Section 4.4 - Polynomials and Their Graphs
- Determine the end behavior of a polynomial function.
- Use multiplicities of zeros and end behavior to graph a polynomial function.
End behavior, zeros, and polynomial graphs
Consider the polynomial , where the coefficients , , , are real numbers with .
The general shape of the graph of can be easily determined from the polynomial's factored form.
The graph crosses the -axis at every zero of multiplicity 1.
The graph flattens and crosses the -axis at every zero of odd multiplicity.
The graph flattens, touches, and bounces off the -axis at every zero of even multiplicity.
The end behavior (the behavior as ) of the graph of is identical to that of , where is the degree and is the leading coefficient:
- even and positive up left and up right
- even and negative down left and down right
- odd and positive down left and up right
- odd and negative up left and down right
- Discuss the zeros and the features of the graph of .
- Imagine you were given the graph of a polynomial function, but you were not given the polynomial itself. Could you make good predictions about the zeros and their multiplicities?