section 5.1 – polynomial functions

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Section 5.1 – Polynomial Functions. Defn : . Polynomial function. In the form of: . The coefficients are real numbers. The exponents are non-negative integers. The domain of the function is the set of all real numbers. Are the following functions polynomials?. yes. - PowerPoint PPT Presentation

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Section 5.1 Polynomial FunctionsDefn: Polynomial function The coefficients are real numbers. The exponents are non-negative integers. The domain of the function is the set of all real numbers.Are the following functions polynomials?yesnoyesnoSection 5.1 Polynomial FunctionsDefn: Degree of a FunctionThe largest degree of the function represents the degree of the function.The zero function (all coefficients and the constant are zero) does not have a degree.35812 State the degree of the following polynomial functionsSection 5.1 Polynomial FunctionsDefn: Power function of Degree n The coefficient is a real number. The exponent is a non-negative integer.Properties of a Power Function w/ n a Positive EVEN integerEven function graph is symmetric with the y-axis.The graph will flatten out for x values between -1 and 1.The domain is the set of all real numbers.The range is the set of all non-negative real numbers.The graph always contains the points (0,0), (-1,1), & (1,1).Section 5.1 Polynomial Functions Properties of a Power Function w/ n a Positive ODD integerOdd function graph is symmetric with the origin.The graph will flatten out for x values between -1 and 1.The domain and range are the set of all real numbers.The graph always contains the points (0,0), (-1,-1), & (1,1).Section 5.1 Polynomial Functions Transformations of Polynomial Functions2222Section 5.1 Polynomial Functions Transformations of Polynomial Functions1541-3Section 5.1 Polynomial FunctionsDefn: Real Zero of a function r is a real zero of the function. r is an x-intercept of the graph of the function. Equivalent Statements for a Real Zerox r is a factor of the function.r is a solution to the function f(x) = 0If f(r) = 0 and r is a real number, then r is a real zero of the function.Section 5.1 Polynomial FunctionsDefn: The graph of the function touches the x-axis but does not cross it. Zero Multiplicity of an Even Number MultiplicityThe number of times a factor (m) of a function is repeated is referred to its multiplicity (zero multiplicity of m).The graph of the function crosses the x-axis. Zero Multiplicity of an Odd NumberSection 5.1 Polynomial Functions3 is a zero with a multiplicity ofIdentify the zeros and their multiplicity3.-2 is a zero with a multiplicity of1.Graph crosses the x-axis.Graph crosses the x-axis.-4 is a zero with a multiplicity of2.7 is a zero with a multiplicity of1.Graph crosses the x-axis.Graph touches the x-axis.-1 is a zero with a multiplicity of1.4 is a zero with a multiplicity of1.Graph crosses the x-axis.Graph crosses the x-axis.2.2 is a zero with a multiplicity ofGraph touches the x-axis.Section 5.1 Polynomial FunctionsIf a function has a degree of n, then it has at most n 1 turning points.Turning PointsThe point where a function changes directions from increasing to decreasing or from decreasing to increasing.If the graph of a polynomial function has t number of turning points, then the function has at least a degree of t + 1 .3-15-18-112-1What is the most number of turning points the following polynomial functions could have? 24711Section 5.1 Polynomial Functions End Behavior of a FunctionSection 5.1 Polynomial Functions End Behavior of a FunctionSection 5.1 Polynomial Functions State and graph a possible function.Line with negative slopeLine with positive slopeParabola opening downSection 5.1 Polynomial Functions State and graph a possible function.42-1

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