Horizontal Translations

There are four types of transformations possible for a graph of a function. They are:

• Rotations
• Translations
• Reflections, and
• Dilations

Further, translation can be divided into two different types, i.e.,

• Horizontal translation
• Vertical translation

In this section, we will learn about horizontal translation in detail and practice solving questions around it. 

Try your hand at solving a few interesting interactive questions at the end of the page.

Lesson Plan

Horizontal Translation: Definition

Horizontal translation refers to the movement toward the left or right of the graph of a function by the given units.

The shape of the function remains the same.

It is also known as the movement/shifting of the graph along the x-axis.

For any base function \(f(x)\), the horizontal translation towards positive x-axis by value \(k\) can be given as:

\(g(x) = f(x-k)\)

In horizontal translation, each point on the graph moves \(k\) units horizontally and the graph is said to translate \(k\) units horizontally.

Horizontal translation for \(f(x) = f(x \pm k)\)

Horizontal Translation: Examples

The following is the graph of \(f\left( x \right) = \left| x \right|\).

The horizontal translation toward the right side by 1 unit in the above function graph can be given as:

\(g\left( x \right) = f\left(x \right) = \left| x - 1 \right|\) 

The graph of the given function after the desired horizontal translation is achieved can be plotted by shifting it toward right by 1 unit. 

Similarly, the graph of \(h\left( x \right)  = f\left( x + 1\right) = \left| x + 1 \right|\) can be obtained by shifting the graph of \(f\) by 1 unit to the left.

Here, we can observe the horizontal translation of the graph by 1 unit.

The following table shows the coordinates of the point on the different curves after translation:

The point on \(f(x)\)	The point on \(f(x + k)\)	The point on \(f(x - k)\)
\( (x, y) \)	\( (x - k, y)\)	\( (x + k, y)\)

Meanwhile, the shape of the function and domain of the function remains the same.

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How Is Horizontal Translation Done?

Let us understand this by an example.

Example

Consider a basic quadratic equation \(f(x) = x^2\).

The graphical representation of this equation is shown below.

Steps

• Select the constant by which we want to translate the function.
  Here, we have selected \( \color{orange}{+2}\).
• Write the new function as \( g(x) = f(x \pm k)\), where \(k\) is the constant.
  Now, our equation of the new function will be:

\[g(x) = f(x \color{orange}{+2}) = (x \color{orange}{+2})^2\]

• Trace the basic function graph again and shift each point in the graph by \(\pm k\), in the horizontal direction (left for +k and right for -k).
  Here, we have shifted the graph of \(x^2\) by \(\color{orange}{+2}\) units in the left direction.

• While tracing the graph for a general horizontal translation, we will simply shift the basic function by \(\pm k\) units:

The domain of the function remains the same in both cases.

Example 2

 

 

Janice has been asked to plot the curve of \(f(x) = e^{x+2}\).

Can you help her?

Solution

Plotting the curve of \(e^x\):

We know,  \(f(x) = e^{x+2}\) is 2 units horizontally translated curve of basic function \(e^x\).

Therefore, shifting the curve of \(e^x\) by 2 units, we get:

\(\therefore\) The curve of  \(f(x) = e^{x+2}\) is plotted.