The line, x, and the parabola, x², are simplest of the family of functions called polynomials.
They range in complexity from linear functions, such as y = a_{1}x^{1}, or y = 3x + 5 to the fourth degree polynomial
y = a_{n}x^{n} + a_{n -1}x^{n -1}a_{n}x^{n} + a_{n -1}x^{n -1} + a_{n - 2}x^{n - 2} + a_{n - 3}x^{n -3} + a_{n - 4}x^{n - 4}
-- or the fourth degree polynomial written in factored form like y = (x^{2} + 1)(x - 3)(x + 4), or to even higher degree polynomials.
They are compositions of other functions and many compositions, translations, dilations, and reflections exist. See Compostition of Functions. or the pages listed below which also demonstrate compositions.
Grab your calculator, perhaps use the coordinate planes on page Graph these Odd and Even Functions, and graph the following functions to see a pattern in these sets of polynomials.
The odd functions like the cube, x^{3}, the fifth power, x^{5}, pass through the origin and look like a curve x passing from the lower left to the upper right. The even functions like the fourth power, x^{4}, or the sixth power, x^{6}, pass through the origin and look like a flattened x² passing from the upper left, through the origin, to the upper right. | ||
1. y = x | 2. y = x² | |
3. y = x^{3} | 4. y = x^{4} | |
5. y = x^{5} | 6. y = x^{6} |
The best way to explore polynomial graphs is through experimentation. The spread sheet poly.xls permits one to do this by varying exponents, factors, and coefficients. Enjoy.
Rational Functions Rational functions are created when a constant or polynomial is divided by a polynomial. Experiment with ratl.xls -- which depicts rational functions graphs | ||
7. y = 1/x | 8. y = 1/x² |
This page is from Exploring Functions Throught the Use of Manipulatives (ISBN: 0-9623593-3-5).
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© 2005, Agnes Azzolino www.mathnstuff.com/math/spoken/here/2class/300/fx/library/polyf.htm |