A-level Maths - How would you start to draw this reciprocal graph...??

Ok, so I'm having a silly moment and am having trouble drawing this graph...

$y=\frac{2x+1}{x-2}$

I know that the bottom part of the fraction translates the graph to the right by 2 as it "tends to infinity" on the x-axis at x=2.
And that the bottom part of the graph crosses the axis at $(-\frac{1}{2},0)$ and $(0,-\frac{1}{2})$.

But I don't know how to draw the rest of the graph properly.
Where on the y-axis does the reciprocal graph "tend to infinity"?

Divide top by bottom to get it in the form y=2 + A/(x-2)

Can you explain the process of dividing the top by the bottom? I don't think I'm following

I ended up with $y=-2+\frac{1}{x-2}$

I think I did something wrong?

Basic long division, you should be able to do since you did chapter 1!