Vertical Translation of a Function with Examples

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.

ALGEBRA
Graph of a quadratic function with vertical transformation

Relevant for

Learning about the vertical translation of functions.

See transformations

ALGEBRA
Graph of a quadratic function with vertical transformation

Relevant for

Learning about the vertical translation of functions.

See transformations

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:

Graph of quadratic function

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:

Graph of a quadratic function with vertical transformation

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.

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 1 Vertical transformation of graph

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EXAMPLE 2

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

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:

graph of cosine with different vertical translation

EXAMPLE 3

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

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:

graph of absolute value 1

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:

graph of absolute value with vertical displacement down

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EXAMPLE 4

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?

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.

If we have the function $latex f(x)=2x^4$, what is the translation of the function $latex g(x)=2x^4-5$ with respect to f?

Choose an answer






We have the function $latex f(x)=x+2$. Which of the following functions has a translation of -3 units parallel to y with respect to the function f?

Choose an answer






Which of the following functions is shifted 6 units up when compared to the function $latex f(x)=x^2-2x-3$?

Choose an answer







See also

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

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