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Section 5.1 – Polynomial Functions Defn: Polynomial function The coefficients are real numbers. The exponents are non-negative integers. The domain of.

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Presentation on theme: "Section 5.1 – Polynomial Functions Defn: Polynomial function The coefficients are real numbers. The exponents are non-negative integers. The domain of."— Presentation transcript:

1 Section 5.1 – Polynomial Functions Defn: 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? yesno yes no

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 Section 5.1 – Polynomial Functions Defn: 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 integer Even 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).

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 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).

5 Section 5.1 – Polynomial Functions Transformations of Polynomial Functions

6 Section 5.1 – Polynomial Functions Transformations of Polynomial Functions

7 Section 5.1 – Polynomial Functions Defn: 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 Zero x – r is a factor of the function. r is a solution to the function f(x) = 0 If f(r) = 0 and r is a real number, then r is a real zero of the function.

8 Section 5.1 – Polynomial Functions Defn: The graph of the function touches the x-axis but does not cross it. Zero Multiplicity of an Even Number Multiplicity The 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 Number

9 Section 5.1 – Polynomial Functions 3 is a zero with a multiplicity of Identify the zeros and their multiplicity 3.-2 is a zero with a multiplicity of 1. Graph crosses the x-axis. -4 is a zero with a multiplicity of 2.7 is a zero with a multiplicity of 1. Graph crosses the x-axis. Graph touches the x-axis. -1 is a zero with a multiplicity of 1.4 is a zero with a multiplicity of 1. Graph crosses the x-axis. 2.2 is a zero with a multiplicity ofGraph touches the x-axis.

10 Section 5.1 – Polynomial Functions If a function has a degree of n, then it has at most n – 1 turning points. Turning Points The 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 What is the most number of turning points the following polynomial functions could have?

11 Section 5.1 – Polynomial Functions End Behavior of a Function

12 Section 5.1 – Polynomial Functions End Behavior of a Function

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

14 Section 5.1 – Polynomial Functions State and graph a possible function. 42


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