Vertical Translation

There are four types of transformation possible for a graph of a function, which are:

• Rotations
• Translations
• Reflections, and
• Scaling

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

• Horizontal translation
• Vertical translation

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

You can check out the interactive simulations to know more about the lesson and try your hand at solving a few interesting interactive questions at the end of the page.

Lesson Plan

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Steps

Step 1: Select the constant by which we want to translate the function.

Here we have selected \( \color{orange}{+2}\).

Step 2: Write the new function as \( g(x) = f(x) \pm C\), where C is the constant.

Here, our equation of the new function will be

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

Step 3: Trace the basic function graph again and shift each point in the graph by \(\pm C\), in the vertical direction( up for +C and down for -C).

Here, we have shifted the graph of \(x^2\) by \(\color{orange}{+2}\) units in an upward direction.

Step 4: Trace the new function, this will be the vertically translated basic function by \(\pm C\) units.

Here, we have got \(\color{orange}{+2} \) units vertically translated graph of \(x^2\) or of the function \(g(x) = x^2 +2\).

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

Try your hand at the following calculator to know more about vertical translation:

Example 2

 

 

Sara 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 unit vertically down 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.

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Let's Summarize

We hope you enjoyed learning about vertical translation with the simulations, interactive questions along vertical translation examples. Now you will be able to easily solve problems on the vertical translation.

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1. What is vertical translation?

Vertical translation definition refers to the up or down movement of the graph of a function.

In this, the shape of the function remains the same.

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

In vertical translation, each point on the graph moves k units vertically and the graph is said to translated k units vertically.

2. How do you know if a translation is vertical?

Vertical translation of function f(x) is given by g(x) = f(x) \(\pm\) C. 

So, any function with a constant term added to the base function represents a vertical translation function.

3. What are vertical shifts?

The movement of the graph of a function along the y-axis is known as a vertical shift.

4. Why are vertical translations opposite?

It depends on the sign of constant term (\(\pm\) C)

5. How do you translate a function vertically?

In vertical translation, each point on the graph moves k units vertically and the graph is said to translated k units vertically.

6. What is the rule of translation?

Rule of translation is to shift graph according to change in function.

7. What is the formula for translation?

The formula for translation or vertical translation equation is g(x) = f(x+k) + C. 

8. 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 translated k units vertically.

Horizontal translation: 

In horizontal translation, each point on the graph moves k units horizontally and the graph is said to translated k units horizontally.