Horizontal Translation

Sliding a Graph Left or Right

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.

If you have a function $f(x)$ , then horizontal translation produces a new function $g(x) = f(x - h)$ where $h$ is the translation constant.

Reading the Sign Rule

Look at where $h$ sits in the rule below. It is written inside the parentheses together with $x$ , so it changes the input before the function runs.

g(x) = f(x - h)

Because $h$ sits inside the parentheses, its sign works the opposite way from the direction you read. Subtracting a positive number from $x$ slides the graph right, while adding a positive number slides it left.

Value of $h$	Shift of the graph	Example of the vertex
$h > 0$	Right by $h \text{ units}$	$(0,0)$ becomes $(h,0)$
$h < 0$	Left by $\lvert h \rvert \text{ units}$	$(0,0)$ becomes $(h,0)$
$h = 0$	The graph does not move	$(0,0)$ stays where it is

All three rows use the vertex $(0,0)$ as a reference so the direction is easy to compare. For a function with a different reference point, every point moves by the same distance in the same direction.

Checking the Rule on a Quadratic Graph

Picture three parabolas drawn on the same plane. They have exactly the same shape, but their vertices sit at different horizontal positions.

Horizontal Translation of Quadratic Function

f(x) = x^2

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

In the graph above, the purple curve is the original function $f(x) = x^2$ with its vertex at $(0,0)$ . The orange curve is $g(x) = (x-3)^2$ . Because $x$ is reduced by $3$ , the graph shifts right by $3 \text{ units}$ and its vertex moves to $(3,0)$ . The green curve is $h(x) = (x+2)^2$ . Here $x$ is increased by $2$ , so the graph shifts left by $2 \text{ units}$ and its vertex moves to $(-2,0)$ .

Notice that all three parabolas keep the same width and the same shape. Only the horizontal position changes, so you can read the direction of the shift straight from the sign inside the parentheses.

Checking the Rule on a Linear Graph

The same rule applies to the linear function $f(x) = 2x + 1$ . On a straight line, a horizontal shift does not change the slope, so checking the sign is easier than on a parabola.

Horizontal Translation of Linear Function

f(x) = 2x + 1

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

The purple line is the original line with slope $2$ . The orange line is $g(x) = 2(x-4)+1$ , which shifts right by $4 \text{ units}$ , and the blue line is $h(x) = 2(x+3)+1$ , which shifts left by $3 \text{ units}$ . All three lines stay parallel because the slope does not change.

In both examples the direction of the shift always comes from the sign inside the parentheses. That is the opposite of a vertical translation, which adds its constant outside the function.

Which Values the Horizontal Shift Moves

Once you know the direction of the shift, you still need to know which parts of the function move with it. A horizontal translation acts on the input, so everything that depends on the position of $x$ moves, while the output values stay the same.

Take a point $(a, b)$ on the graph of $f(x)$ . After a translation by $h$ , that point moves to $(a + h, b)$ on the graph of $f(x - h)$ . The $y$ value does not change, only the $x$ coordinate moves.

Part of the function	Before the translation	After the translation by $h$
Graph shape	The original shape	The same shape, moved horizontally
Point $(a, b)$	On the graph of $f(x)$	Moves to $(a + h, b)$
Domain $[c, d]$	The original interval	Moves to $[c + h, d + h]$
Range	The original output values	The same values as the original function

The domain moves with the graph because it describes the allowed values of $x$ , while the range stays the same because the height of the graph at each point is unchanged. A vertical translation swaps those roles, keeping the domain and moving the range.

Checking the Rule on an Exponential Graph

Exponential functions behave differently from lines and parabolas. Their shape and asymptote stay the same, but the point where the curve crosses the $y$ -axis moves with the graph.

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.

The horizontal asymptote stays at $y = 0$ , because a horizontal shift does not change how the curve behaves when $x$ grows very large or very small. The function $g(x) = 2^{x-2}$ shifts right by $2 \text{ units}$ , while $h(x) = 2^{x+1}$ shifts left by $1 \text{ unit}$ .

The $y$ -intercept moves as well, because that point sits at $x = 0$ and no longer keeps its original height. This is where a horizontal shift differs from a vertical one, which moves the output values without changing the horizontal position of the intercept.

Telling Horizontal and Vertical Shifts Apart

Horizontal and vertical translations both move a graph, but they act on different parts of the rule. That difference decides which coordinate changes.

Property	Horizontal translation	Vertical translation
Rule	$g(x) = f(x - h)$	$g(x) = f(x) + k$
Part that changes	The function input	The function output
Direction of the shift	Along the $x$ coordinate	Along the $y$ coordinate
Sign of the constant	Opposite to the direction you read	The same as the sign of $k$
Domain and range	Domain moves, range stays	Domain stays, range moves

The main difference is where the constant sits. A constant inside the parentheses changes the input, so the shift is horizontal and its sign works in reverse. A constant outside the parentheses changes the output, so the shift is vertical and its sign works as written.

Changes that stretch or compress a graph are covered separately in horizontal dilation and vertical dilation, while the order of steps for combining several transformations is covered in combined function transformations.

Exercises

The three problems below use the sign rule you have just learned. Try them without the answer key first, then compare your steps with the worked answers.

Given the function $f(x) = x^2 + 2x + 1$ . Determine the equation of the function resulting from horizontal translation to the right by $3 \text{ units}$ .
The graph of the function $g(x) = \sqrt{x}$ is translated horizontally to the left by $4 \text{ units}$ . Determine the equation of the resulting translated function, then determine 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

Every worked answer starts by replacing $x$ correctly, because that is where the sign rule lives. The equation is then simplified and the new graph is checked to confirm the direction of the shift.

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 $4 \text{ units}$ : $g'(x) = g(x + 4) = \sqrt{x + 4}$
- 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 $4 \text{ units}$ .
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: $h'(x) = 3^{x-2}$

Function $h(x) = 3^x$ and Its Translation Result
Original exponential function and the result of horizontal translation to the right by $2 \text{ units}$ .