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 order3x3 + 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 – 1It degree is the highest degree

Degree 5Since 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


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



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 =



f(4) = 3(4)2 – 3(4) + 1 = 37f(5) = 3(5)2 – 3(5) + 1 = 61

= 3(25) – 15 + 1 = 75 – 15 + 1 = 61



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 =



f(4) = 3(4)2 – 3(4) + 1 = 37f(5) = 3(5)2 – 3(5) + 1 = 61f(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


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


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


even or odda is positive and the degree is odd


even or odd


even or odda is positive and the degree is even


even or odd


even or odda 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

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