A rational function is a ratio of polynomials:
What can go wrong? We've got a denominator: perhaps it's zero, which would be bad....
This function is interesting: it is odd (possesses symmetry), it is its own inverse, and its graph is made up of special functions called hyperbolas:
Each one of their graphs can be obtained from that of 1/x by transformations. Let's see how -- by long-division.
Here's two of the most important things about rational functions:
The end-behavior asymptote is the polynomial that the function approaches as .
This might be a horizontal asymptote (which means that , a constant function):
This might be a slant asymptote (which means that , a linear function):
Or this may be more complicated (e.g. a quadratic asymptote, where , a quadratic function):
It turns out that every rational function can be expressed as the sum of a polynomial and rational function where the numerator degree is less than the denominator degree:
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The inner behaviour (near zero) | The end behavior (as | The transition zone, where both terms are clearly visible |
controls the end behavior, and controls the near behavior. |
The way we find the end-behavior asymptote is via long-division (I hear you groan!). Let's try one....
The graphing strategy for rational functions is the same as for polynomials, only we include the asymptotes.