How do you graph y=cos(x+pi/6)?

Oct 14, 2017

Graph $y = \cos \left(x\right)$ and shift everything to the left by $\frac{\pi}{6}$

Explanation:

We know that $\sin$ and $\cos$ has a period of $2 \pi$. That is to say that it repeats itself every $2 \pi$ units.
I would assume you know how to graph a $f \left(x\right) = \cos \left(x\right)$ functions, if not, it should look like this:

Now, you need to graph $f \left(x\right) = \cos \left(x + \frac{\pi}{6}\right)$.

Imagine you have a function $f \left(x\right)$ and another function $g \left(x\right) = f \left(x + 1\right)$.

What this means is that for any point $\left(x , y\right)$ on the graph $g \left(x\right)$, it will take $x + 1$ units for $f \left(x\right)$ to reach that same $y$ value.
That is what this $g \left(x\right) = f \left(x + 1\right)$ is saying.

This means that all points on $g \left(x\right)$ is occurring 1 unit earlier than $f \left(x\right)$ so we shift $f \left(x\right)$ to the left by 1 unit to obtain $g \left(x\right)$.

To generalize:
If $g \left(x\right) = f \left(x + n\right)$ we shift $f \left(x\right)$n$u n i t s \to t h e \ast \le f t \ast \to \ge t$g(x). If g(x)=f(x-n)$w e s h \mathmr{if} t$f(x) $n$ units to the right to get $g \left(x\right)$.

Now, we can apply it to this question:

We have $f \left(x\right) = \cos \left(x + \frac{\pi}{6}\right)$ which is basically saying we should shift $\cos \left(x\right)$ to the left by $\frac{\pi}{6}$ units.

The blue curve is your $y = \cos \left(x + \frac{\pi}{6}\right)$
The red curve is your $y = \cos \left(x\right)$