describe how to sketch the graph of y = -tan(2x) + 3 using the parent function.

Explanation:

Step1: Identify parent - function

The parent function of $y =-\tan(2x)+3$ is $y = \tan(x)$. The graph of $y=\tan(x)$ has vertical asymptotes at $x=\frac{\pi}{2}+n\pi$, $n\in\mathbb{Z}$, and passes through the origin $(0,0)$ with a period of $\pi$.

Step2: Consider horizontal compression

For the function $y = \tan(2x)$, the period is changed. The general formula for the period of $y = \tan(bx)$ is $\frac{\pi}{|b|}$. Here $b = 2$, so the period is $\frac{\pi}{2}$. The vertical asymptotes of $y=\tan(2x)$ are at $2x=\frac{\pi}{2}+n\pi$, or $x=\frac{\pi}{4}+\frac{n\pi}{2}$, $n\in\mathbb{Z}$.

Step3: Account for reflection

The negative sign in front of $\tan(2x)$ reflects the graph of $y = \tan(2x)$ about the $x$-axis.

Step4: Consider vertical shift

The $+ 3$ in $y=-\tan(2x)+3$ shifts the graph of $y =-\tan(2x)$ upward by 3 units.

Answer:

First, start with the graph of the parent - function $y = \tan(x)$ which has a period of $\pi$ and vertical asymptotes at $x=\frac{\pi}{2}+n\pi$, $n\in\mathbb{Z}$. Compress the graph horizontally by a factor of 2 to get $y = \tan(2x)$ with a period of $\frac{\pi}{2}$ and vertical asymptotes at $x=\frac{\pi}{4}+\frac{n\pi}{2}$, $n\in\mathbb{Z}$. Then, reflect the graph of $y = \tan(2x)$ about the $x$-axis to obtain $y=-\tan(2x)$. Finally, shift the graph of $y =-\tan(2x)$ upward by 3 units to get the graph of $y=-\tan(2x)+3$.