There are four types of transformations possible for a graph of a function. They are:
Further, translation can be divided into two different types, i.e.,
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.
1. | What Is a Horizontal Translation? |
2. | Important Notes on Horizontal Translation |
3. | Solved Examples on Horizontal Translation |
4. | Challenging Questions on Horizontal Translation |
5. | Interactive Questions on Horizontal Translation |
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:
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 \(f(x) = f(x \pm k)\) |
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.
Let us understand this by an example.
Consider a basic quadratic equation \(f(x) = x^2\).
The graphical representation of this equation is shown below.
Now if we want to translate this graph horizontally, we have to follow the given steps:
The domain of the function remains the same in both cases.
Important NotesExample 1 |
Jonas was given a task to plot the curve of the basic function \(f(x) = x^3\) that is translated horizontally by -4 units.
Can you help him with this?
Solution
We know that curve of \(f(x)=x^3\) is:
Since the curve is translated horizontally by -4 units, we can write the equation of the new curve as:
Plotting the curve of \(g(x)\):
The new function is \(g(x) = f(x-4)^3\). |
Example 2 |
Janice has been asked to plot the curve of \(f(x) = e^\).
Can you help her?
Solution
Plotting the curve of \(e^x\):
We know, \(f(x) = e^\) 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 \(f(x) = e^\) is plotted. |
Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.
The mini-lesson targeted the fascinating concept of horizontal translation. The math journey around horizontal translation started with what a student already knew and went on to creatively crafting a fresh concept in the young minds. Done in a way that not only it is relatable and easy to grasp, but also will stay with them forever.
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Horizontal translation refers to the movement of the graph of a function to the left or right by a certain number of units.
The shape of the function remains the same.
It is also known as the movement/shifting of the graph along the x-axis.
If the base of the original function experiences a shift along the x-axis, the translation spotted is a horizontal translation.
The movement of the graph of a function along the x-axis is known as a horizontal shift.
While translating a graph horizontally, it might occur that the procedure is opposite or counter-intuitive. That means:
This concept can be understood by analyzing the fact that the horizontal shift in the graph is done to restore the graph's base back to the same origin.
Hence, it is shifted toward the opposite direction of the desired translation.
Horizontal translation refers to the movement of the graph of a function to the left or right by a certain number of units.
The shape of the function remains the same.
It is also known as the movement/shifting of the graph along the x-axis.
The formula for translation or the translation equation is \(g(x) = f(x \pm k) + C\).
Vertical translation:
In vertical translation, each point on the graph moves \(k\) units vertically and the graph is said to be translated \(k\) units vertically.
Horizontal translation:
In horizontal translation, each point on the graph moves \(k\) units horizontally and the graph is said to be translated \(k\) units horizontally.