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Graphs of Rational Functions

Graphs of functions with x in the denominator of a fraction

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Rolling in Dough!

Saving for a rainy day is something that most people try to do when they start earning money. You never know when some extra dough might come in handy, and there are lots of strategies for saving. One of the most aggressive strategies is to attempt to double your investments as frequently as possible.

Amazing But True

Let's say you started saving today, and you took a penny from your pocket and put it in the bank. Say tomorrow you deposit two more pennies, giving you a total of three cents in the bank. In two days you'd put four pennies in the bank, bringing your savings to seven cents, and so on... How much money would you have saved in a week? A month? A year?

Would you believe that your savings would be $7.28 in one week and $10,737,418.24 in a month, and that you'd be the richest person on Earth after one year, with more than one trillion dollars in the bank?! So why don't we all just start saving this way? Then everyone would be rich!

The problem is the asymptote. While it's nice to think that you could keep doubling your investments with each day, it's not a realistic savings strategy. Why not? Because after Day 10 (you'd put $10.24 in the bank on Day 10), it starts to become more difficult for most people to continue the pattern of deposits: $20.48 on Day 11, $40.96 on Day 12, $81.92 on Day 13, $163.84 on Day 14, $327.68 on Day 15, and so on... As you can imagine, if you work an hourly job making minimum wage ($7.25/hour), your take-home pay is roughly $250 per week. So, even if you wanted to save $327.68 in a single day, you wouldn't make enough money to do so. An asymptote is similar: it's the physical limit of a function, in much the same way as how the amount of money you earn is a physical limit on the amount of money you can save. Since it's impossible to save more than you earn, your earnings act as an asymptote on your savings.

See for yourself: http://en.wikipedia.org/wiki/Asymptote

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Describe the asymptote for the function \begin{align*}f(x)=\frac{8}{x^3}\end{align*}f(x)=8x3.

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