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URL: https://nakafa.com/en/subjects/mathematics/function-transformation/horizontal-translation
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/material/lesson/mathematics/function-transformation/horizontal-translation/en.mdx
Learn how horizontal translation shifts a function's graph left or right without altering its shape, enhancing your understanding of function transformations.
---
## Basic Concepts of Horizontal Translation
Horizontal translation is a geometric transformation that shifts the graph of a function left or right along the $$x$$-axis without changing the shape of the graph. Imagine sliding an object horizontally on a table, its shape remains the same, only its position changes.
Visible text: Horizontal translation is a geometric transformation that shifts the graph of a function left or right along the -axis without changing the shape of the graph. Imagine sliding an object horizontally on a table, its shape remains the same, only its position changes.
If we have a function $$f(x)$$, then horizontal translation produces a new function $$g(x) = f(x - h)$$ where $$h$$ is the translation constant.
Visible text: If we have a function , then horizontal translation produces a new function where is the translation constant.
### Rules of Horizontal Translation
For any function $$f(x)$$, horizontal translation is defined as:
Visible text: For any function , horizontal translation is defined as:
```math
g(x) = f(x - h)
```
Where:
- If $$h > 0$$, the graph shifts to the **right** by $$h \text{ units}$$
- If $$h < 0$$, the graph shifts to the **left** by $$|h| \text{ units}$$
- If $$h = 0$$, there is no translation (graph remains the same)
Visible text: - If , the graph shifts to the **right** by
- If , the graph shifts to the **left** by
- If , there is no translation (graph remains the same)
## Visualization of Horizontal Translation
Let's see how horizontal translation works on the quadratic function $$f(x) = x^2$$.
Visible text: Let's see how horizontal translation works on the quadratic function .
Component: LineEquation
Props:
- title: Horizontal Translation of Quadratic Function $$f(x) = x^2$$
Visible text: Horizontal Translation of Quadratic Function
- description: Notice how the graph shifts horizontally without changing the parabola shape.
- 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: [0.5, 1, 0] }],
showPoints: false,
},
{
points: Array.from({ length: 41 }, (_, i) => {
const x = (i - 20) * 0.25;
return { x, y: (x - 3) * (x - 3), z: 0 };
}),
color: getColor("ORANGE"),
labels: [{ text: "g(x) = (x - 3)²", offset: [0.5, 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("TEAL"),
labels: [{ text: "h(x) = (x + 2)²", offset: [0.5, 1, 0] }],
showPoints: false,
},
]
From the visualization above, we can observe:
- The original function $$f(x) = x^2$$ has its vertex at $$(0, 0)$$
- Function $$g(x) = (x - 3)^2$$ shifts right by $$3 \text{ units}$$ with vertex at $$(3, 0)$$
- Function $$h(x) = (x + 2)^2$$ shifts left by $$2 \text{ units}$$ with vertex at $$(-2, 0)$$
Visible text: - The original function has its vertex at
- Function shifts right by with vertex at
- Function shifts left by with vertex at
## Horizontal Translation on Linear Functions
Now let's apply the same concept to the linear function $$f(x) = 2x + 1$$.
Visible text: Now let's apply the same concept to the linear function .
Component: LineEquation
Props:
- title: Horizontal Translation of Linear Function $$f(x) = 2x + 1$$
Visible text: Horizontal Translation of Linear Function
- description: The line maintains the same slope, only its horizontal position changes.
- showZAxis: false
- cameraPosition: [10, 6, 10]
- data: [
{
points: Array.from({ length: 21 }, (_, i) => {
const x = (i - 10) * 0.5;
return { x, y: 2 * x + 1, z: 0 };
}),
color: getColor("VIOLET"),
labels: [{ text: "f(x) = 2x + 1", offset: [1, 0.5, 0] }],
showPoints: false,
},
{
points: Array.from({ length: 21 }, (_, i) => {
const x = (i - 10) * 0.5;
return { x, y: 2 * (x - 4) + 1, z: 0 };
}),
color: getColor("AMBER"),
labels: [{ text: "g(x) = 2(x - 4) + 1", 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) + 1, z: 0 };
}),
color: getColor("CYAN"),
labels: [{ text: "h(x) = 2(x + 3) + 1", offset: [1, 0.5, 0] }],
showPoints: false,
},
]
Notice that:
- All lines have the same slope of $$2$$
- Function $$g(x) = 2(x - 4) + 1$$ shifts right by $$4 \text{ units}$$
- Function $$h(x) = 2(x + 3) + 1$$ shifts left by $$3 \text{ units}$$
Visible text: - All lines have the same slope of
- Function shifts right by
- Function shifts left by
## Important Properties of Horizontal Translation
### Graph Shape Remains Unchanged
Horizontal translation preserves the original shape of the graph. The vertical distance between points on the graph remains the same, only the horizontal position changes.
