chris wanted to transform the graph of the parent function (y = cot(x)) by horizontally compressing it so…

Answer:

He made a mistake in the horizontal - translation part. The correct function for the given transformations should be $y = \cot\left(2\left(x-\frac{\pi}{4}\right)\right)+1=\cot\left(2x - \frac{\pi}{2}\right)+1$, not $y=\cot\left(2x-\frac{\pi}{4}\right)+1$.

Explanation:

Step1: Recall the period - change formula

The period of the cotangent function $y = A\cot(Bx - C)+D$ is $T=\frac{\pi}{|B|}$. Given the desired period $T = \frac{\pi}{2}$, we have $\frac{\pi}{|B|}=\frac{\pi}{2}$, so $B = 2$ for horizontal compression.

Step2: Recall the horizontal - translation formula

For a horizontal translation of $h$ units to the right of the function $y=\cot(Bx)$, the function becomes $y=\cot(B(x - h))$. Here, $h=\frac{\pi}{4}$ and $B = 2$, so the function after horizontal compression and translation should be $y=\cot\left(2\left(x-\frac{\pi}{4}\right)\right)$.

Step3: Recall the vertical - translation formula

For a vertical translation of $k$ units up of the function $y=\cot\left(2\left(x-\frac{\pi}{4}\right)\right)$, the final function is $y=\cot\left(2\left(x-\frac{\pi}{4}\right)\right)+1=\cot\left(2x-\frac{\pi}{2}\right)+1$. Chris did not apply the horizontal - translation correctly as he did not factor out the $B = 2$ when applying the horizontal - shift.