1. Do Now

2. 5.1 Polynomial Functions
Target: I can classify polynomials and recognize their leading coefficients and describe end behavior

3. The degree of a polynomial is determined by the greatest exponent when there is only one variable (x) in the polynomial
Polynomial functions where the degree is n and a is the coefficient look like this:
f(x) = anxn + an-1xn-1 + … +a1x + a0
Example: f(x) = 4x4 – 3x3 + 12x2 – 7x + 2 (degree 4)

4. Degree
The degree of the function tells the maximum number of real zeros the function has, or the number of times the graph of the function crosses the x-axis
(ex: degree 4 function means there are at most 4 real zeros)
f(x) = 4x4 – 3x3 + 12x2 – 7x + 2

5. Leading Coefficient
The leading coefficient is the coefficient of the first term in a polynomial when the terms are written in descending order by degrees.
For example, the quartic function
f(x) = -2x4 + x3 – 5x2 – 10 has a leading coefficient of -2.

6. A polynomial function is a function of the form:
Remember integers are … –2, -1, 0, 1, 2 … (no decimals or fractions) so positive integers would be 0, 1, 2 …
A polynomial function is a function of the form:
n must be a positive integer
All of these coefficients are real numbers
The degree of the polynomial is the largest power on any x term in the polynomial.

7. Graphs of polynomials are smooth and continuous.
No gaps or holes, can be drawn without lifting pencil from paper
No sharp corners or cusps
This IS the graph of a polynomial
This IS NOT the graph of a polynomial

8. Let’s look at the graph of where n is an even integer.
and grows steeper on either side
Notice each graph looks similar to x2 but is wider and flatter near the origin between –1 and 1
The higher the power, the flatter and steeper

9. Let’s look at the graph of where n is an odd integer.
Notice each graph looks similar to x3 but is wider and flatter near the origin between –1 and 1
and grows steeper on either side
The higher the power, the flatter and steeper

10. Polynomial Function in General Form
Polynomial Functions
Polynomial Function in General Form
Degree
Name of Function
Constant 1
Linear 2
Quadratic 3
Cubic 4
Quartic 5
Quintic
Teachers: This definition for ‘degree’ has been simplified intentionally to help students understand the concept quickly and easily.
The largest exponent within the polynomial determines the degree of the polynomial.

11. Write the polynomials in standard form. Classify by degree and terms.

12. The degree of a polynomial function affects the shape of its graph and determines the maximum number of turning points, or places the graph changes directions.
It also affects the end behavior, or the directions of the graph to the far left and the far right.

13. POLYNOMIAL FUNCTIONS f(x) = 3 Constant Function Degree = 0
GENERAL SHAPES OF
POLYNOMIAL FUNCTIONS
f(x) = 3
Constant Function
Degree = 0
Max. Zeros: 0

14. POLYNOMIAL FUNCTIONS f(x) = x + 2 Linear Function Degree = 1
GENERAL SHAPES OF
POLYNOMIAL FUNCTIONS
f(x) = x + 2
Linear
Function
Degree = 1
Max. Zeros: 1

15. POLYNOMIAL FUNCTIONS f(x) = x2 + 3x + 2 Quadratic Function Degree = 2
GENERAL SHAPES OF
POLYNOMIAL FUNCTIONS
f(x) = x2 + 3x + 2
Quadratic
Function
Degree = 2
Max. Zeros: 2

16. POLYNOMIAL FUNCTIONS f(x) = x3 + 4x2 + 2 Cubic Function Degree = 3
GENERAL SHAPES OF
POLYNOMIAL FUNCTIONS
f(x) = x3 + 4x2 + 2
Cubic
Function
Degree = 3
Max. Zeros: 3

17. POLYNOMIAL FUNCTIONS f(x) = x4 + 4x3 – 2x – 1 Quartic Function
GENERAL SHAPES OF
POLYNOMIAL FUNCTIONS
f(x) = x4 + 4x3 – 2x – 1
Quartic
Function
Degree = 4
Max. Zeros: 4

