Question

Which transformation should be applied to the graph of the function y=cot(x) to obtain the graph of the function y=6 cot(3x-pi/2)+4

Answer 1

Given the graph of $y=\cot x$, 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 $y=\cot3x$.

Now, shifting the graph by $\frac{\pi}{2}$ units to the right gives the graph of $y=\cot\left(3x- \frac{\pi}{2} \right)$ 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 $y=6\cot\left(3x- \frac{\pi}{2} \right)$

Finally, shifting the graph by 4 units up gives the graph of $y=6\cot\left(3x- \frac{\pi}{2} \right)+4$