02.16polyfuns

# 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:** 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, 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 = ax
^{1}, 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 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

.