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