graph each function.\n$f(x)=\\frac{3}{x-2}+4$

Explanation:

Step1: Identify parent function

The parent function is $g(x)=\frac{3}{x}$, a hyperbola.

Step2: Find vertical asymptote

Set denominator $x-2=0$, so $x=2$.

Step3: Find horizontal asymptote

For $f(x)=\frac{3}{x-2}+4$, horizontal asymptote is $y=4$.

Step4: Check a test point

Use $x=3$: $f(3)=\frac{3}{3-2}+4=3+4=7$. So the point $(3,7)$ lies on the graph.

Step5: Match to options

The first graph has vertical asymptote $x=2$, horizontal asymptote $y=4$, and passes through $(3,7)$.

Answer:

The first (top-most) graph option: the hyperbola with vertical asymptote $x=2$, horizontal asymptote $y=4$, left branch below $y=4$, right branch above $y=4$.