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

Learn vertical reflection by flipping function graphs across the horizontal axis through examples and exercises.

---

## Basic Concepts of Vertical Reflection

Vertical reflection is a geometric transformation that reflects the graph of a function across the $$x$$-axis, like seeing the reflection of an object on the surface of calm water. Imagine an object reflected in a horizontal mirror, its shape remains the same but its position is flipped vertically.

Visible text: Vertical reflection is a geometric transformation that reflects the graph of a function across the -axis, like seeing the reflection of an object on the surface of calm water. Imagine an object reflected in a horizontal mirror, its shape remains the same but its position is flipped vertically.

If we have a function $$f(x)$$, then vertical reflection produces a new function $$g(x) = -f(x)$$ which is the reflection of the original function across the $$x$$-axis.

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

### Rules of Vertical Reflection

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

Visible text: For any function , vertical 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 Vertical Reflection

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

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

Component: LineEquation
Props:
- title: Vertical Reflection of Quadratic Function $$f(x) = x^2$$
  Visible text: Vertical Reflection of Quadratic Function
- description: Notice how the graph reflects across the $$x$$-axis, forming an inverted reflection.
  Visible text: Notice how the graph reflects across the -axis, forming an inverted reflection.
- showZAxis: false
- 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 * x), z: 0 };
}),
color: getColor("ORANGE"),
labels: [{ text: "g(x) = -x²", offset: [0.5, -1, 0] }],
showPoints: false,
},
]

From the visualization above, we can observe:

- The original function $$f(x) = x^2$$ opens upward with vertex at $$(0, 0)$$
- The reflected function $$g(x) = -x^2$$ opens downward with vertex still at $$(0, 0)$$
- Both graphs are symmetric across the $$x$$-axis

Visible text: - The original function opens upward with vertex at 
- The reflected function opens downward with vertex still at 
- Both graphs are symmetric across the -axis

## Vertical 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: Vertical Reflection of Linear Function $$f(x) = 2x + 3$$
  Visible text: Vertical Reflection of Linear Function
- description: The reflected line has opposite slope and opposite $$y$$-intercept.
  Visible text: The reflected line has opposite slope and opposite -intercept.
- showZAxis: false
- cameraPosition: [12, 6, 12]
- 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: [1, 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) = -2x - 3", offset: [1, -0.5, 0] }],
showPoints: false,
},
]

Notice that:

- The original function $$f(x) = 2x + 3$$ has positive slope and intersects the $$y$$-axis at $$(0, 3)$$
- The reflected function $$g(x) = -2x - 3$$ has negative slope and intersects the $$y$$-axis at $$(0, -3)$$
- Both lines intersect at the $$x$$-axis

Visible text: - The original function has positive slope and intersects the -axis at 
- The reflected function has negative slope and intersects the -axis at 
- Both lines intersect at the -axis

## Important Properties of Vertical Reflection

### Horizontal Axis as Mirror Line

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

Visible text: Vertical 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**: Does not change after vertical reflection
- **Range**: Changes to the opposite of the original range

If the range of the original function is $$[c, d]$$, then the range after vertical reflection becomes $$[-d, -c]$$.

Visible text: If the range of the original function is , then the range after vertical reflection becomes .

## Application Examples

### Exponential Function Example

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

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

Component: LineEquation
Props:
- title: Vertical Reflection of Exponential Function $$f(x) = 2^x$$
  Visible text: Vertical Reflection of Exponential Function
- description: The exponential curve reflects to become a decreasing curve with horizontal asymptote below the $$x$$-axis.
  Visible text: The exponential curve reflects to become a decreasing curve with horizontal asymptote below the -axis.
- 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: [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), z: 0 };
}),
color: getColor("EMERALD"),
labels: [{ text: "g(x) = -2^x", offset: [0.5, -1, 0] }],
showPoints: false,
},
]

For exponential functions:

- The horizontal asymptote remains at $$y = 0$$ for both functions (since the $$x$$-axis reflects onto itself)
- The $$y$$-intercept changes from $$(0, 1)$$ to $$(0, -1)$$
- The function that was originally increasing becomes decreasing

Visible text: - The horizontal asymptote remains at for both functions (since the -axis reflects onto itself)
- The -intercept changes from to 
- The function that was originally increasing becomes decreasing

## Vertical Reflection on Trigonometric Functions

Let's see how vertical reflection affects the $$\sin$$ function.

