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May 3, 2015

Posted by **mai** on Monday, October 10, 2011 at 7:59pm.

- Math -
**Steve**, Monday, October 10, 2011 at 11:56pmWhenever 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.

Things like

(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.

For example,

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.

- Math -
**Steve**, Monday, October 10, 2011 at 11:57pmAs for transformations, do some google searches for translation and scaling, and there will be all kinds of good articles and pictures.