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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 3x 3 + 2 – x 5 + 7x 2 + x Just move the terms around -x 5 + 3x 3 + 7x 2 + 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 -x 5 + 3x 3 + 7x 2 + 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) = 3x 2 – 3x + 1Let x = 4, 5, and 6

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Evaluate a Polynomial To Evaluate replace the variable with a given value. f(x) = 3x 2 – 3x + 1Let x = 4, 5, and 6 f(4) = 3(4) 2 – 3(4) + 1 = 37 = 3(16) – 12 + 1 = 48 – 12 + 1 = 36 + 1 = 37

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Evaluate a Polynomial To Evaluate replace the variable with a given value. f(x) = 3x 2 – 3x + 1Let 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) = 3x 2 – 3x + 1Let 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) – 15 + 1 = 75 – 15 + 1 = 61

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Evaluate a Polynomial To Evaluate replace the variable with a given value. f(x) = 3x 2 – 3x + 1Let 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) = 3x 2 – 3x + 1Let 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) – 18 + 1 = 91

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Find p(y 3 ) if p(x) = 2x 4 – x 3 + 3x

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p(y 3 ) = 2(y 3 ) 4 – (y 3 ) 3 + 3(y 3 ) p(y 3 ) = 2y 12 – y 9 + 3y 3

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Find b(2x – 1) – 3b(x) if b(m) = 2m 2 + m - 1 Do this problem in two parts b(2x – 1) =

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Find b(2x – 1) – 3b(x) if b(m) = 2m 2 + 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(4x 2 – 2x -2x + 1) + (2x -1) – 1 = 2(4x 2 – 4x + 1) + (2x – 1) -1 = 8x 2 – 8x + 2 + 2x -1 – 1 = 8x 2 - 6x

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

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End Behavior We understand the end behavior of a quadratic equation. y = ax 2 + bx + cboth 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 = 6x 8 – 5x 3 + 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 = 6x 7 – 5x 3 + 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 = -6x 7 – 5x 3 + 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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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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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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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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