How do you graph #y = Arctan(x/3) #?

1 Answer
Nov 26, 2015

Start with a graph of #y=arctan(x)#.
graph{y=arctan(x) [-10, 10, -5, 5]}
Then stretch it horizontally by a factor of #3#.
graph{y=arctan(x/3) [-10, 10, -5, 5]}

Explanation:

Consider a graph of a function #y=f(x)# as given.
Let's see how this graph is related to a graph of a function #y=f(x/k)#, where #k>1#.

Assume, point #(a,b)# belongs to a graph of function #y=f(x)#. It means that #b=f(a)#.
Then #b=f((ak)/k)#, which means that point #(ak,b)# belongs to a graph of function #y=f(x/k)#.

We see now that for every point #(a,b)# that belongs to a graph of function #y=f(x)#, graph of function #y=f(x/k)# contains a point #(ak,b)#.

Think now about a transformation of stretching a graph horizontally by a factor of #k>1#. It means that every point with coordinates #(a,b)# will be transformed into a point #(ak,b)# - exactly as happens with a graph of function #y=f(x/k)#, if compared with a graph of function #y=f(x)#.

Therefore, you can graph a function #y=f(x/k)# by starting from a graph of #y=f(x)# and stretching it horizontally by a factor of #k#.

I can recommend the Web-based course of advanced mathematics at Unizor, where a chapter linked to menu items Algebra - Graphs explains this in details.
You can also refer to chapters on Trigonometry with a relatively detailed description of all trigonometric functions and their graphs.