# How do I sketch the graph y= (2x+5)/(x-1)

How do I sketch the graph y= (2x+5)/(x-1)

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Say you want to simplify
$$2+\frac{3}{2x+1}$$
then you would multiply the first constant by the highest common denominator so you can put it all in one fraction as so:
$$\frac{2(2x+1)+3}{2x+1}$$
and then simplify and you get
$$\frac{4x+5}{2x+1}$$.
But what if you started with this expression and wanted to go back to the start now?

You need to cancel out the x in the numerator so you need to convert it into the form
$$\frac{a(2x+1)+b}{2x+1}$$
Notice that the numerator has 4x in it so a=2 so that we can have this 4x.
$$\frac{2(2x+1)+b}{2x+1}$$
Now in order to keep the numerator the same we need a number b so that it makes $2(2x+1)+b=4x+5$ and once you solve this you get b=3.
Then we get
$$\frac{2(2x+1)+3}{2x+1}=\frac{2(2x+1)}{2x+1}+\frac{3}{2x+1}=2+\frac{3}{2x+1}$$

From this expression we can easily sketch the graph
$$y=2+\frac{3}{2x+1}$$
by noting that there is a vertical asymptote at 2x+1=0, x=-1/2 and there is a horizontal asymptote at y=2 (since the fraction never equals 0).

Now see if you can apply the same idea to your problem.