There are four types of transformations possible for a graph of a function. Theyare:
- 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
| 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 |
What Is a Horizontal Translation?
Horizontal Translation: Definition
Horizontaltranslation refers to themovement toward the leftor rightof 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 horizontaltranslation, 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 towardthe 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 givenfunction after thedesired horizontal translation is achieved can be plotted by shifting it towardright by 1 unit.
Similarly, the graph of \(h\left( x \right) = f\left( x + 1\right)= \left| x + 1\right|\)can be obtained byshifting the graph of\(f\) by 1 unit to the left.
Here, we can observethe 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.
How IsHorizontal Translation Done?
Let us understand this by an example.
Example
Consider a basic quadratic equation \(f(x) = x^2\).
The graphical representation of this equationis shown below.
Now if we want to translate this graph horizontally, we have to follow the given steps:
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 (leftfor +kand rightfor -k).
Here, we haveshiftedthe graph of \(x^2\) by \(\color{orange}{+2}\) units in theleftdirection.
- While tracing the graph for a general horizontal translation,we will simply shift thebasic function by \(\pm k\) units:
The domain of the function remains the same in both cases.
Important Notes
- Horizontal translation refers to the shifting of the curve along the x-axis by some specific units without changing the shape and domain of the function.
- Horizontal translation of function f(x) is given byg(x) = f(x \(\pm\) k).
Solved Examples
Example 1
Jonas was given a task to plot the curve of the basic function \(f(x) = x^3\) that is translated horizontallyby-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:
\[g(x) = f(x-4)^3\]
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 \(f(x) = e^{x+2}\).
Can you help her?
Solution
Plotting the curve of \(e^x\):
We know, \(f(x) = e^{x+2}\) is 2 units horizontallytranslated 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.
Challenging Questions
- What are the properties of horizontal translation?
- If the function f(x) has coordinates as (x,y), can you find the coordinates of the point on the curve after \(k\) units of horizontaltranslation?
Interactive Questions
Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.
Let's Summarize
The mini-lesson targetedthe fascinating concept of horizontal translation. The math journey around horizontal translation startedwith what a student already knewand wenton 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.
About Cuemath
AtCuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students!
Through an interactive and engaging learning-teaching-learning approach, the teachers explore all angles of a topic.
Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.
Frequently Asked Questions(FAQs) on Horizontal Translation
1. What is horizontal translation?
Horizontaltranslation refers to themovement of the graph of a function tothe leftor rightby 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.
2. How do you know if a translation is horizontal?
If the base of the original function experiences a shift along the x-axis, the translation spotted is a horizontal translation.
3. What are horizontal shifts?
The movement of the graph of a function along the x-axis is known as a horizontal shift.
4. Why are horizontal translations opposite?
While translating a graph horizontally, it might occur that the procedure is opposite or counter-intuitive. That means:
- For negative horizontal translation, we shift the graph towards the positivex-axis.
- For positive horizontal translation, we shift the graph towards the negative x-axis.
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 towardthe opposite direction of the desired translation.
5. How do you translate a function horizontally?
Horizontaltranslation refers to the movement of the graph of a function to theleftor rightby a certain number ofunits.
The shape of the function remains the same.
It is also known as the movement/shifting of the graph along the x-axis.
6. What is the formula for translation?
The formula for translation or the translation equation is \(g(x) = f(x \pm k) + C\).
7. What is the difference between horizontal and vertical translation?
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 horizontaltranslation, each point on the graph moves \(k\) units horizontally and the graph is said to be translated \(k\) units horizontally.
