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# Transformations of Functions – Explanation & Examples

When graphing functions, youâ€™ll be asked to **transform and translate functions** in various ways. Ever wondered how graphs can suddenly be transformed into a different one so that it represents a different function? Itâ€™s all thanks to the various forms of transformations we can perform on a functionâ€™s graph.

*Functions transformations are the different ways we can change the form of a functionâ€™s graph so that it becomes a different function.*

Amazing, right? If you want to save time when graphing different functions, youâ€™ve reached the right article! Weâ€™ll learn about **transformations done on functions** and **focus on translations**.

Before we begin, though, since weâ€™re working on graph transformations, weâ€™d recommend reviewing your resources on parent functions. Check out our article on parent functions, too, if you want to take a refresher.

## What are transformations of functions?

Weâ€™ve learned about parent functions and how a family of functions shares a similar shape. We can extend this knowledge by learning about the transformations of functions.

Transformations of functions are the processes that can be performed on an existing graph of a function to return a modified graph. We normally refer to the parent functions to describe the transformations done on a graph.

As can be seen from the example, transformations on a function can come in different forms and affect the graphs in different ways. This article and the next four ones will focus on the different transformations we can perform on a given function.

Below is a list of the common transformations performed on a graph:

- Horizontal and vertical transformations (or translations)
- Horizontal and vertical stretches
- Horizontal and vertical compressions
- Reflections and rotations

Our article will focus on the horizontal and vertical transformations that we can apply to a function.

## How to do transformations of functions?Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â

We can perform transformations based on the rule that we are provided for the transformation. Some transformations will require us to flip the graph over the y-axis or reflect it about the origin.

For now, weâ€™ll focus on two transformations: vertical and horizontal.

### Horizontal transformation or translation on a function

When we transform or translate a graph horizontally, we either **shift the graph to certain units to the right or to the left**. This will be a rigid transformation, meaning the shape of the graph remains the same.

Letâ€™s try translating the parent function y = x^{3} three units to the right and three units to the left.

When we translate a graph three units to the right, we subtract 3 from the input variable, x. Similarly, we add 3 to x when we translate three units to the left.

The table below generalizes horizontal transformations for all types of function f(x).

Translate f(x) | f(x + h) , when h > 0 |

Translate f(x) | f(x – h) , when h > 0 |

### Vertical transformation or translation on a function

Now, what happens if we translate three units upward or downwards instead? We call this **vertical transformation**. This type of transformation also retains the shape of the graph but **shifts it either upward or downward**.

Letâ€™s what happens if we shift y = x^{2} two units upward and downward.

When we translate a graph two units downward, we subtract 2 from the output value, y. Similarly, we add 2 to y when we translate it two units to the upward.

The table below generalizes vertical transformations for all types of function f(x).

Translate f(x) | f(x) + k, when k > 0 |

Translate f(x) | f(x) â€“ k, when k > 0 |

Weâ€™ve now learned the general rules for horizontal and vertical transformations, so how do we apply these when we graph functions?

**How to graph transformations?**

When working with functions resulting from multiple transformations, we always **go back to the functionâ€™s parent function.** Below are some important pointers to remember when graphing transformations:

- Identify the transformations performed on the parent function.
- Graph the parent function as a guide (this is optional).
- Perform each transformation on the graph until we complete all the identified transformations.

Why donâ€™t we start graphing f(x) = (x + 1)^{2} â€“ 3 by first identifying its transformations?

Since the graph is a quadratic function, we start with the parent function y = x^{2}.

The first terms, (x + 1)^{2}, show that the function y = x^{2} is **translated 1 unit to the left**.

The last term, -3, indicates that the resulting function is translated **3 units downward**.

This means that final graph for the function **f(x) = (x + 1) ^{2} â€“ 3 is as shown by the red graph**. As you can see, just by shifting the graphs vertically and horizontally, we can already modify them to represent a different function.

Weâ€™ll now try different questions that involve horizontal and vertical translations in the examples shown below.

*Example 1*

What happens when f(x) = x^{3} is translated 4 units to the right and 2 units downward?

__Solution__

Refer to the two tables that summarize the vertical and horizontal transformations as shown from the previous sections.

If we want to translate the cubic function, f(x) = x^{3}, 4 units to the right,** we add 4 to the input value, x. Hence, we have (x + 4) ^{3}.**

Since we still need to translate 2 units downward, letâ€™s subtract two units from the resulting function. The resulting **function now becomes ****(x + 4) ^{3} â€“ 2**

**.**

*Example 2*

The table of values for f(x) and g(x) are as shown below.

x | -4 | -2 | 0 | 2 | 4 |

f(x) | 16 | 4 | 0 | 4 | 16 |

x | -4 | -2 | 0 | 2 | 4 |

g(x) | 19 | 7 | 3 | 7 | 19 |

Use the information above and select which of the following best describes g(x) in terms of f(x).

- The function g(x) is the result when f(x) is translated 3 units upward.
- The function g(x) is the result when f(x) is translated 3 units downward.
- The function g(x) is the result when f(x) is translated 3 units to the right.
- The function g(x) is the result when f(x) is translated 3 units to the left.

__Solution__

Observe how for each output value, g(x) is always 3 units greater than f(x). This means that g(x) = f(x) + 3. Remember that for f(x) + k, we translate k units upward.

We have k = 3, so we can have g(x) when we **translate f(x) 3 units upward**.

*Example 3*

The graphs of y = âˆšx, g(x), and h(x) are shown below.

Describe the transformations done on each function and find their algebraic expressions as well.

__Solution__

Find the horizontal and vertical transformations done on the two functions using their shared parent function, y = âˆšx.

From the graph, we can see that g(x) is equivalent to **y = âˆšx but translated 3 units to the right and 2 units upward**. From this, we can construct the expression for h(x):

- The function y = âˆšx is translated 3 units to the left, so we have h(x) = âˆš(x + 3).
- Since we also need to translate the resulting function 2 units upward, we have
**h(x) = âˆš(x+3) + 2**.

We can apply the same process for g(x). The g(x) graph is equivalent to **y = âˆšx being translated one unit to the right and 3 units downward**. From this, we can find the expression for g(x):

- The function y = âˆšx is translated 1 unit to the left, so we have g(x) = âˆš(x + 1).
- Since we also need to translate the resulting function 3 units downward, we have
**g(x) = âˆš(x+1) – 3**.

*Example 4*

The function g(x) can be attained by translating y = 3^{x} by 3 units to the left and 2 units upward. Find the expression for g(x) and graph the resulting function.

__Solution__

When we translate y = 3^{x} by three units to the left, we subtract 3 from the input value or x. Hence, we have y = 3^{(x â€“ 3)}. We still need to translate it 2 units upward, hence, we have **g(x) = 3 ^{(x â€“ 3) }+ 2**.

We know how the parent function y = 3^{x} looks like. Letâ€™s use its graph and translate the graph vertically and horizontally.

Hence, we have the final graph shown below.

*Example 5*

Describe the translations applied on y = x^{3 }to attain the function h(x) = (x â€“ 1)^{3} â€“ 1. Use the transformations to graph h(x) as well.

__Solution__

Letâ€™s break down h(x) first: h(x) = (x **â€“ 1**)^{3} **â€“ 1**. Hence, we need to translate x^{3} **one unit to the right and one unit downward**. Letâ€™s go ahead and graph x^{3} first. We then apply the transformations.

Letâ€™s go ahead and remove the parent function to show h(x) by itself.

### Practice Questions

*Images/mathematical drawings are created with GeoGebra.*