Characteristics Of Power And Polynomial Functions | College Algebra

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Module 9: Power and Polynomial Functions

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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 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.

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)=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] turning point the location where the graph of a function changes direction

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• College Algebra. Authored by: Abramson, Jay et al.. Provided by: OpenStax. Located at: http://cnx.org/contents/[email protected]. License: CC BY: Attribution. License Terms: Download for free at http://cnx.org/contents/[email protected]

Licenses and Attributions CC licensed content, Original

• Revision and Adaptation. Provided by: Lumen Learning. License: CC BY: Attribution

CC licensed content, Shared previously

• College Algebra. Authored by: Abramson, Jay et al.. Provided by: OpenStax. Located at: http://cnx.org/contents/[email protected]. License: CC BY: Attribution. License Terms: Download for free at http://cnx.org/contents/[email protected]

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