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October 5th copyright2009merrydavidson Warm up Prove inverses by composition f(x)= -3x + 8 g(x)= 8 – x 3 Happy Birthday to: Emily Wiegmann 2.1 Polynomial Functions Examples of polynomial functions f(x) = 3x + 5 f(x) = 4 f(x) = x2 f(x) = 5x3 Linear Constant Quadratic Cubic 2.1 Polynomial Functions What do they all have in common? Non-negative integer exponents of the variables f(x) = 3x + 5 f(x) = 4 Leading coefficient 2 f(x) = x not = 0 f(x) = 5x3 Smooth & continuous graphs Linear Constant Quadratic Cubic Polynomial Functions are classified by degree. Constant Functions; degree 0 Linear Functions; degree 1 Quadratic Functions;degree 2 Cubic Functions; degree 3 What is a Polynomial? Vocabulary: Degree of a term f ( x ) 2 x y 3 x y 5 xy y 5 3 2 6 5 The sum of the powers on the variables in one term Degree of a polynomial The term with the highest degree 6 Leading Coefficient The coefficient of the highest degree term. Not always the first term. -2 3 4 4 4 Examples: Find the leading coefficient and degree of each polynomial function. Polynomial Function Leading Coefficient Degree f ( x ) 2 x 5 3 x 3 5 x 1 –2 5 f ( x) x3 6 x 2 x 7 1 3 14 0 f ( x ) 14 Practice 1: Given the following equations determine the following: 1. Determine if the function is a polynomial in 1 variable. Why or why not? 2. If the function is a polynomial; what is the degree of each term, and of the polynomial. A. f x 3x2 x 1 Yes, notice powers on the x are positive integers and coefficients are real numbers. 2,1, 0 2 C. f x B. No, notice power on x is -1 3 3x 1 x f x 3 x No, notice power on the x is the fraction 1/2 xx E. f x 3x 2 1 2 No, there are 2 different 2 y 1 variables. 3 4x x D. 3x 2 f x 1 5 Yes, notice powers on the x are positive integers and coefficients are real numbers. 3/5,1 2 Where the function touches or crosses the x-axis. What are zero’s of a function? y = -x^3+4x (-2,0),(0,0),(2,0) Zero’s are listed as ordered pairs. What are the zero’s of the given function. y = x^4-5x^2+4 (-2,0),(-1,0),(1,0),(2,0) Zero’s are listed as ordered pairs. What are the zero’s of the given function. y = -2x^4+2x^2 When the graph is “sitting” on the x-axis, it is a “double” root. Multiplicity of 2 (-1,0),(0,0),(1,0) (-1,0)(0,0)MP2,(1,0) This is an even degree function so we need an even number (4) of roots Summary: The degree of the polynomial tells the number of zero’s (x-intercepts). There is one less “turning point” than the degree. y = -x^3+4x Degree 3 2 turning points VOCABULARY Roots/Solutions refer to the algebraic answer of the polynomial equation and is expressed as x = answer. Ex: What is the solution to f(x)=x2 – 9? You would put: x = 3, -3 VOCABULARY Zero’s/x-intercept’s refer to the graph of the polynomial equation and are expressed as ordered pairs. Ex: What are the zero’s of f(x)=x2 – 9? You would put: (-3,0), (3,0) 2.2 Polynomial Functions of higher degree y = x5 y = x^5 Opposite end behavior Down f(x) as x f(x) as x UP 2.2 Polynomial Functions of higher degree y = -x^5 y = -x5 as x f(x) as x Opposite end behavior Up Down f(x) 2.2 Polynomial Functions of higher degree The power tells your left arm what to do. Summary: The coefficient tells your right arm what to do. Pos xodd down/up y = 3x^7 ex: 3x7 Neg xodd up/down y = -4x^9 ex: - 4x9 END in the direction of the coefficient. 2.2 Polynomial Functions of higher degree y = x6 as x f(x) as x f(x) y = x^6 Same end behavior UP UP 2.2 Polynomial Functions of higher degree y = -x6 y = -x^6 as x f(x) as x Same end behavior DOWN DOWN f(x) 2.2 Polynomial Functions of higher degree The power Summary: The coefficient tells your right arm what to do. Pos xeven tells your left arm what to do. up/up y = 3x^8 ex: 3x8 Neg xeven down/down y = -4x^8 ex: - 4x8 END in the direction of the coefficient. Closure: Positive coefficient even power odd power Negative coefficient LIMIT NOTATION again lim f ( x) x lim f ( x) x Practice What is the right hand and left hand end behavior for the following polynomial functions? a) -3x3 + 5x2 – 2x + 3 up/down b) 5x6 + 4x4 + 7x – 10 up/up c) -2x8 + 3x7 + 4x – 2 down/down d) 5x7 + 3x2 - 3 down/up e) (x+2)2(2x-1) down/up Now write them in limit notation…. Homework: WS 3-3