Section 4.4 - Polynomials and Their Graphs Section Objectives
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 Example
Discuss the zeros and the features of the graph of . ( Solution ) 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? (Here are some examples from an old review packet: first and second .) For further help... Do a Google search for "graphing polynomials". Video on zeros of polynomials. Video on polynomial end behavior. Use Desmos or Geogebra to graph polynomials.