02.16polyfuns

Family of Polynomial Functions

The graph of y = xn 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: as n increases, the base becomes flatter (compared to the base of a parabola) and the arms become steeper.

For example

• y = x2 has a minimum turning point and both arms go up to positive infinity.
• y = x4 has a minimum turning point and both arms go up to positive infinity.
• y = –x4 has a maximum turning point and both arms go down to negative infinity.

The graph of y = xn 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, the right arm will go up and the left arm will go down.
• If the coefficient of the largest power of x is negative, the right arm will go down and the left arm will go up.
• The directions will be the same as for a straight line y = ax (ie y = ax1, n = 1) with positive or negative gradient.

For example

• y = x3 the left arm goes to negative infinity and the right arm goes to positive infinity
• y = x5 the left arm goes to negative infinity and the right arm goes to positive infinity
• y = –x5 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 linear factor repeated twice (ie a squared factor) indicates a turning point on the x-axis at that x-value
• a linear factor repeated three times (ie a cubed factor) indicates a stationary point of inflection on the x-axis at that x-value
• a linear 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

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