What is a transformation? Well, it’s something that transforms one function into another! To see what I mean and how that looks, check out this lesson!

## Transformations

A lot of people are of the opinion that math doesn’t make any sense. Some of my students even seem to think that long ago, we mathematicians met in a secret chamber somewhere and came up with all these crazy rules for no good reason other than to drive people crazy.

While I can assure you that this is not the case, I do understand why people can feel this way. Teaching things like 50 = 1 does make it especially hard for me to defend my position that math isn’t there purely for us teachers to laugh at you when you mess up.But maybe this will help! It’s a topic where we picked a word that accurately describes what it means in math. **Transformations** kinda, like, transform! You don’t have to be an English scholar to know that this word implies one thing changing into another thing.

Kinda like how a Camaro can transform into a big evil fighting robot, a mathematical function can transform into…

a different mathematical function! Not as cool as the robot, I know, but at least it’s simple. If it was transforming into a robot, I bet the math would be way harder.

## Types of Transformations

Anyways, let’s take a look at a few of the different kinds of transformations that we can have in math. First off we’ve got **rotations**. These simply take a function, maybe this one, and rotate it! We can also have **reflections**. These are transformations that act like a mirror and reflect a function to a new place, maybe like this, or like this! The next kind of transformation is called a **dilation**. These stretch or shrink the graph, maybe from something like this to this!

Let’s summarize what we just learned. Translations are accomplished by adding or subtracting values from the function. Adding outside the *f*(*x*) shifts the graph up, which implies that subtracting outside the *f*(*x*) would shift it down. Subtracting inside the *f*(*x*) shifts the graph to the right, which implies that adding inside the *f*(*x*) would shift it left. This is the opposite of what you might think.

The way I remember it is by asking myself ‘if it says *f*(*x* + 2), how would I make it zero?’ The answer is, ‘by putting in a -2,’ thus shifting it to the left.What this means is that if we have a function ** f(x – h) + k**,

*h*tells us how many units to slide the function left or right, and

*k*tells us how many units to slide it up or down. Let’s look a few quick examples.How about this one: given that this graph is

*f*(

*x*), graph

*f*(

*x*+ 1) – 3.

Okay, well, the stuff inside the *f*(*x*) shifts the function left and right and does the opposite of what I’d expect it to. That means the +1 will shift everything in the graph one to the left. The -3 on the outside represents the up/down shift and follows the pattern you’d expect, which means it will pull the entire graph down three spots, which makes our new graph look like this!One last example: The function *g* here is a translation of the function *f*. Write the equation for *g* in terms of *f*. Yikes, this one has pretty crazy directions. Let’s not worry about that too much and focus on what we’ve learned. So, *g* looks basically the same as *f*, only it’s been shifted over two to the right and up four.

That means we have a -2 on the inside of the function and a +4 on the outside. Therefore, *g*(*x*) = *f*(*x* – 2) + 4

## Lesson Summary

Let’s review. **Transformations** of functions change one function to a slightly different one. **Rotations** spin the function, **reflections** flip the function across a line (kind of like a mirror), **dilations** stretch or shrink the function and **translations** slide the function around.

To perform a function translation, you must add or subtract values to either the inside or the outside of the *f*(*x*). Values on the inside will shift the graph in the *x* direction, left or right, while values on the outside will shift in the *y* direction, up or down. Shifting the graph up or down follows the pattern you might guess, with adding shifting it up and subtracting shifting it down, but shifting it left or right is the opposite – adding shifts it left, while subtracting shifts it right.

## Lesson Objectives

Once you’ve completed this lesson you’ll understand the different ways functions can be transformed and how to perform these transformations.