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Horizontal Translation

Horizontal Translation

Horizontal translation slides a function graph left or right along the $x$ -axis without changing its shape. Every point moves by the same horizontal distance, as if the graph were an object sliding across a table.

Horizontal TranslationShift function graphs left or right, derive the sign rule, and track how the domain and range change.

On this page

Horizontal Translation
- Rules of Horizontal Translation
Visualization of Horizontal Translation
Horizontal Translation on Linear Functions
Properties of Horizontal Translation
Moving an Exponential Graph Horizontally
Difference from Vertical Translation
- Horizontal Translation
- Vertical Translation
Exercises
- Answer Key

Comments

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:

g(x) = f(x - h)

The direction of the shift follows these rules:

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)

Visualization of Horizontal Translation

For the quadratic function $f(x) = x^2$ , the new vertex shows how the sign inside the input controls the direction of the shift.

Horizontal Translation of Quadratic Function

f(x) = x^2

Notice how the graph shifts horizontally without changing the parabola shape.

The graph shows:

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 $(-2, 0)$

Horizontal Translation on Linear Functions

The same translation rule applies to the linear function $f(x) = 2x + 1$ .

Horizontal Translation of Linear Function

f(x) = 2x + 1

The line maintains the same slope, only its horizontal position changes.

The three lines show:

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 $3 \text{ units}$

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 $h \text{ units}$
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 $[c + h, d + h]$ .

Moving an Exponential Graph Horizontally

The exponential function $f(x) = 2^x$ retains its shape and asymptote under a horizontal translation.

Horizontal Translation of Exponential Function

f(x) = 2^x

The exponential curve maintains its characteristics after horizontal translation.

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 $1 \text{ unit}$

Difference from Vertical Translation

Horizontal and vertical translations act on different parts of the function rule:

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

Given the function $f(x) = x^2 + 2x + 1$ . Determine the equation of the function resulting from horizontal translation to the right by $3 \text{ units}$ .
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
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

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

Function $f(x) = x^2 + 2x + 1$ and Its Translation Result
Original quadratic function and the result of horizontal translation to the right by $3 \text{ units}$ .
Equation of the resulting translated function:
- Translation to the left by $4 \text{ units}$ : $g'(x) = g(x + 4) = \sqrt{x + 4}$
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:

Horizontal Reflection

Rotation

Published: May 26, 2025. Updated: Sep 1, 2026.

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 \text{ units}

.

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 \text{ units}

.