Polynomial Functions of Higher Degree

Graphs of Polynomial Functions

The graph of a polynomial function is continuous, which means that it has no breaks, holes, or gaps. Also, they have smooth rounded graphs, without any sharp turns.

D=0 D=1

D=2 D=3

D=4 D=5

Zeros of Polynomial Functions

For a polynomial function of degree n, the following are true:

• The graph of has at most n real zeros
• The function has at most n-1 relative extrema (relative minimums or maximums)

Minima and Maxima are the plural form of minimum and maximum.

End Behavior

End behavior is what happens at the left and the right of the graph. Instead of simply using words, notation is used.

For the following graph:

Translation:

As x approaches infinity, approaches infinity

As x approaches negative infinity, approaches negative infinity

For a polynomial function of degree n, if n is...

• Even, the end behaviors are the same
• Odd, the end behaviors are different
• Even and the leading coefficient is positive, the graph rises to the left and right
• Even and the leading coefficient is negative, the graph falls to the left and right
• Odd and the leading coefficient is positive, the graph falls to the left and rises to the right
• Odd and the leading coefficient is negative, the graph rises to the left and falls to the right

ex.

Both ends of the graph would rise or

because n is even, and the leading coefficient is positive.

The graph would rise to the left and fall to the right or

because n is odd and the leading coefficient is negative.