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

Nov 26, 2015

Start with a graph of $y = \arctan \left(x\right)$.
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 $y = f \left(x\right)$ as given.
Let's see how this graph is related to a graph of a function $y = f \left(\frac{x}{k}\right)$, where $k > 1$.

Assume, point $\left(a , b\right)$ belongs to a graph of function $y = f \left(x\right)$. It means that $b = f \left(a\right)$.
Then $b = f \left(\frac{a k}{k}\right)$, which means that point $\left(a k , b\right)$ belongs to a graph of function $y = f \left(\frac{x}{k}\right)$.

We see now that for every point $\left(a , b\right)$ that belongs to a graph of function $y = f \left(x\right)$, graph of function $y = f \left(\frac{x}{k}\right)$ contains a point $\left(a k , b\right)$.

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

Therefore, you can graph a function $y = f \left(\frac{x}{k}\right)$ by starting from a graph of $y = f \left(x\right)$ 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.