Question

consider the following.
$f(x)=\frac{8}{x}$
(a) sketch the graph of $f$.

Explanation:

Step1: Analyze function properties

The function $f(x)=\frac{8}{x}$ is a hyper - bolic function. The domain is $x
eq0$. When $x>0$, as $x$ increases, $y = \frac{8}{x}$ decreases. When $x<0$, as $x$ increases (moves towards 0 from the left), $y=\frac{8}{x}$ decreases (negative values with larger magnitude).

Step2: Identify asymptotes

The vertical asymptote is $x = 0$ since the function is undefined at $x = 0$. The horizontal asymptote is $y=0$ because as $x
ightarrow\pm\infty$, $y=\frac{8}{x}
ightarrow0$.

Step3: Plot key points

For $x = 1$, $y = 8$; for $x=-1$, $y=-8$.

Answer:

The graph of $y = \frac{8}{x}$ has two branches, one in the first - quadrant and one in the third - quadrant, approaching the $x$ - axis ($y = 0$) and the $y$ - axis ($x = 0$) as asymptotes.