# 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**

**What Is a Vertical Translation?**

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

Here, 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.

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

Now, if we shift this exact graph 1 unit down, we will get the following graph:

This is the graph of the function \(g\left( x \right) = \left| x \right|- 1\)

This means that we can obtain the graph of *g(x)* by simply down-shifting the graph of *f* by 1 unit.

Similarly, the graph of \(h\left( x \right) = \left| x \right| + 1 = f\left( x\right) + 1\) can be obtained by up-shifting the graph of *f* by 1 unit:

Here, we have seen the vertical 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) + C | The point on f(x) - C |
---|---|---|

\( (x, y) \) |
\( (x, y + C)\) |
\( (x, y - C)\) |

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

**How to Do Vertical Translation?**

Let us understand this by an example:

**Example**

Suppose we have a basic quadratic equation \(f(x) = x^2\) and the graphical representation of the graph is shown below.

Now if we want to translate this graph vertically we have to follow the given steps:

**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:

- Vertical translation refers to the shifting of the curve along the y-axis by some specific units without changing the shape and domain of the function.
- Vertical translation of function f(x) is given by g(x) = f(x) \(\pm\) C.

**Solved Examples**

Example 1 |

Harry was given a task to plot the curve of the basic function \(f(x) = x^3\) that is translated up to 4 units. Can you help him with this?

**Solution**

We know that curve of \(f(x)=x^3\) is:

Since the curve is translated up, we can write the equation of the new curve as:

\[g(x) = f(x) + 4\]

Plotting the curve of \(g(x)\):

\(\therefore\)The new function is \(g(x) = f(x) + 4\). |

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. |

- What are the properties of horizontal translation?
- If the function f(x) has coordinates as (x,y), can you tell what will be coordinates of the point on the curve after C units of vertical translation?

**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**

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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**Frequently Asked Questions (FAQs)**

## 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:

Horizontal translation:

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

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