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]

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 non-negative integer powers.
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.

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

end behavior: the behavior of the graph of a function as the input decreases without bound and increases without bound

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)=a{x}^{n}[/latex] where a is a constant, the base is a variable, and the exponent is n, is a smooth curve represented by 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]

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