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

2. 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

3. 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

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

5. 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) –
= 48 –
= = 37

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
f(5) = 3(5)2 – 3(5) + 1 =

7. 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) –
= 75 – = 61

8. 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 =

9. 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) – = 91

10. Find p(y3) if p(x) = 2x4 – x3 + 3x

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

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

13. 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 x -1 – 1
= 8x2 - 6x

14. 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

15. 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

16. 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.

17. 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.

18. 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)

19. 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

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

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

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

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

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

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

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

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