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

1 Answer
Oct 14, 2017

Answer:

Graph #y=cos(x)# and shift everything to the left by #pi/6#

Desmos

Explanation:

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

Desmos

Now, you need to graph #f(x)=cos(x+pi/6)#.

Imagine you have a function #f(x)# and another function #g(x)=f(x+1)#.

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

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

To generalize:
If #g(x)=f(x+n)# we shift #f(x) #n# units to the **left** to get #g(x)#. If #g(x)=f(x-n)# we shift #f(x) #n# units to the right to get #g(x)#.

Now, we can apply it to this question:

We have #f(x)=cos(x+pi/6)# which is basically saying we should shift #cos(x)# to the left by #pi/6# units.

Desmos

The blue curve is your #y=cos(x+pi/6)#
The red curve is your #y=cos(x)#