Summary: Characteristics of Power and Polynomial Functions

Key Equations

general form of a polynomial function [latex]f\left(x\right)={a}_{n}{x}^{n}+\dots+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[/latex]

Key Concepts

  • A power function is a variable base raised to a number power.
  • The behavior of a graph as the input decreases beyond bound and increases beyond bound is called the end behavior.
  • The end behavior depends on whether the power is even or odd.
  • A polynomial function is the sum of terms, each of which consists of a transformed power function with positive whole number power.
  • The degree of a polynomial function is the highest power of the variable that occurs in a polynomial. The term containing the highest power of the variable is called the leading term. The coefficient of the leading term is called the leading coefficient.
  • The end behavior of a polynomial function is the same as the end behavior of the power function represented by the leading term of the function.
  • A polynomial of degree n will have at most n x-intercepts and at most n – 1 turning points.

Glossary

coefficient
a nonzero real number multiplied by a variable raised to an exponent
continuous function
a function whose graph can be drawn without lifting the pen from the paper because there are no breaks in the graph
degree
the highest power of the variable that occurs in a polynomial
end behavior
the behavior of the graph of a function as the input decreases without bound and increases without bound
leading coefficient
the coefficient of the leading term
leading term
the term containing the highest power of the variable
polynomial function
a function that consists of either zero or the sum of a finite number of non-zero terms, each of which is a product of a number, called the coefficient of the term, and a variable raised to a non-negative integer power.
power function
a function that can be represented in the form [latex]f\left(x\right)=k{x}^{p}[/latex] where k is a constant, the base is a variable, and the exponent, p, is a constant smooth curve a graph with no sharp corners
term of a polynomial function
any [latex]{a}_{i}{x}^{i}[/latex] of a polynomial function in the form [latex]f\left(x\right)={a}_{n}{x}^{n}+\dots+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[/latex]
turning point
the location at which the graph of a function changes direction