Trig Graphs

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:

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

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