# 7.1 polynomial functions degree and lead coefficient end behavior

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7.1 Polynomial Functions Degree and Lead Coefficient End Behavior Slide 2 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 Slide 3 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 Slide 4 Evaluate a Polynomial To Evaluate replace the variable with a given value. f(x) = 3x 2 3x + 1Let x = 4, 5, and 6 Slide 5 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 Slide 6 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 = Slide 7 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 Slide 8 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 = Slide 9 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 Slide 10 Find p(y 3 ) if p(x) = 2x 4 x 3 + 3x Slide 11 p(y 3 ) = 2(y 3 ) 4 (y 3 ) 3 + 3(y 3 ) p(y 3 ) = 2y 12 y 9 + 3y 3 Slide 12 Find b(2x 1) 3b(x) if b(m) = 2m 2 + m - 1 Do this problem in two parts b(2x 1) = Slide 13 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 Slide 14 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 Slide 15 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 Slide 16 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. Slide 17 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

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