Question

How do you graph `y= tan (2x)`?

Answer 1

Here's the graph (mousewheel to zoom):
graph{tan(2x) [-5, 5, -2.5, 2.5]}

Explanation:

The graph is just like tan(x), but 2 times faster. It has period `pi/2`. The roots are at `npi/2` for all integers `n` and graph has slope 2 at these points. The asymptotes are at `(n+1/2)pi/2`. [More info here.](http://www.wolframalpha.com/input/?i=tan%282x%29)

Generally for any fancy function `f(x)` we can think of its internal clock (as if it is function of time)

For real number `a>1`

• graph of `f(ax)` is squeezed horizontally (clock is faster)
• graph of `f(x/a)` is stretched horizontally (clock is slower)
• graph of `af(x)` is stretched vertically
• graph of `f(x)/a` is squeezed vertically

And for positive real number `b`

• graph of `f(x+b)` is shifted left (clock is ahead of time)
• graph of `f(x-b)` is shifted right (clock is delayed)
• graph of `f(x)+b` is shifted up
• graph of `f(x)-b` is shifted down

Please ask if any clarifications are needed.