Given the graph of [tex]y=\cot x[/tex], horizontally stretching the graph by a factor of 3 (i.e. reducing the period of the graph or increasing the number of turns in a given interval by a factor of 3) gives the graph [tex]y=\cot3x[/tex].
Now, shifting the graph by [tex] \frac{\pi}{2} [/tex] units to the right gives the graph of [tex]y=\cot\left(3x- \frac{\pi}{2} \right)[/tex] and vertically stretching the graph by a factor of 6 (i.e. increasing the height of each point on the graph by a factor of 6 relative to the original position) gives the graph of [tex]y=6\cot\left(3x- \frac{\pi}{2} \right)[/tex]
Finally, shifting the graph by 4 units up gives the graph of [tex]y=6\cot\left(3x- \frac{\pi}{2} \right)+4[/tex]
