7.1 Polynomial Functions

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 a is positive and the degree is even

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

7.1 Polynomial Functions