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Section 5.1 โ Polynomial Functions Defn: Polynomial function In the form of: ๐ ๐ฅ = ๐๐ ๐ฅ ๐ + ๐๐โ1 ๐ฅ ๐โ1 + โฏ ๐1 ๐ฅ + ๐0 . 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? ๐ ๐ฅ = 5๐ฅ + 2๐ฅ 2 โ 6๐ฅ 3 + 3 yes โ ๐ฅ = 2๐ฅ 3 (4๐ฅ 5 + 3๐ฅ) yes ๐ ๐ฅ = 2๐ฅ 2 โ 4๐ฅ + ๐ฅ โ 2 no 2๐ฅ 3 + 3 no ๐ ๐ฅ = 5 4๐ฅ + 3๐ฅ Section 5.1 โ 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 ๐ ๐ฅ = 2๐ฅ 5 โ 4๐ฅ 3 + ๐ฅ โ 2 ๐ ๐ฅ = 5๐ฅ + 2๐ฅ 2 โ 6๐ฅ 3 + 3 3 5 ๐ ๐ฅ = 4๐ฅ 3 + 6๐ฅ 11 โ ๐ฅ 10 + ๐ฅ 12 โ ๐ฅ = 2๐ฅ 3 (4๐ฅ 5 + 3๐ฅ) 12 8 Section 5.1 โ Polynomial Functions Defn: Power function of Degree n In the form of: ๐ ๐ฅ = ๐๐ฅ ๐ . 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 ๏ฎ 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. Section 5.1 โ Polynomial Functions Properties of a Power Function w/ n a Positive ODD integer Odd function ๏ฎ 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. Section 5.1 โ Polynomial Functions Transformations of Polynomial Functions ๐ ๐ฅ = ๐ฅ2 + 2 ๐ ๐ฅ = (๐ฅ โ 2)2 2 ๐ ๐ฅ = (๐ฅ โ 2)2 +2 2 2 2 Section 5.1 โ Polynomial Functions Transformations of Polynomial Functions ๐ ๐ฅ = (๐ฅ + 1)5 ๐ ๐ฅ = โ(๐ฅ โ 4)3 โ 3 ๐ ๐ฅ = โ(๐ฅ โ 1)2 + 5 5 1 4 -3 1 Section 5.1 โ 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 โ r is a factor of the function. r is a solution to the function f(x) = 0 Section 5.1 โ 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. Section 5.1 โ Polynomial Functions Identify the zeros and their multiplicity ๐ ๐ฅ = ๐ฅโ3 ๐ฅ+2 3 3 is a zero with a multiplicity of 1. Graph crosses the x-axis. -2 is a zero with a multiplicity of 3. Graph crosses the x-axis. ๐ ๐ฅ =5 ๐ฅ+4 ๐ฅโ7 2 -4 is a zero with a multiplicity of 1. Graph crosses the x-axis. 7 is a zero with a multiplicity of 2. Graph touches the x-axis. ๐ ๐ฅ = ๐ฅ + 1 (๐ฅ โ 4) ๐ฅ โ 2 2 -1 is a zero with a multiplicity of 1. 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. Section 5.1 โ 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 most n โ 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? ๐ ๐ฅ = 2๐ฅ 5 โ 4๐ฅ 3 + ๐ฅ โ 2 ๐ ๐ฅ = 5๐ฅ + 2๐ฅ 2 โ 6๐ฅ 3 + 3 3-1 5-1 2 4 ๐ ๐ฅ = 4๐ฅ 3 + 6๐ฅ 11 โ ๐ฅ 10 + ๐ฅ 12 โ ๐ฅ = 2๐ฅ 3 (4๐ฅ 5 + 3๐ฅ) 8-1 12-1 11 7 Section 5.1 โ Polynomial Functions End Behavior of a Function If ๐ ๐ฅ = ๐๐ ๐ฅ ๐ + ๐๐โ1 ๐ฅ ๐โ1 + โฏ ๐1 ๐ฅ + ๐0 , then the end behaviors of the graph will depend on the first term of the function, ๐๐ฅ ๐ . If ๐ ๐ฅ = ๐๐ฅ ๐ and n is even, then both ends will approach +๏ฅ. If ๐ ๐ฅ = โ๐๐ฅ ๐ and n is even, then both ends will approach โ๏ฅ. If ๐ ๐ฅ = ๐๐ฅ ๐ and n is odd, then as x ๏ฎ โ ๏ฅ, ๐ ๐ฅ ๏ฎ โ๏ฅ and as x ๏ฎ ๏ฅ, ๐ ๐ฅ ๏ฎ ๏ฅ. If ๐ ๐ฅ = โ๐๐ฅ ๐ and n is odd, then as x ๏ฎ โ ๏ฅ, ๐ ๐ฅ ๏ฎ ๏ฅ and as x ๏ฎ ๏ฅ, ๐ ๐ฅ ๏ฎ โ๏ฅ. Section 5.1 โ Polynomial Functions End Behavior of a Function ๐ ๐ฅ = โ๐๐ฅ ๐ and n is even ๐ ๐ฅ = ๐๐ฅ ๐ and n is even ๐ ๐ฅ = ๐๐ฅ ๐ and n is odd ๐ ๐ฅ = โ๐๐ฅ ๐ and n is odd Section 5.1 โ Polynomial Functions State and graph a possible function. ๐ง๐๐๐๐ : โ1, 2 ๐ค๐๐กโ ๐๐ข๐๐ก๐๐๐๐๐๐๐ก๐ฆ 2, 4 ๐๐๐๐๐๐ 4 ๐ฅ = โ1 ๐ฅ = 2 ๐ฅ = 4 ๐ฅ+1=0 ๐ฅโ2=0 ๐ฅโ4=0 ๐ ๐ฅ = ๐ฅ + 1 (๐ฅ โ 4) ๐ฅ โ 2 ๐ฅ = โ1 2 ๐ฅ=4 ๐ฅ + 1 (โ1 โ 4) โ1 โ 2 2 4 + 1 (๐ฅ โ 4) 4 โ 2 2 ๐ฅ + 1 (โ)(+) โ๐ฅ โ 1 + (๐ฅ โ 4)(+) ๐ฅโ4 Line with negative slope Line with positive slope ๐ฅ=2 2 + 1 (2 โ 4) ๐ฅ โ 2 2 โ + (โ)(๐ฅ โ 2)2 โ โ(๐ฅ โ 2)2 Parabola opening down Section 5.1 โ Polynomial Functions State and graph a possible function. ๐ ๐ฅ = ๐ฅ + 1 (๐ฅ โ 4) ๐ฅ โ 2 -1 2 4 2