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

Learn vertical dilation of functions with clear examples. Learn how to stretch or compress graphs vertically using scale factors and apply transformations.

---

## Basic Concepts of Vertical Dilation

Vertical dilation is a geometric transformation that changes the size of a function graph vertically, like stretching or compressing a rubber band up and down. Imagine pulling a photo with both hands, one above and one below, then stretching or compressing it vertically without changing the width of the photo.

If we have a function $$f(x)$$, then vertical dilation produces a new function $$g(x) = k \cdot f(x)$$ where $$k$$ is the scale factor that determines how much the vertical size changes.

Visible text: If we have a function , then vertical dilation produces a new function where is the scale factor that determines how much the vertical size changes.

### Rules of Vertical Dilation

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

Visible text: For any function , vertical dilation is defined as:

```math
g(x) = k \cdot f(x)
```

Where $$k$$ is the scale factor that affects the transformation:

Visible text: Where is the scale factor that affects the transformation:

- If $$k > 1$$, the graph is stretched vertically (enlarged)
- If $$0 < k < 1$$, the graph is compressed vertically (reduced)
- If $$k = 1$$, the graph does not change
- If $$k < 0$$, the graph undergoes reflection as well as dilation

Visible text: - If , the graph is stretched vertically (enlarged)
- If , the graph is compressed vertically (reduced)
- If , the graph does not change
- If , the graph undergoes reflection as well as dilation

## Visualization of Vertical Dilation

Let's see how vertical dilation works on the quadratic function $$f(x) = x^2$$ with various scale factors.

Visible text: Let's see how vertical dilation works on the quadratic function with various scale factors.

Component: LineEquation
Props:
- title: Vertical Dilation of Quadratic Function $$f(x) = x^2$$
  Visible text: Vertical Dilation of Quadratic Function
- description: Notice how the graph is stretched or compressed vertically with different scale factors.
- showZAxis: false
- cameraPosition: [12, 8, 12]
- 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: [3.5, 1, 0] }],
showPoints: false,
},
{
points: Array.from({ length: 41 }, (_, i) => {
const x = (i - 20) * 0.25;
return { x, y: 2 * x * x, z: 0 };
}),
color: getColor("ORANGE"),
labels: [{ text: "g(x) = 2x²", offset: [3.5, 2, 0] }],
showPoints: false,
},
{
points: Array.from({ length: 41 }, (_, i) => {
const x = (i - 20) * 0.25;
return { x, y: 0.5 * x * x, z: 0 };
}),
color: getColor("SKY"),
labels: [{ text: "h(x) = 0.5x²", offset: [3.5, 0.5, 0] }],
showPoints: false,
},
]

From the visualization above, we can observe:

- The original function $$f(x) = x^2$$ as reference
- Function $$g(x) = 2x^2$$ is vertically stretched by factor $$2$$
- Function $$h(x) = 0.5x^2$$ is vertically compressed by factor $$0.5$$
- All graphs have the same vertex at $$(0, 0)$$

Visible text: - The original function as reference
- Function is vertically stretched by factor 
- Function is vertically compressed by factor 
- All graphs have the same vertex at

## Vertical Dilation on Linear Functions

Now let's apply the same concept to the linear function $$f(x) = x + 1$$.

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

Component: LineEquation
Props:
- title: Vertical Dilation of Linear Function $$f(x) = x + 1$$
  Visible text: Vertical Dilation of Linear Function
- description: The dilated line has slope that changes according to the scale factor.
- showZAxis: false
- cameraPosition: [10, 6, 10]
- data: [
{
points: Array.from({ length: 21 }, (_, i) => {
const x = (i - 10) * 0.5;
return { x, y: x + 1, z: 0 };
}),
color: getColor("AMBER"),
labels: [{ text: "f(x) = x + 1", offset: [2, 1.5, 0] }],
showPoints: false,
},
{
points: Array.from({ length: 21 }, (_, i) => {
const x = (i - 10) * 0.5;
return { x, y: 3 * (x + 1), z: 0 };
}),
color: getColor("TEAL"),
labels: [{ text: "g(x) = 3(x + 1)", offset: [1, 2, 0] }],
showPoints: false,
},
{
points: Array.from({ length: 21 }, (_, i) => {
const x = (i - 10) * 0.5;
return { x, y: 0.5 * (x + 1), z: 0 };
}),
color: getColor("ROSE"),
labels: [{ text: "h(x) = 0.5(x + 1)", offset: [2, 0.5, 0] }],
showPoints: false,
},
]

Notice that:

- The original function $$f(x) = x + 1$$ has slope $$1$$
- Function $$g(x) = 3(x + 1)$$ has slope $$3$$ (stretched)
- Function $$h(x) = 0.5(x + 1)$$ has slope $$0.5$$ (compressed)
- All lines still intersect the $$y$$-axis, but at different points

Visible text: - The original function has slope 
- Function has slope (stretched)
- Function has slope (compressed)
- All lines still intersect the -axis, but at different points

## Important Properties of Vertical Dilation

### Effect on Coordinate Points

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

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 dilation
- **Range**: Changes according to the scale factor $$k$$

Visible text: - **Domain**: Does not change after vertical dilation
- **Range**: Changes according to the scale factor

If the range of the original function is $$[c, d]$$, then the range after vertical dilation with factor $$k > 0$$ becomes $$[k \cdot c, k \cdot d]$$.

