Summary: Graphs of Polynomial Functions

Key Concepts

  • Polynomial functions of degree 2 or more are smooth, continuous functions.
  • To find the zeros of a polynomial function, if it can be factored, factor the function and set each factor equal to zero.
  • Another way to find the x-intercepts of a polynomial function is to graph the function and identify the points where the graph crosses the x-axis.
  • The multiplicity of a zero determines how the graph behaves at the x-intercept.
  • The graph of a polynomial will cross the x-axis at a zero with odd multiplicity.
  • The graph of a polynomial will touch and bounce off the x-axis at a zero with even multiplicity.
  • The end behavior of a polynomial function depends on the leading term.
  • The graph of a polynomial function changes direction at its turning points.
  • A polynomial function of degree n has at most n – 1 turning points.
  • To graph polynomial functions, find the zeros and their multiplicities, determine the end behavior, and ensure that the final graph has at most n – 1 turning points.
  • Graphing a polynomial function helps to estimate local and global extremas.
  • The Intermediate Value Theorem tells us that if [latex]f\left(a\right) \text{and} f\left(b\right)[/latex] have opposite signs, then there exists at least one value c between a and b for which [latex]f\left(c\right)=0[/latex].


global maximum
highest turning point on a graph; [latex]f\left(a\right)[/latex] where [latex]f\left(a\right)\ge f\left(x\right)[/latex] for all x.
global minimum
lowest turning point on a graph; [latex]f\left(a\right)[/latex] where [latex]f\left(a\right)\le f\left(x\right)[/latex] for all x.
Intermediate Value Theorem
for two numbers a and b in the domain of f, if [latex]a<b[/latex] and [latex]f\left(a\right)\ne f\left(b\right)[/latex], then the function f takes on every value between [latex]f\left(a\right)[/latex] and [latex]f\left(b\right)[/latex]; specifically, when a polynomial function changes from a negative value to a positive value, the function must cross the x-axis
the number of times a given factor appears in the factored form of the equation of a polynomial; if a polynomial contains a factor of the form [latex]{\left(x-h\right)}^{p}[/latex], [latex]x=h[/latex] is a zero of multiplicity p.