## Transformations of Functions

Published: Jul 20th, 2020

Learning Objectives
By the end of this section, you should be able to:
1. Classify the different types of function transformations
2. Apply transformations to a function

Consider the function in red. Let's call it $$f(x)$$ (how original, right?) How would we determine the exact shape of this function after a translation six units upwards? How about a reflection along the x axis? Now, you might say that we could write a table of values and calculate a few points by hand. While this is one way to determine the shape, having some knowledge of the transformations of functions can save you a lot of time, and can make graphing more challenging functions a cinch.

In this section, we will discuss three types of transformations and apply them to $$f(x)$$.

1. Translations: A translation is arguably among the simplest of all transformations. There are two types of translations possible: vertical and horizontal. We denote vertical translations with $$f(x) + k$$, where $$\left\{k\epsilon\mathbb{R}\right\}$$. If $$k > 0$$, the function is shifted $$k$$ units upwards, and if $$k < 0$$, the function is shifted $$k$$ units downwards. If $$k = 0$$, there is, of course, no translation at all. Horizontal translations are a bit more interesting. We denote horizontal translations as $$f(x+k)$$, where $$\left\{k\epsilon\mathbb{R}\right\}$$. If $$k >0$$, the function is shifted to the LEFT by $$k$$ units, and if $$k < 0$$, it is shifted to the RIGHT by $$k$$ units. Observe below the graph of $$f(x)$$, overlayed with a few vertical and horizontal translations:

2. Dilatations: Nope, that's not a typo. Dilatations. These are stretches and contractions to functions, and may occur in the horizontal or vertical direction. In the vertical direction, we denote dilatations of some function $$f(x)$$ by $$k \times f(x)$$, where $$\left\{k\epsilon\mathbb{R}\right\}$$. If $$k > 1$$, the graph is vertically stretched by k units, and if $$0 < k < 1$$, the graph is squeezed by k units. That's not too bad, right? The most important skill with dilatations is understanding what they question is asking. For example, if you were asked to show the result of a function being stretched by a factor of 2, you would draw $$2f(x)$$. If you were asked for the function contracted by a factor of two, then you would draw $$\frac{1}{2}f(x)$$. In the horizontal direction, we denote dilatations by $$f(kx)$$, where $$\left\{k\epsilon\mathbb{R}\right\}$$. If $$k > 1$$, the graph is horizontally squeezed, but if $$0 < k < 1$$, it is horizontally stretched. The same terminology that applies in the vertical direction applies in the horizontal. Here are a few examples (yes, there should be notation for translations as well, but for the purpose of clarity, these have been omitted).

3. Reflections: In his famous song, Mirrors, Justin Timberlake says "It's like you're my mirror (Oh), My mirror staring back at me (Oh)". Little did he know, Justin Timberlake was telling us about functions. More specifically, he was commenting on the reflections of functions. Reflections of functions can either occur along the x or y axis, as illustrated in the animation below. We denote vertical reflections by $$\pm f(x)$$. If $$+f(x) = f(x)$$, the graph is in its normal orientation. If $$-f(x)$$, the graph is reflected along the x axis. In the horizontal direction, this is denoted by $$f(\pm x)$$. If $$f(+x) = f(x)$$, the graph is in its normal orientation, and if $$f(-x)$$, the graph is reflected along the y axis.

You didn't think that was it, right? Well, if you did, you'd be wrong. It's DEMO TIME! Please enjoy the interactive demo below as a supplement to your learning!

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