A Quick Reminder

Here’s the \textcolor{blue}{\sin x} graph…

and the \textcolor{limegreen}{\cos x} graph…

… and also the \textcolor{red}{\tan x} graph.

So, the \textcolor{blue}{\sin x} and \textcolor{limegreen}{\cos x} graphs have a periodicity of 360°, while the \textcolor{red}{\tan x} has a periodicity of 180°.

To help you remember, we have these three rules:

• \textcolor{blue}{\sin x} = \sin (x + 360°) = \sin (x - 360°)
• \textcolor{limegreen}{\cos x} = \cos (x + 360°) = \cos (x - 360°)
• \textcolor{red}{\tan x} = \tan (x + 180°) = \tan (x - 180°)

Now, it’s time to take a look at some transformations.

Vertical Translation

We’ll start with something nice and simple – a vertical translation.

So, below we’ve got a graph of \textcolor{blue}{y = \sin x}. Nothing out of the ordinary there.

We also have two vertical translations, \textcolor{limegreen}{y = (\sin x) + 1} and \textcolor{red}{y = (\sin x) - 1}.

Horizontal Translation

We also have horizontal translations, where the transformation acts directly on x.

Again, we’ll start with \textcolor{blue}{y = \sin x}.

Now we’ve got a horizontal translation, \textcolor{limegreen}{y = \sin (x + 90°)}.

In short, a transformation y = \sin (x + c) is a translation to the left of c. So, if you have \sin (x - 45°), for example, it is a translation 45° to the right.

In other words, a transformation of y = \sin (x + c) is a translation of -c along the x-axis

Vertical Stretching

Now, we’ll introduce vertical stretching to our repertoire.

We have three transformations of \textcolor{blue}{y = \sin x} here:

• \textcolor{purple}{y = 2\sin x}
• \textcolor{limegreen}{y = \dfrac{1}{2}\sin x}
• \textcolor{red}{y = -\sin x}

Horizontal Stretching

We’ll take a look at horizontal stretching now, too.

We have three transformations of \textcolor{blue}{y = \sin x}:

• \textcolor{purple}{y = \sin (2x)}
• \textcolor{limegreen}{y = \sin \left( \dfrac{1}{2}x \right)}
• \textcolor{red}{y = \sin (-x)}

A Handy Table