### Effect on Coordinate Points
If point $$(a, b)$$ is on the graph of $$f(x)$$, then after horizontal translation by $$h$$, that point becomes $$(a + h, b)$$ on the graph of $$f(x - h)$$.
Visible text: If point is on the graph of , then after horizontal translation by , that point becomes on the graph of .
### Domain and Range
- **Domain**: Shifts by $$h \text{ units}$$
- **Range**: Does not change after horizontal translation
Visible text: - **Domain**: Shifts by
- **Range**: Does not change after horizontal translation
If the domain of the original function is $$[c, d]$$, then the domain after horizontal translation $$h$$ becomes $$[c + h, d + h]$$.
Visible text: If the domain of the original function is , then the domain after horizontal translation becomes .
## Application Examples
### Exponential Function Example
Let's look at horizontal translation on the exponential function $$f(x) = 2^x$$.
Visible text: Let's look at horizontal translation on the exponential function .
Component: LineEquation
Props:
- title: Horizontal Translation of Exponential Function $$f(x) = 2^x$$
Visible text: Horizontal Translation of Exponential Function
- description: The exponential curve maintains its characteristics after horizontal translation.
- showZAxis: false
- cameraPosition: [8, 5, 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 - 2), z: 0 };
}),
color: getColor("EMERALD"),
labels: [{ text: "g(x) = 2^(x-2)", 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 + 1), z: 0 };
}),
color: getColor("ROSE"),
labels: [{ text: "h(x) = 2^(x+1)", offset: [0.5, 1, 0] }],
showPoints: false,
},
]
For exponential functions:
- The horizontal asymptote remains at $$y = 0$$ for all functions
- The $$y$$-intercept changes due to horizontal shift
- Function $$g(x) = 2^{x-2}$$ shifts right by $$2 \text{ units}$$
- Function $$h(x) = 2^{x+1}$$ shifts left by $$1 \text{ unit}$$
Visible text: - The horizontal asymptote remains at for all functions
- The -intercept changes due to horizontal shift
- Function shifts right by
- Function shifts left by
## Difference from Vertical Translation
It's important to understand the difference between horizontal and vertical translation:
### Horizontal Translation
- Changes the function input: $$f(x - h)$$
- Affects the x position of each point
- Domain changes, range remains
Visible text: - Changes the function input:
- Affects the x position of each point
- Domain changes, range remains
### Vertical Translation
- Changes the function output: $$f(x) + k$$
- Affects the y position of each point
- Domain remains, range changes
Visible text: - Changes the function output:
- Affects the y position of each point
- Domain remains, range changes
## Exercises
1. Given the function $$f(x) = x^2 + 2x + 1$$. Determine the equation of the function resulting from horizontal translation to the right by $$3 \text{ units}$$.
2. If the graph of function $$g(x) = \sqrt{x}$$ is translated horizontally to the left by $$4 \text{ units}$$, determine:
- The equation of the resulting translated function
- The domain of the function after translation
3. Function $$h(x) = 3^x$$ undergoes horizontal translation such that point $$(0, 1)$$ becomes $$(2, 1)$$. Determine the translation constant value and the equation of the resulting translated function.
Visible text: 1. Given the function . Determine the equation of the function resulting from horizontal translation to the right by .
2. If the graph of function is translated horizontally to the left by , determine:
- The equation of the resulting translated function
- The domain of the function after translation
3. Function undergoes horizontal translation such that point becomes . Determine the translation constant value and the equation of the resulting translated function.
### Answer Key
1. Horizontal translation to the right by $$3 \text{ units}$$:
Function $$f(x) = x^2 + 2x + 1$$ and Its Translation Result}
description={
<>
Original quadratic function and the result of horizontal translation to
the right by $$3 \text{ units}$$.
