The horizontal and vertical translation is a transformation that allows us to modify the graph of the original function. Given the function *f*(*x*), we can translate the function horizontally with the transformation *f*(*x*+*a*) and vertically with the transformation *f*+*a* where *a* is a value that can be positive or negative.

Here, we will learn everything related to the horizontal and vertical translation of a function. We will look at some examples to illustrate the concepts.

##### ALGEBRA

**Relevant for**…

Learning about the horizontal and vertical translation of functions.

##### ALGEBRA

**Relevant for**…

Learning about the horizontal and vertical translation of functions.

## Determining the horizontal translation of a function

The horizontal translation in a function is a transformation that produces a shift to the left or to the right of the original function. That is, the translation occurs parallel to the *x*-axis.

We can understand the horizontal translation of a function by taking the function $latex f(x)=2x-1$ as an example. When we graph this function, we get the following line:

Now, we are going to apply the transformations (i) $latex f(x+2)$ and (ii) $latex f(x-2)$. Therefore, using the original function $latex f(x)=2x-1$ and simplifying the transformations, we have:

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

(i) $latex f(x+2)=2x+3~$ and (ii) $latex f(x-2)=2x-5$

We can then graph functions (i) and (ii) using the same coordinate plane as the original function to compare their graphs. Therefore, we have:

In case (i), the transformation $latex f(x+2)$ produced a translation of 2 units to the left. That is, -2 units parallel to the *x*-axis.

In case (ii), the transformation $latex f(x-2)$ produced a translation of 2 units to the right. That is, 2 units parallel to the *x* axis.

In short, we have:

- The transformation $latex f(x+a)$ results in a shift in the original graph of
*f*of $latex a$ units to the left. - The transformation $latex f(x-a)$ results in a shift in the original graph of
*f*of $latex a$ units to the right.

## 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 horizontal and vertical translation in functions

**EXAMPLE 1**

Sketch the graph of $latex f(x)=x^2-1$. Then, find the equation of the transformation $latex g(x)=f(x+2)$ and graph it.

##### Solution

Starting with the graph of $latex f(x)$, we have:

Now, we can find the equation of the function $latex g(x)$ by applying the transformation on the original function and simplifying:

$latex g(x)=f(x+2)$

$latex =(x+2)^2-1$

$latex =x^2+4x+4-1$

$latex =x^2+4x+3$

We can graph the function $latex g(x)$ by considering that the graph of *g* can be obtained by translating the graph of *f* by 2 units to the left, that is, -2 units on the *x*-axis.

.

**EXAMPLE **2

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 **3

**EXAMPLE**

Graph the cosine function in its base form. Then graph two cosine functions that are shifted 1 unit and 2 units to the right from the base shape.

##### Solution

The base cosine function, $latex f(x)=cos(x)$, has a value of 1 when *x* equals 0. Also, it passes through the point (*π*/2, 0) and has a period of *π*.

To apply a horizontal translation of 1 unit and 2 units to the right, we have to apply the transformations $latex g(x)=f(x-1)$ and $latex h(x)=f(x-2)$ respectively.

When we graph the three functions, we have:

**EXAMPLE **4

**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 **5

**EXAMPLE**

Obtain the graph of $latex g(x)=|x-2|$.

##### Solution

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

Therefore, the graph of $latex g(x)=|x-2|$ can be obtained by shifting the graph of the absolute value function 2 units to the right when compared to the base form:

.

**EXAMPLE **6

**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 **7

**EXAMPLE**

What transformation do we need to apply to shift the function $latex f(x)=\tan(5x-2)$, -4 units parallel to the *x*-axis?

##### Solution

A shift of -4 units parallel to the *x*-axis is equivalent to applying a horizontal translation of 4 units to the left.

We can accomplish this translation by applying the $latex f(x-4)$ transformation. In this case, we have the function $latex f(x)=\tan(5x-2)$. Therefore, we have:

$latex f(x-4)=\tan(5(x-4)-2)$

$latex f(x-4)=\tan(5x-20-2)$

$latex f(x-4)=\tan(5x-22)$

**EXAMPLE **8

**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:

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

## Horizontal and vertical translation of functions – Practice problems

#### Which function has a translation of 6 units up with respect to the function $latex f(x)=-2x-3$?

Write the answer in the input box.

## See also

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

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