Function and Relation Library

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Function and Relation Library

Polynomial Functions

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 = a₁x¹, or y = 3x + 5 to the fourth degree polynomial

y = a_nxⁿ + a_{n -1}x^{n -1}a_nxⁿ + 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 = (x² + 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.

Notes on Polynomial Functions

Dilations Create Functions -- Dilation by A Constant
Dilation Creates Polynomial & Rational Functions
Notes on Polynomial Functions
Graphing Polynomial Functions
Graphing Rational Functions
Curve Shifting because of Addition & Subtraction

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.
Odd Functions Even Functions
The odd functions like the cube, x³, the fifth power, x⁵, 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, x⁴, or the sixth power, x⁶, 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 = x³ 4. y = x⁴

5. y = x⁵ 6. y = x⁶

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

Odd Functions	Even 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

Visit the Interactive Sketch Pad Material
On Polynomial & Rational Functions

parabola -- Multi page: 2 parabolas: 1 in general form, 1 in standard forms,
* both generated by equations defined by parameters.
* WITH SHOW/HIDE ACTION BUTTON
* graphs of 2 parabolas: 1 in general form, 1 in standard forms,
* 1 parabola, set a, then generat parabola w/2 points.
* 1 parabola controlled by 3 points, used for determinant analysis
* 1 vertical axis of symmetry, 1 horizontal "axis of symmetry
TracePoly -- drag point f(x) to state and show new ( x, f(x))
recipfx -- graph of parameter-defined reciprocal function,
* with slide-able tangent line,
* VERTICAL ASYMPTOTE MUST BE POSITIONED BY USER!

This page is from Exploring Functions Throught the Use of Manipulatives (ISBN: 0-9623593-3-5).

You are hereby granted permission to make ONE printed copy of this page and its picture(s) for your PERSONAL and not-for-profit use. YOU MAY NOT MAKE ANY ADDITIONAL COPIES OF THIS PAGE, ITS PICTURE(S), ITS SOUND CLIP(S), OR ITS ANIMATED GIFS WITHOUT PERMISSION.

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Odd Functions	Even Functions	The odd functions like the cube, x³, the fifth power, x⁵, 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, x⁴, or the sixth power, x⁶, 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 = x³	4. y = x⁴
5. y = x⁵	6. y = x⁶