Horizontal Translation

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.

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.

Rules of Horizontal Translation

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

Where:

• If $h > 0$, the graph shifts to the right by $h$ units
• If $h < 0$, the graph shifts to the left by $|h|$ units
• If $h = 0$, there is no translation (graph remains the same)

Visualization of Horizontal Translation

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

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 units with vertex at $(3, 0)$
• Function $h(x) = (x + 2)^2$ shifts left by 2 units with vertex at $(-2, 0)$

Horizontal Translation on Linear Functions

Now let's apply the same concept to the linear function $f(x) = 2x + 1$.

Notice that:

• All lines have the same slope of 2
• Function $g(x) = 2(x - 4) + 1$ shifts right by 4 units
• Function $h(x) = 2(x + 3) + 1$ shifts left by 3 units

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 $f(x - h)$.

Domain and Range

• Domain: Shifts by $h$ units
• 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 $[c + h, d + h]$.

Application Examples

Exponential Function Example

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

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 units
• Function $h(x) = 2^{x+1}$ shifts left by 1 unit

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

Vertical Translation

• Changes the function output: $f(x) + k$
• 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 units.

2. If the graph of function $g(x) = \sqrt{x}$ is translated horizontally to the left by 4 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.

Answer Key

1. Horizontal translation to the right by 3 units: $f'(x) = f(x - 3) = (x - 3)^2 + 2(x - 3) + 1 = x^2 - 4x + 4$

   Function $f(x) = x^2 + 2x + 1$ and Its Translation Result

   Original quadratic function and the result of horizontal translation to the right by 3 units.

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2. Equation of the resulting translated function:

   • Translation to the left by 4 units: $g'(x) = g(x + 4) = \sqrt{x + 4}$
   • Domain after translation: $x + 4 \geq 0$, so $x \geq -4$ or $[-4, \infty)$

   Visualization:

   Function $g(x) = \sqrt{x}$ and Its Translation Result

   Original square root function and the result of horizontal translation to the left by 4 units.

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3. Point $(0, 1)$ on $h(x) = 3^x$ becomes $(2, 1)$, meaning horizontal translation by $h = 2$ units to the right. Equation of the translation result: $h'(x) = 3^{x-2}$

   Function $h(x) = 3^x$ and Its Translation Result

   Original exponential function and the result of horizontal translation to the right by 2 units.

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