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**Degree and Lead Coefficient End Behavior**

7.1 Polynomial Functions Degree and Lead Coefficient End Behavior

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**Polynomial should be written in descending order**

The polynomial is not in the correct order 3x3 + 2 – x5 + 7x2 + x Just move the terms around -x5 + 3x3 + 7x2 + x + 2 Now it is in correct form

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**When the polynomial is in the correct order**

Finding the lead coefficient is the number in front of the first term -x5 + 3x3 + 7x2 + x + 2 Lead coefficient is – 1 It degree is the highest degree Degree 5 Since it only has one variable, it is a Polynomial in One Variable

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Evaluate a Polynomial To Evaluate replace the variable with a given value. f(x) = 3x2 – 3x + 1 Let x = 4, 5, and 6

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Evaluate a Polynomial To Evaluate replace the variable with a given value. f(x) = 3x2 – 3x + 1 Let x = 4, 5, and 6 f(4) = 3(4)2 – 3(4) + 1 = 37 = 3(16) – = 48 – = = 37

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Evaluate a Polynomial To Evaluate replace the variable with a given value. f(x) = 3x2 – 3x + 1 Let x = 4, 5, and 6 f(4) = 3(4)2 – 3(4) + 1 = 37 f(5) = 3(5)2 – 3(5) + 1 =

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Evaluate a Polynomial To Evaluate replace the variable with a given value. f(x) = 3x2 – 3x + 1 Let x = 4, 5, and 6 f(4) = 3(4)2 – 3(4) + 1 = 37 f(5) = 3(5)2 – 3(5) + 1 = 61 = 3(25) – = 75 – = 61

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Evaluate a Polynomial To Evaluate replace the variable with a given value. f(x) = 3x2 – 3x + 1 Let x = 4, 5, and 6 f(4) = 3(4)2 – 3(4) + 1 = 37 f(5) = 3(5)2 – 3(5) + 1 = 61 f(6) = 3(6)2 – 3(6) + 1 =

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Evaluate a Polynomial To Evaluate replace the variable with a given value. f(x) = 3x2 – 3x + 1 Let x = 4, 5, and 6 f(4) = 3(4)2 – 3(4) + 1 = 37 f(5) = 3(5)2 – 3(5) + 1 = 61 f(6) = 3(6)2 – 3(6) + 1 = 91 = 3(36) – = 91

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**Find p(y3) if p(x) = 2x4 – x3 + 3x**

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**Find p(y3) if p(x) = 2x4 – x3 + 3x**

p(y3) = 2(y3)4 – (y3)3 + 3(y3) p(y3) = 2y12 – y9 + 3y3

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**Find b(2x – 1) – 3b(x) if b(m) = 2m2 + m - 1**

Do this problem in two parts b(2x – 1) =

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**Find b(2x – 1) – 3b(x) if b(m) = 2m2 + m - 1**

Do this problem in two parts b(2x – 1) = 2(2x – 1)2 + (2x -1) – 1 =2(2x – 1)(2x – 1) + (2x – 1) – 1 =2(4x2 – 2x -2x + 1) + (2x -1) – 1 = 2(4x2 – 4x + 1) + (2x – 1) -1 = 8x2 – 8x x -1 – 1 = 8x2 - 6x

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**Find b(2x – 1) – 3b(x) if b(m) = 2m2 + m - 1**

Do this problem in two parts b(2x – 1) = 8x2 - 6x -3b(x) = -3(2x2 + x – 1) = -6x2 – 3x + 3 b(2x – 1) – 3b(x) = (8x2 – 6x) + (-6x2 – 3x + 3) = 2x2 – 9x + 3

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**End Behavior We understand the end behavior of a quadratic equation.**

y = ax2 + bx + c both sides go up if a> 0 both sides go down a < 0 If the degree is an even number it will always be the same. y = 6x8 – 5x3 + 2x – 5 go up since 6>0 and 8 the degree is even

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End Behavior If the degree is an odd number it will always be in different directions. y = 6x7 – 5x3 + 2x – 5 Since 6>0 and 7 the degree is odd raises up as x goes to positive infinite and falls down as x goes to negative infinite.

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End Behavior If the degree is an odd number it will always be in different directions. y = -6x7 – 5x3 + 2x – 5 Since -6<0 and 7 the degree is odd falls down as x goes to positive infinite and raises up as x goes to negative infinite.

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End Behavior If a is positive and degree is even, then the polynomial raises up on both ends (smiles) If a is negative and degree is even, then the polynomial falls on both ends (frowns)

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End Behavior If a is positive and degree is odd, then the polynomial raises up as x becomes larger, and falls as x becomes smaller If a is negative and degree is odd, then the polynomial falls as x becomes larger, and rasies as x becomes smaller

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**Tell me if a is positive or negative and if the degree is even or odd**

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**Tell me if a is positive or negative and if the degree is even or odd**

a is positive and the degree is odd

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**Tell me if a is positive or negative and if the degree is even or odd**

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**Tell me if a is positive or negative and if the degree is even or odd**

a is positive and the degree is even

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**Tell me if a is positive or negative and if the degree is even or odd**

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**Tell me if a is positive or negative and if the degree is even or odd**

a is negative and the degree is odd

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Homework Page 350 – 351 # 17 – 27 odd, 31, 34, 37, 39 – 43 odd

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Homework Page 350 – 351 # 16 – 28 even, 30, 35, 40 – 44 even

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Notes Over 3.2 Graphs of Polynomial Functions Continuous Functions Non-Continuous Functions Polynomial functions are continuous.

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