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

### Axis Intercepts

- **$$x$$-intercept**: Does not change (except if $$k = 0$$)
- **$$y$$-intercept**: Changes according to the scale factor

Visible text: - **-intercept**: Does not change (except if )
- **-intercept**: Changes according to the scale factor

## Application Examples

### Exponential Function

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

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

Component: LineEquation
Props:
- title: Vertical Dilation of Exponential Function $$f(x) = 2^x$$
  Visible text: Vertical Dilation of Exponential Function
- description: The exponential curve undergoes height changes according to the scale factor.
- 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: [2, 1, 0] }],
showPoints: false,
},
{
points: Array.from({ length: 31 }, (_, i) => {
const x = (i - 15) * 0.3;
return { x, y: 2 * Math.pow(2, x), z: 0 };
}),
color: getColor("EMERALD"),
labels: [{ text: "g(x) = 2 · 2^x", offset: [1, 2, 0] }],
showPoints: false,
},
]

For exponential functions:

- The horizontal asymptote remains at $$y = 0$$ for both functions
- The $$y$$-intercept changes from $$(0, 1)$$ to $$(0, 2)$$
- The growth rate of the function increases according to the scale factor

Visible text: - The horizontal asymptote remains at for both functions
- The -intercept changes from to 
- The growth rate of the function increases according to the scale factor

## Vertical Dilation with Negative Factor

Let's see what happens when the scale factor is negative.

Component: LineEquation
Props:
- title: Vertical Dilation with Negative Factor $$f(x) = x^2$$
  Visible text: Vertical Dilation with Negative Factor
- description: Negative scale factor causes reflection as well as dilation.
- showZAxis: false
- cameraPosition: [10, 6, 10]
- data: [
{
points: Array.from({ length: 41 }, (_, i) => {
const x = (i - 20) * 0.25;
return { x, y: x * x, z: 0 };
}),
color: getColor("CYAN"),
labels: [{ text: "f(x) = x²", offset: [1, 1, 0] }],
showPoints: false,
},
{
points: Array.from({ length: 41 }, (_, i) => {
const x = (i - 20) * 0.25;
return { x, y: -2 * x * x, z: 0 };
}),
color: getColor("PINK"),
labels: [{ text: "g(x) = -2x²", offset: [1, -2, 0] }],
showPoints: false,
},
]

When the scale factor is negative:

- The graph undergoes reflection across the $$x$$-axis
- Simultaneously undergoes dilation according to the absolute value of the scale factor
- A parabola that opens upward becomes one that opens downward

Visible text: - The graph undergoes reflection across the -axis
- Simultaneously undergoes dilation according to the absolute value of the scale factor
- A parabola that opens upward becomes one that opens downward

## Exercises

1. Given the function $$f(x) = x^2 - 4x + 3$$. Determine the equation of the function resulting from vertical dilation with scale factor $$3$$.

2. If the graph of function $$g(x) = \sqrt{x}$$ undergoes vertical dilation with factor $$\frac{1}{2}$$, determine:
   - The equation of the resulting dilated function
   - The range of the function after dilation

3. Function $$h(x) = |x - 1| + 2$$ undergoes vertical dilation with factor $$-1$$. Determine the vertex of the resulting dilated function.

Visible text: 1. Given the function . Determine the equation of the function resulting from vertical dilation with scale factor .

2. If the graph of function undergoes vertical dilation with factor , determine:
 - The equation of the resulting dilated function
 - The range of the function after dilation

3. Function undergoes vertical dilation with factor . Determine the vertex of the resulting dilated function.

### Answer Key

1. Vertical dilation with factor $$3$$: <InlineMath math="f'(x) = 3f(x) = 3(x^2 - 4x + 3) = 3x^2 - 12x + 9" />

   <LineEquation
     title={<>Function $$f(x) = x^2 - 4x + 3$$ and Its Dilation Result</>}
     description={
       <>
         The original parabola is vertically stretched by factor{" "}
         $$3$$ producing a taller parabola.
       </>
     }
     showZAxis={false}
     cameraPosition={[12, 8, 12]}
     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: 3 * (x * x - 4 * x + 3), z: 0 };
         }),
         color: getColor("ORANGE"),
         labels: [{ text: "f'(x) = 3x² - 12x + 9", offset: [1, -3, 0] }],
         showPoints: false,
       },
     ]}
   />

2. Equation of the resulting dilated function:

   - Vertical dilation: <InlineMath math="g'(x) = \frac{1}{2}g(x) = \frac{1}{2}\sqrt{x}" />
   - Range after dilation: $$[0, \infty)$$ becomes $$[0, \infty)$$ but with smaller maximum values

   Visualization:

