Degree and Lead Coefficient End Behavior 7.1 Polynomial Functions Degree and Lead Coefficient End Behavior

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

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

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

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) – 12 + 1 = 48 – 12 + 1 = 36 + 1 = 37

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 =

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) – 15 + 1 = 75 – 15 + 1 = 61

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 =

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) – 18 + 1 = 91

Find p(y3) if p(x) = 2x4 – x3 + 3x

Find p(y3) if p(x) = 2x4 – x3 + 3x p(y3) = 2(y3)4 – (y3)3 + 3(y3) p(y3) = 2y12 – y9 + 3y3

Find b(2x – 1) – 3b(x) if b(m) = 2m2 + m - 1 Do this problem in two parts b(2x – 1) =

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 + 2 + 2x -1 – 1 = 8x2 - 6x

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

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

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.

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.

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)

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

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

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

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

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

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

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

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

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