Function graph transformations allow us to modify the graph of the original function. One of these transformations is vertical translation. When we have a function *f*, we can apply a vertical shift by adding or subtracting a constant to the function *f*.

In this article, we will learn everything related to the vertical translation of a function. We will use several examples to illustrate important concepts.

## Determining the vertical translation of a function

The vertical translation of a function is a transformation that causes the graph of the original function to be moved up or down. That is, the translation occurs parallel to the *y*-axis.

To understand the vertical translation of a function, we can consider the function $latex f(x)=x^2$ as an example. If we graph this function, we get the following curve:

If we now add and subtract 1 unit from the original function, we have the functions (i) $latex f(x)+1$ and (ii) $latex f(x)-1$. Simplifying, we have:

(i) $latex f(x)+1=x^2+1~$ and (ii) $latex f(x)-1=x^2-1$

Using the same Cartesian plane as the original function $latex f(x)$, we can graph functions (i) and (ii) to obtain the following:

We can see that, in case (i), the graph of *f* has been moved up 1 unit. That is, 1 unit parallel to the *y*-axis.

On the other hand, the graph of the function (ii) is equal to the graph of *f* moved down 1 unit. That is, -1 unit parallel to the *y*-axis.

In short, we have the following:

- The transformation $latex f(x)+a$ produces a shift in the original graph of $latex f(x)$ of $latex a$ units up.
- The transformation $latex f(x)-a$ produces a shift in the original graph of $latex f(x)$ of $latex a$ units down.

## Examples of vertical translation in functions

The vertical translation of functions is used to solve the following problems. Try to solve the problems yourself before looking at the solution.

**EXAMPLE 1**

We have the function $latex f(x)=x^3$. If we have the function $latex g(x)=x^3-3$, get the graphs of *g* and *f*.

##### Solution

Function *f* is the base cubic function. This function is plotted on the left side of the following diagram.

In the case of function *g*, we can see that this function is equivalent to the function *f* with a vertical translation of -3 units, as we can see on the right-hand side of the graph in the following diagram.

.

**EXAMPLE **2

**2**

**EXAMPLE**Graph the function $latex f(x)=\cos(x)+2$.

##### Solution

By comparing the given function to the standard cosine function $latex f(x)=\cos(x)$, we can deduce that a vertical translation of 2 units up was applied.

Therefore, we can graph the function $latex f(x)=\cos(x)+2$ by graphing a basic cosine function and moving it 2 units up:

**EXAMPLE **3

**3**

**EXAMPLE**What is the graph of $latex g(x)=|x|-2$?

##### Solution

Here, we have the absolute value function. In its base form, $latex f(x)=|x|$, the graph of the absolute value function is as follows:

Therefore, we can get the graph of $latex g(x)=|x|-2$ if we move the graph of the base absolute value function down 2 units:

.

**EXAMPLE **4

**4**

**EXAMPLE**What transformation do we need to apply to the function $latex f(x)=\tan(5x-2)$ if we want to shift it -5 units parallel to the *y*-axis?

##### Solution

A shift of -5 units parallel to the *y*-axis is the same as applying a shift of 5 units down on the original function.

To perform this shift, we apply the transformation $latex f(x)-5$. In this case, we have the function $latex f(x)=\tan(5x-2)$. Therefore, we have:

$latex f(x)-5=\tan(5x-2)-5$

## Vertical translation of functions- Practice problems

Apply what you have learned about the vertical translation of functions to solve the following practice problems.

## See also

Interested in learning more about function transformations? Take a look at these pages:

### Learn mathematics with our additional resources in different topics

**LEARN MORE**