}
showZAxis={false}
cameraPosition={[12, 8, 12]}
data={[
{
points: Array.from({ length: 41 }, (_, i) => {
const x = (i - 20) * 0.25;
return { x, y: x * x + 2 * x + 1, z: 0 };
}),
color: getColor("PURPLE"),
labels: [{ text: "f(x) = x² + 2x + 1", 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 + 4, z: 0 };
}),
color: getColor("ORANGE"),
labels: [{ text: "f'(x) = x² - 4x + 4", offset: [1, 1, 0] }],
showPoints: false,
},
]}
/>
2. Equation of the resulting translated function:
- Translation to the left by $$4 \text{ units}$$:
- Domain after translation: $$x + 4 \geq 0$$, so $$x \geq -4$$ or $$[-4, \infty)$$
Visualization:
Function $$g(x) = \sqrt{x}$$ and Its Translation Result}
description={
<>
Original square root function and the result of horizontal translation
to the left by $$4 \text{ units}$$.
}
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: [1, 0.5, 0] }],
showPoints: false,
},
{
points: Array.from({ length: 21 }, (_, i) => {
const x = i * 0.25 - 4;
if (x + 4 >= 0) {
return { x, y: Math.sqrt(x + 4), z: 0 };
}
return null;
}).filter(Boolean),
color: getColor("TEAL"),
labels: [{ text: "g'(x) = √(x + 4)", offset: [1, 0.5, 0] }],
showPoints: false,
},
]}
/>
3. Point $$(0, 1)$$ on $$h(x) = 3^x$$ becomes $$(2, 1)$$, meaning horizontal translation by $$h = 2 \text{ units}$$ to the right.
Equation of the translation result:
Function $$h(x) = 3^x$$ and Its Translation Result}
description={
<>
Original exponential function and the result of horizontal translation
to the right by $$2 \text{ units}$$.
}
showZAxis={false}
cameraPosition={[10, 6, 10]}
data={[
{
points: Array.from({ length: 31 }, (_, i) => {
const x = (i - 15) * 0.2;
return { x, y: Math.pow(3, x), z: 0 };
}),
color: getColor("INDIGO"),
labels: [{ text: "h(x) = 3^x", 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("EMERALD"),
labels: [{ text: "h'(x) = 3^(x-2)", offset: [0.5, 1, 0] }],
showPoints: false,
},
]}
/>
Visible text: 1. Horizontal translation to the right by :
Function and Its Translation Result}
description={
<>
Original quadratic function and the result of horizontal translation to
the right by .
}
showZAxis={false}
cameraPosition={[12, 8, 12]}
data={[
{
points: Array.from({ length: 41 }, (_, i) => {
const x = (i - 20) * 0.25;
return { x, y: x * x + 2 * x + 1, z: 0 };
}),
color: getColor("PURPLE"),
labels: [{ text: "f(x) = x² + 2x + 1", 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 + 4, z: 0 };
}),
color: getColor("ORANGE"),
labels: [{ text: "f'(x) = x² - 4x + 4", offset: [1, 1, 0] }],
showPoints: false,
},
]}
/>
2. Equation of the resulting translated function:
- Translation to the left by :
- Domain after translation: , so or
Visualization:
Function and Its Translation Result}
description={
<>
Original square root function and the result of horizontal translation
to the left by .
}
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: [1, 0.5, 0] }],
showPoints: false,
},
{
points: Array.from({ length: 21 }, (_, i) => {
const x = i * 0.25 - 4;
if (x + 4 >= 0) {
return { x, y: Math.sqrt(x + 4), z: 0 };
}
return null;
}).filter(Boolean),
color: getColor("TEAL"),
labels: [{ text: "g'(x) = √(x + 4)", offset: [1, 0.5, 0] }],
showPoints: false,
},
]}
/>
3. Point on becomes , meaning horizontal translation by to the right.
Equation of the translation result:
Function and Its Translation Result}
description={
<>
Original exponential function and the result of horizontal translation
to the right by .
}
showZAxis={false}
cameraPosition={[10, 6, 10]}
data={[
{
points: Array.from({ length: 31 }, (_, i) => {
const x = (i - 15) * 0.2;
return { x, y: Math.pow(3, x), z: 0 };
}),
color: getColor("INDIGO"),
labels: [{ text: "h(x) = 3^x", 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("EMERALD"),
labels: [{ text: "h'(x) = 3^(x-2)", offset: [0.5, 1, 0] }],
showPoints: false,
},
]}
/>