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

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 = 37f(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 = 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

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 odda 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 odda 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 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