Would someone please explain how to sketch this graph

This is the graph:
http://img543.imageshack.us/img543/9978/…
I don't get how they did it. I know how to find asymptotes because the denominator cannot equal 0, but I don't understand how they went from a normal quadratic function to a reciprocal. How did they know what the graph was going to look like without using a calculator to sketch it?

Hi,

So, you know how to find the asymptotes. Then, you know they are:

x = 0 and x = 4, y = 0.

Since the function has been reciprocated, reciprocate the vertex point y-value.
That is, -4 becomes -1/4.

Now, try to approach the vertical asymptotes from the point (2,-1/4).

Decreasing x, you approach x = 0, so try something like x = 0.1. You should see that this is negative (y = -100/39), so you know the graph tends to -∞ on that side.

Now, increasing x, you approach x = 4, so try something like x = 3.9. You should see that this is also negative (y = -100/39), so you know the graph tends to -∞ on that side too.

[ If you inferred that by symmetry, good job. Else, there's no shame in checking. ]

You now have the arc shape in the middle. What happens on either side of the vertical asymptotes?
Well, you know these must tend to the x-axis, from your asymptote.

Try something like x = -0.1 and x = 4.1. The sign of these (y = 100/41) will tell you that the graph begins at +∞ and descends towards the x-axis on both sides.

[ Again, after finding one, the other follows from symmetry of a quadratic, but there's no harm in checking. ]

This is exactly what has been sketched in your link.