Question

How do you graph `y=-3tan(x-(pi/4))` over the interval `[-pi, 2pi]`?

Answer 1

A general tangent function has the form;

`y = Atan(Bx - C) + D`

`A` changes the amplitude of the graph by stretching it in the vertical direction. `B` changes the period of the graph by stretching it in the horizontal direction. `C` tells you how far to move the graph in the `x` direction, and `D` tells you how far to move the graph in the `y` direction.

First we should start with a basic tangent function. This is a graph of the function `y = tan(x)` on the interval `[-pi,2pi]`.

Here `A` and `B` are `1` and `C` and `D` are 0. Lets change the amplitude of our graph by including the `-3` from above. The negative will flip the image and the `3` will stretch it vertically, so a graph of `-3tan(x)` would look like the red below.

Now we can change the `C` value to get our final graph. The `pi/4` term will move the graph `pi/4` radians to the right, so our final graph will look like the red below.