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As many know, rational functions have weird shapes like hyperbolas placed diagonally. They are in couple parts, for example $\frac{1}{x}$. Why does that happen?

Trup Krup Drób Grób
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    Try to precise that "weird". Hyperbolas are beautiful. –  Apr 29 '16 at 21:45
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    If hyperbolas are "weird" shapes, wait till you get into polar graphing... – imranfat Apr 29 '16 at 22:01
  • In fact, the graph of $\frac{1}{x}$ is a hyperbola. – Travis Willse Apr 29 '16 at 22:25
  • What on earth is weird about the shape?!?!? they look exactly like I would expect!! They have to go from zero to infinity as x goes from 1 to 0 and as x gets incrementally smaller 1/x must get huge. Meanwhile the exact opposite occurs when x goes from 1 to infinity 1/x goes from 1 to 0 with huger and huger increments in x required to make small decreases in 1/x. The shape exactly reflect the nature of 1/x. You might as well ask why do circles look so round. – fleablood Apr 29 '16 at 22:27
  • Uh, it's weird you can't divide by zero? That's basically why $y=1/x$ comes in two pieces. – Oscar Lanzi Apr 30 '16 at 00:16
  • I believe that the OP means to ask why the graph of $1/x$ does not look something like $-x$, as it is the inverse of $x$. I remember making that mistake also. – zz20s Apr 30 '16 at 00:23
  • 1/(1/x) = x is why it is symetric. As x approaches 0 very small decrements in x correspond to huge increments in 1/x (and equivalently as x approaches infinity it takes larger and larger increments in x to correspond to small decrements in 1/x) so that accounts for the bowed hyperbolic shape. 1/0 is undefined so the accounts for the horizontal assymptote. And finally x >0 iff 1/x >0 so that accounts for the "jump" from pos and neg infinity on either side of zero. To be honest, mx+b and 1/x are the two least weird and most expected shapes I can think of. More so than,$x^2$ or $y^2+x^2=c$, imo. – fleablood Apr 30 '16 at 02:15

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I would rather consider the shape of this graph beautiful than weird. Rational funcions have shapes like this because of their nature where variable is in the denominator. When denominator approaches $0$ from both sides, the value rapidly increases or decreases into infinity. This is called an asymptote.

KKZiomek
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