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Section 5.1 – Polynomial Functions

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.

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Section 5.1 – Polynomial Functions

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  1. 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 no no yes

  2. Section 5.1 – Polynomial Functions Defn: Degree of a Function • The 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. State the degree of the following polynomial functions 3 5 12 8

  3. Section 5.1 – Polynomial Functions Defn: Power function of Degree n • In the form of: . • The coefficient is a real number. • The exponent is a non-negative integer. Properties of a Power Function w/ n a Positive EVEN integer • Even function  graph is symmetric with the y-axis. • 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). • The graph will flatten out for x values between -1 and 1.

  4. Section 5.1 – Polynomial Functions Properties of a Power Function w/ n a Positive ODD integer • Odd function  graph is symmetric with the origin. • The domain and range are the set of all real numbers. • The graph always contains the points (0,0), (-1,-1), & (1,1). • The graph will flatten out for x values between -1 and 1.

  5. Section 5.1 – Polynomial Functions Transformations of Polynomial Functions • 2 • 2 • 2 • 2

  6. Section 5.1 – Polynomial Functions Transformations of Polynomial Functions • 5 • 1 • 4 • 1 • -3

  7. Section 5.1 – Polynomial Functions Defn: Real Zero of a function • If f(r) = 0 and r is a real number, then r is a real zero of the function. Equivalent Statements for a Real Zero • r is a real zero of the function. • r is an x-intercept of the graph of the function. • x – r is a factor of the function. • r is a solution to the function f(x) = 0

  8. Section 5.1 – Polynomial Functions Defn: Multiplicity • The number of times a factor (m) of a function is repeated is referred to its multiplicity (zero multiplicity of m). Zero Multiplicity of an Even Number • The graph of the function touches the x-axis but does not cross it. Zero Multiplicity of an Odd Number • The graph of the function crosses the x-axis.

  9. Section 5.1 – Polynomial Functions Identify the zeros and their multiplicity • 1. • 3 is a zero with a multiplicity of • Graph crosses the x-axis. • -2 is a zero with a multiplicity of • 3. • Graph crosses the x-axis. • 1. • -4 is a zero with a multiplicity of • Graph crosses the x-axis. • 7 is a zero with a multiplicity of • 2. • Graph touches the x-axis. • 1. • -1 is a zero with a multiplicity of • Graph crosses the x-axis. • 4 is a zero with a multiplicity of • 1. • Graph crosses the x-axis. • 2 is a zero with a multiplicity of • 2. • Graph touches the x-axis.

  10. Section 5.1 – Polynomial Functions Turning Points • The point where a function changes directions from increasing to decreasing or from decreasing to increasing. • If a function has a degree of n, then it has at mostn – 1 turning points. • If the graph of a polynomial function has t number of turning points, then the function has at least a degree of t + 1 . What is the most number of turning points the following polynomial functions could have? 3-1 5-1 2 4 8-1 12-1 11 7

  11. Section 5.1 – Polynomial Functions End Behavior of a Function • If , then the end behaviors of the graph will depend on the first term of the function, . • If and n is even, then both ends will approach +. • If and n is even, then both ends will approach –. • If and n is odd, • then as x  – ,  – and as x ,  . • If and n is odd, • then as x  – , and as x ,  –.

  12. Section 5.1 – Polynomial Functions End Behavior of a Function • and n is even • and n is even • and n is odd • and n is odd

  13. Section 5.1 – Polynomial Functions State and graph a possible function. • Line with negative slope • Line with positive slope • Parabola opening down

  14. Section 5.1 – Polynomial Functions State and graph a possible function. • 2 • 4 • -1

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