Describing Transformations of Rational Functions

Using the form $g(x)=a\left(\frac{1}{x-h}\right)+k,$ we want to write a rational function that matches the given graph.

We can see that the vertical asymptote is the line $x=3.$ Therefore, we have that $h=3.$ Similarly, we see that the horizontal asymptote is the line $y=4.$ With this, we know that $k=4.$ We can write a partial equation of the function. $$\begin{array}{c}g(x)=a\left(\frac{1}{x-3}\right)+4\end{array}$$ To find the value of $a,$ we can use any of the points on the curve. For simplicity, we will use the point $(2,1).$ Therefore, we will substitute $2$ for $x$ and $1$ for $g(x),$ and solve for $a.$ Finally, we can write the desired equation. $$\begin{array}{c}g(x)=3\left(\frac{1}{x-3}\right)+4\end{array}$$