consider the following function: $y = \\frac{1}{(x + 5)}+2$. how does the graph of this function compare…

Explanation:

Step1: Recall function - shift rules

For a function $y = f(x - h)+k$, $h$ represents horizontal shift and $k$ represents vertical shift. If $h>0$, the graph shifts right by $h$ units; if $h < 0$, the graph shifts left by $|h|$ units. If $k>0$, the graph shifts up by $k$ units; if $k < 0$, the graph shifts down by $|k|$ units. The parent function is $y=\frac{1}{x}$, and the given function is $y=\frac{1}{x + 5}+2$. We can rewrite the given function in the form $y=f(x - h)+k$ as $y=\frac{1}{x-(- 5)}+2$.

Step2: Determine horizontal and vertical shifts

Here, $h=-5$ and $k = 2$. Since $h=-5<0$, the graph of the function shifts left by $| - 5|=5$ units. Since $k = 2>0$, the graph of the function shifts up by 2 units.

Answer:

It is shifted left 5 units and up 2 units from the parent function.