suppose you want to transform the graph of the function $y = \\tan(x+\frac{\\pi}{4})-1$ into the graph of…

Explanation:

Step1: Analyze sign - change

The coefficient of $\tan$ changes from $1$ to $- 1$ in $y = \tan(x+\frac{\pi}{4})-1$ and $y=-\tan(x + \frac{\pi}{2})+1$. Reflecting the graph of $y = f(x)$ across the $x -$axis gives $y=-f(x)$. So, the first step is to reflect $y = \tan(x+\frac{\pi}{4})-1$ across the $x -$axis to get $y=-\tan(x+\frac{\pi}{4}) + 1$.

Step2: Analyze horizontal - shift

We want to change the argument from $x+\frac{\pi}{4}$ to $x+\frac{\pi}{2}$. Let $y =-\tan(x+\frac{\pi}{4}) + 1$ and we want to get to $y=-\tan(x+\frac{\pi}{2})+1$. If we replace $x$ with $x-\frac{\pi}{4}$ in $y =-\tan(x+\frac{\pi}{4}) + 1$, we have $y=-\tan((x-\frac{\pi}{4})+\frac{\pi}{4})+1=-\tan(x)+1$. To get $y =-\tan(x+\frac{\pi}{2})+1$, we need to shift the graph of $y =-\tan(x+\frac{\pi}{4}) + 1$ $\frac{\pi}{4}$ units to the left.

Step3: Analyze vertical - shift

The constant term changes from $1$ to $1$. The original function after the first two steps is $y=-\tan(x+\frac{\pi}{2})+1$. To get from $y =-\tan(x+\frac{\pi}{2})+1$ (after first two steps) to $y=-\tan(x+\frac{\pi}{2})+1$ (target function), we need to shift the graph $2$ units up.

Answer:

Reflect the graph of the first function across the x - axis, translate it $\frac{\pi}{4}$ units to the left, and translate it 2 units up.