Visible text: Let's see how vertical reflection affects the function.

Component: LineEquation
Props:
- title: Vertical Reflection of Sine Function $$f(x) = \sin(x)$$
  Visible text: Vertical Reflection of Sine Function
- description: The reflected $$\sin$$ wave produces a wave that moves in opposite phase.
  Visible text: The reflected wave produces a wave that moves in opposite phase.
- showZAxis: false
- cameraPosition: [0, 0, 12]
- data: [
{
points: Array.from({ length: 61 }, (_, i) => {
const x = (i - 30) * 0.2;
return { x, y: Math.sin(x), z: 0 };
}),
color: getColor("CYAN"),
labels: [{ text: "f(x) = sin(x)", offset: [1, 1.5, 0] }],
showPoints: false,
},
{
points: Array.from({ length: 61 }, (_, i) => {
const x = (i - 30) * 0.2;
return { x, y: -Math.sin(x), z: 0 };
}),
color: getColor("ROSE"),
labels: [{ text: "g(x) = -sin(x)", offset: [1, -1.5, 0] }],
showPoints: false,
},
]

Notice that:

- The amplitude remains the same but the wave direction is inverted
- The period and frequency do not change
- Maximum points become minimum points and vice versa

## Exercises

1. Given the function $$f(x) = x^2 + 4x + 3$$. Determine the equation of the function resulting from vertical reflection.

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

3. Function $$h(x) = \sqrt{x + 1}$$ undergoes vertical reflection. Determine the $$y$$-intercept of the resulting reflected function.

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

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

3. Function undergoes vertical reflection. Determine the -intercept of the resulting reflected function.

### Answer Key

1. Vertical reflection: <InlineMath math="f'(x) = -f(x) = -(x^2 + 4x + 3) = -x^2 - 4x - 3" />

   <LineEquation
     title={<>Function $$f(x) = x^2 + 4x + 3$$ and Its Reflection Result</>}
     description="The original parabola opening upward is reflected to become a parabola opening downward."
     showZAxis={false}
     data={[
       {
         points: Array.from({ length: 41 }, (_, i) => {
           const x = (i - 20) * 0.25;
           return { x, y: x * x + 4 * x + 3, z: 0 };
         }),
         color: getColor("PURPLE"),
         labels: [{ text: "f(x) = x² + 4x + 3", 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 + 3), z: 0 };
         }),
         color: getColor("ORANGE"),
         labels: [{ text: "f'(x) = -x² - 4x - 3", offset: [1, -1, 0] }],
         showPoints: false,
       },
     ]}
   />

2. Equation of the resulting reflected function:

   - Vertical reflection: <InlineMath math="g'(x) = -(3^x + 2) = -3^x - 2" />
   - Range after reflection: Since the original range of $$g(x) = 3^x + 2$$ is $$(2, \infty)$$, the range after reflection is $$(-\infty, -2)$$

   Visualization:

   <LineEquation
     title={<>Function $$g(x) = 3^x + 2$$ and Its Reflection Result</>}
     description="The increasing exponential curve is reflected to become a decreasing curve with a new horizontal 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) + 2, z: 0 };
         }),
         color: getColor("VIOLET"),
         labels: [{ text: "g(x) = 3^x + 2", 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("TEAL"),
         labels: [{ text: "g'(x) = -3^x - 2", offset: [0.5, -1, 0] }],
         showPoints: false,
       },
     ]}
   />

3. The original function $$h(x) = \sqrt{x + 1}$$ has a $$y$$-intercept at $$(0, 1)$$ because $$h(0) = \sqrt{0 + 1} = 1$$.
   After vertical reflection: <InlineMath math="h'(x) = -\sqrt{x + 1}" />, the $$y$$-intercept becomes $$(0, -1)$$.