   <LineEquation
     title={<>Function $$g(x) = \sqrt{x}$$ and Its Dilation Result</>}
     description={
       <>
         The square root curve is vertically compressed by factor{" "}
         $$0.5$$ producing a lower curve.
       </>
     }
     showZAxis={false}
     cameraPosition={[8, 6, 8]}
     data={[
       {
         points: Array.from({ length: 21 }, (_, i) => {
           const x = i * 0.25;
           return { x, y: Math.sqrt(x), z: 0 };
         }),
         color: getColor("VIOLET"),
         labels: [{ text: "g(x) = √x", offset: [2, 1, 0] }],
         showPoints: false,
       },
       {
         points: Array.from({ length: 21 }, (_, i) => {
           const x = i * 0.25;
           return { x, y: 0.5 * Math.sqrt(x), z: 0 };
         }),
         color: getColor("TEAL"),
         labels: [{ text: "g'(x) = 0.5√x", offset: [2, 0.5, 0] }],
         showPoints: false,
       },
     ]}
   />

3. The original function $$h(x) = |x - 1| + 2$$ has its vertex at $$(1, 2)$$.
   After vertical dilation with factor $$-1$$: <InlineMath math="h'(x) = -1 \cdot (|x - 1| + 2) = -|x - 1| - 2" />, the vertex becomes $$(1, -2)$$.

   <LineEquation
     title={<>Function $$h(x) = |x - 1| + 2$$ and Its Dilation Result</>}
     description={<>The absolute value function undergoes reflection and dilation with factor $$-1$$.</>}
     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 - 1) + 2, z: 0 };
         }),
         color: getColor("INDIGO"),
         labels: [{ text: "h(x) = |x - 1| + 2", offset: [1, -1.5, 0] }],
         showPoints: false,
       },
       {
         points: Array.from({ length: 41 }, (_, i) => {
           const x = (i - 20) * 0.25;
           return { x, y: -(Math.abs(x - 1) + 2), z: 0 };
         }),
         color: getColor("EMERALD"),
         labels: [{ text: "h'(x) = -|x - 1| - 2", offset: [1, 1.5, 0] }],
         showPoints: false,
       },
     ]}
   />

Visible text: 1. Vertical dilation with factor : <InlineMath math="f'(x) = 3f(x) = 3(x^2 - 4x + 3) = 3x^2 - 12x + 9" />

 <LineEquation
 title={<>Function and Its Dilation Result</>}
 description={
 <>
 The original parabola is vertically stretched by factor{" "}
 producing a taller parabola.
 </>
 }
 showZAxis={false}
 cameraPosition={[12, 8, 12]}
 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: 3 * (x * x - 4 * x + 3), z: 0 };
 }),
 color: getColor("ORANGE"),
 labels: [{ text: "f'(x) = 3x² - 12x + 9", offset: [1, -3, 0] }],
 showPoints: false,
 },
 ]}
 />

2. Equation of the resulting dilated function:

 - Vertical dilation: <InlineMath math="g'(x) = \frac{1}{2}g(x) = \frac{1}{2}\sqrt{x}" />
 - Range after dilation: becomes but with smaller maximum values

 Visualization:

 <LineEquation
 title={<>Function and Its Dilation Result</>}
 description={
 <>
 The square root curve is vertically compressed by factor{" "}
 producing a lower curve.
 </>
 }
 showZAxis={false}
 cameraPosition={[8, 6, 8]}
 data={[
 {
 points: Array.from({ length: 21 }, (_, i) => {
 const x = i * 0.25;
 return { x, y: Math.sqrt(x), z: 0 };
 }),
 color: getColor("VIOLET"),
 labels: [{ text: "g(x) = √x", offset: [2, 1, 0] }],
 showPoints: false,
 },
 {
 points: Array.from({ length: 21 }, (_, i) => {
 const x = i * 0.25;
 return { x, y: 0.5 * Math.sqrt(x), z: 0 };
 }),
 color: getColor("TEAL"),
 labels: [{ text: "g'(x) = 0.5√x", offset: [2, 0.5, 0] }],
 showPoints: false,
 },
 ]}
 />

3. The original function has its vertex at .
 After vertical dilation with factor : <InlineMath math="h'(x) = -1 \cdot (|x - 1| + 2) = -|x - 1| - 2" />, the vertex becomes .

 <LineEquation
 title={<>Function and Its Dilation Result</>}
 description={<>The absolute value function undergoes reflection and dilation with factor .</>}
 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 - 1) + 2, z: 0 };
 }),
 color: getColor("INDIGO"),
 labels: [{ text: "h(x) = |x - 1| + 2", offset: [1, -1.5, 0] }],
 showPoints: false,
 },
 {
 points: Array.from({ length: 41 }, (_, i) => {
 const x = (i - 20) * 0.25;
 return { x, y: -(Math.abs(x - 1) + 2), z: 0 };
 }),
 color: getColor("EMERALD"),
 labels: [{ text: "h'(x) = -|x - 1| - 2", offset: [1, 1.5, 0] }],
 showPoints: false,
 },
 ]}
 />