2, 0)$$
- Both graphs are symmetric across the $$y$$-axis

Visible text: - The original function has its vertex at 
- The reflected function has its vertex at 
- Both graphs are symmetric across the -axis

## Horizontal Reflection on Linear Functions

Now let's apply the same concept to the linear function $$f(x) = 2x + 3$$.

Visible text: Now let's apply the same concept to the linear function .

Component: LineEquation
Props:
- title: Horizontal Reflection of Linear Function $$f(x) = 2x + 3$$
  Visible text: Horizontal Reflection of Linear Function
- description: The reflected line has opposite slope but the same $$y$$-intercept.
  Visible text: The reflected line has opposite slope but the same -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 $$f(x) = 2x + 3$$ has a positive slope of $$2$$
- The reflected function $$g(x) = -2x + 3$$ has a negative slope of $$2$$
- Both lines intersect the $$y$$-axis at the same point $$(0, 3)$$

Visible text: - The original function has a positive slope of 
- The reflected function has a negative slope of 
- Both lines intersect the -axis at the same point

## Important Properties of Horizontal Reflection

### Vertical Axis as Mirror Line

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.

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

### Effect on Coordinate Points

If point $$(a, b)$$ is on the graph of $$f(x)$$, then the corresponding point on the graph of $$f(-x)$$ is $$(-a, b)$$.

Visible text: If point is on the graph of , then the corresponding point on the graph of is .

### 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 $$[c, d]$$, then the domain after horizontal reflection becomes $$[-d, -c]$$.

Visible text: If the domain of the original function is , then the domain after horizontal reflection becomes .

## Application Examples

### Exponential Function Example

Let's look at horizontal reflection on the exponential function $$f(x) = 2^x$$.

Visible text: Let's look at horizontal reflection on the exponential function .

Component: LineEquation
Props:
- title: Horizontal Reflection of Exponential Function $$f(x) = 2^x$$
  Visible text: Horizontal Reflection of Exponential Function
- 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 $$y = 0$$ for both functions
- The $$y$$-intercept remains the same at $$(0, 1)$$
- The function that was originally increasing becomes decreasing after reflection

Visible text: - The horizontal asymptote remains at for both functions
- The -intercept remains the same at 
- 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.

Component: LineEquation
Props:
- title: Horizontal Reflection of Square Root Function $$f(x) = \sqrt{x}$$
  Visible text: Horizontal Reflection of Square Root Function
- 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 $$f(x) = \sqrt{x}$$ is $$[0, \infty)$$
- The domain of the reflected function $$g(x) = \sqrt{-x}$$ is $$(-\infty, 0]$$
- Both curves meet at the origin $$(0, 0)$$

Visible text: - The domain of the original function is 
- The domain of the reflected function is 
- Both curves meet at the origin

## Exercises

1. Given the function $$f(x) = x^2 + 3x + 2$$. Determine the equation of the function resulting from horizontal reflection.

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

3. Function $$h(x) = |x + 2|$$ undergoes horizontal reflection. Determine the vertex of the resulting reflected function.

Visible text: 1. Given the function . Determine the equation of the function resulting from horizontal reflection.

2. If the graph of function is reflected across the -axis, determine:
 - The equation of the resulting reflected function
 - The domain of the function after reflection

3. Function 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 $$f(x) = x^2 + 3x + 2$$ 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 $$(-\infty, \infty)$$ because exponential functions are defined for all real numbers

   Visualization:

   <LineEquation
     title={<>Function $$g(x) = 3^x + 1$$ 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 $$h(x) = |x + 2|$$ has its vertex at $$(-2, 0)$$.
   After horizontal reflection: <InlineMath math="h'(x) = |-x + 2| = |-(x - 2)| = |x - 2|" />, the vertex becomes $$(2, 0)$$.

   <LineEquation
     title={<>Function $$h(x) = |x + 2|$$ 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,
       },
     ]}
   />

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

 <LineEquation
 title={<>Function and Its Reflection Result</>}
 description={<>The original parabola is reflected across the -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 because exponential functions are defined for all real numbers

 Visualization:

 <LineEquation
 title={<>Function 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 has its vertex at .
 After horizontal reflection: <InlineMath math="h'(x) = |-x + 2| = |-(x - 2)| = |x - 2|" />, the vertex becomes .

 <LineEquation
 title={<>Function and Its Reflection Result</>}
 description={<>The absolute value function is reflected across the -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,
 },
 ]}
 />