A021polyfuns


 * Family of Polynomial Functions **


 * The graph of y = x n where n is even **

Any polynomial function with an even degree will have a turning point and arms going off in the same direction.
 * If the coefficient of the largest power of x is positive, both arms will go up and there will be a minimum turning point.
 * If the coefficient of the largest power of x is negative, both arms will go down and there will be a maximum turning point.


 * Note that as n increases, the base becomes flatter (compared to the base of a parabola) and the arms become steeper.

For example
 * y = x 2 has a minimum turning point and both arms go up to positive infinity.
 * y = x 4 has a minimum turning point and both arms go up to positive infinity.
 * y = –x 4 has a maximum turning point and both arms go down to negative infinity.


 * The graph of y = x n where n is odd **

Any polynomial function with an odd degree will have arms going off in the opposite directions.
 * If the coefficient of the largest power of x is positive, both arms will go up and there will be a minimum turning point.
 * If the coefficient of the largest power of x is negative, both arms will go down and there will be a maximum turning point.
 * The directions will be the same as for a straight line (n = 1) with positive or negative gradient.

For example
 * y = x 3 the left arm goes to negative infinity and the right arm goes to positive infinity
 * y = x 5 the left arm goes to negative infinity and the right arm goes to positive infinity
 * y = –x 5 the left arms goes to positive infinity and the right arm goes to negative infinity


 * Factorised Polynomials **

If the polynomial can be factorised then:
 * Any linear factors will give the x-intercepts
 * a factor repeated twice (ie a squared factor) indicates a turning point on the x-axis at that x-value
 * a factor repeated three times (ie a cubed factor) indicates a stationary point of inflection on the x-axis at that x-value
 * a factor repeated four times (ie a factor raised to the power of 4) indicates a turning point on the x-axis at that x-value
 * etc