   <LineEquation
     title={<>Function $$h(x) = \sqrt{x + 1}$$ and Its Reflection Result</>}
     description={<>The square root curve is reflected across the $$x$$-axis resulting in a curve opening downward.</>}
     showZAxis={false}
     data={[
       {
         points: Array.from({ length: 21 }, (_, i) => {
           const x = i * 0.25;
           return { x, y: Math.sqrt(x + 1), z: 0 };
         }),
         color: getColor("INDIGO"),
         labels: [{ text: "h(x) = √(x + 1)", offset: [1, 0.5, 0] }],
         showPoints: false,
       },
       {
         points: Array.from({ length: 21 }, (_, i) => {
           const x = i * 0.25;
           return { x, y: -Math.sqrt(x + 1), z: 0 };
         }),
         color: getColor("EMERALD"),
         labels: [{ text: "h'(x) = -√(x + 1)", offset: [1, -0.5, 0] }],
         showPoints: false,
       },
     ]}
   />

Visible text: 1. Vertical reflection: <InlineMath math="f'(x) = -f(x) = -(x^2 + 4x + 3) = -x^2 - 4x - 3" />

 <LineEquation
 title={<>Function and Its Reflection Result</>}
 description="The original parabola opening upward is reflected to become a parabola opening downward."
 showZAxis={false}
 data={[
 {
 points: Array.from({ length: 41 }, (_, i) => {
 const x = (i - 20) * 0.25;
 return { x, y: x * x + 4 * x + 3, z: 0 };
 }),
 color: getColor("PURPLE"),
 labels: [{ text: "f(x) = x² + 4x + 3", 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 + 3), z: 0 };
 }),
 color: getColor("ORANGE"),
 labels: [{ text: "f'(x) = -x² - 4x - 3", offset: [1, -1, 0] }],
 showPoints: false,
 },
 ]}
 />

2. Equation of the resulting reflected function:

 - Vertical reflection: <InlineMath math="g'(x) = -(3^x + 2) = -3^x - 2" />
 - Range after reflection: Since the original range of is , the range after reflection is 

 Visualization:

 <LineEquation
 title={<>Function and Its Reflection Result</>}
 description="The increasing exponential curve is reflected to become a decreasing curve with a new horizontal 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) + 2, z: 0 };
 }),
 color: getColor("VIOLET"),
 labels: [{ text: "g(x) = 3^x + 2", 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("TEAL"),
 labels: [{ text: "g'(x) = -3^x - 2", offset: [0.5, -1, 0] }],
 showPoints: false,
 },
 ]}
 />

3. The original function has a -intercept at because .
 After vertical reflection: <InlineMath math="h'(x) = -\sqrt{x + 1}" />, the -intercept becomes .

 <LineEquation
 title={<>Function and Its Reflection Result</>}
 description={<>The square root curve is reflected across the -axis resulting in a curve opening downward.</>}
 showZAxis={false}
 data={[
 {
 points: Array.from({ length: 21 }, (_, i) => {
 const x = i * 0.25;
 return { x, y: Math.sqrt(x + 1), z: 0 };
 }),
 color: getColor("INDIGO"),
 labels: [{ text: "h(x) = √(x + 1)", offset: [1, 0.5, 0] }],
 showPoints: false,
 },
 {
 points: Array.from({ length: 21 }, (_, i) => {
 const x = i * 0.25;
 return { x, y: -Math.sqrt(x + 1), z: 0 };
 }),
 color: getColor("EMERALD"),
 labels: [{ text: "h'(x) = -√(x + 1)", offset: [1, -0.5, 0] }],
 showPoints: false,
 },
 ]}
 />