Whenever you have a rational function, vertical asymptotes are possible. A rational function is a fraction where one polynomial is divided by another. If the denominator is zero and the numerator is not zero, then you have a vertical asymptote.
Consider a simple function
y = 1/x
when x = 0, no value of y is defined. There is a vertical asymptote at x=0.
There are lots of free online graphing web sites. Find one, and play around with rational functions.
(x^2-5x + 2)/(x-4)
and so on.
Now, when you have a rational function, there is always the possibility of a horizontal asymptote. If you have a function like
(3x^2 - 9x - 2)/(x^3 + x + 1)
Then as x gets huge, x^3 grows much faster than x^2 or x.
x=10 x^2=100 x^3=1000
x=100 x^2 = 10000 x^3 = 1000000
So, for large values of x, the above function looks just like
3x^2/x^3 = 3/x
As x gets huge, the quotient gets small, so the horizontal asymptote is y=0.
If you play around with the graphing tools, you'll see both of these kinds of asymptotes appearing.
As for transformations, do some google searches for translation and scaling, and there will be all kinds of good articles and pictures.