18. POLYNOMIAL FUNCTIONS f(x) = x5 + 4x4 – 2x3 – 4x2 + x – 1 Quintic
GENERAL SHAPES OF
POLYNOMIAL FUNCTIONS
f(x) = x5 + 4x4 – 2x3 – 4x2 + x – 1
Quintic
Function
Degree = 5
Max. Zeros: 5

19. POLYNOMIAL FUNCTIONS f(x) = x2 Degree: Even Leading Coefficient: +
END BEHAVIOR
f(x) = x2
Degree: Even
Leading Coefficient: +
End Behavior:
Up and up

20. POLYNOMIAL FUNCTIONS f(x) = -x2 Degree: Even Leading Coefficient: –
END BEHAVIOR
f(x) = -x2
Degree: Even
Leading Coefficient: –
End Behavior:
Down and down

21. POLYNOMIAL FUNCTIONS f(x) = x3 Degree: Odd Leading Coefficient: +
END BEHAVIOR
f(x) = x3
Degree: Odd
Leading Coefficient: +
End Behavior:
Down and up

22. POLYNOMIAL FUNCTIONS f(x) = -x3 Degree: Odd Leading Coefficient: –
END BEHAVIOR
f(x) = -x3
Degree: Odd
Leading Coefficient: –
End Behavior:
Up and down

23. End Behavior Summary Degree Even Odd Leading Coefficient Positive
Up and Up
Down and Up
Negative
Down and Down
Up and Down

24. Assignment Homework– p. 285 #9 – 37 odds
Challenge - #47-49 (must do all 3)

25. Cubic Polynomials
Look at the two graphs and discuss the questions given below.
Graph B
Graph A
1. How can you check to see if both graphs are functions?
2. How many x-intercepts do graphs A & B have?
3. What is the end behaviour for each graph?
4. Which graph do you think has a positive leading coeffient? Why?
5. Which graph do you think has a negative leading coefficient? Why?

26. Quartic Polynomials
Look at the two graphs and discuss the questions given below.
Graph B
Graph A
1. How can you check to see if both graphs are functions?
2. How many x-intercepts do graphs A & B have?
3. What is the end behaviour for each graph?
4. Which graph do you think has a positive leading coeffient? Why?
5. Which graph do you think has a negative leading coefficient? Why?

27. Factored form & Standard form Sign of Leading Coefficient
Cubic Polynomials
The following chart shows the properties of the graphs on the left.
Equation
Factored form & Standard form
X-Intercepts
Sign of Leading Coefficient
End Behaviour
Domain and Range
Factored
y=(x+1)(x+4)(x-2)
Standard
y=x3+3x2-6x-8
-4, -1, 2
Positive
As x, y and x-, y-
Domain
{x| x Є R}
Range
{y| y Є R}
y=-(x+1)(x+4)(x-2)
y=-x3-3x2+6x+8
Negative
As x, y- and
x-, y

28. Factored form & Standard form Sign of Leading Coefficient
Cubic Polynomials
The following chart shows the properties of the graphs on the left.
Equation
Factored form & Standard form
X-Intercepts
Sign of Leading Coefficient
End Behaviour
Domain and Range
Factored
y=(x+3)2(x-1)
Standard
y=x3+5x2+3x-9
-3, 1
Positive
As x, y and x-, y-
Domain
{x| x Є R}
Range
{y| y Є R}
y=-(x+3)2(x-1)
y=-x3-5x2-3x+9
Negative
As x, y- and
x-, y

29. Factored form & Standard form Sign of Leading Coefficient
Cubic Polynomials
The following chart shows the properties of the graphs on the left.
Equation
Factored form & Standard form
X-Intercepts
Sign of Leading Coefficient
End Behaviour
Domain and Range
Factored
y=(x-2)3
Standard
y=x3-6x2+12x-8 2
Positive
As x, y and x-, y-
Domain
{x| x Є R}
Range
{y| y Є R}
y=-(x-2)3
y=-x3+6x2-12x+8
Negative
As x, y- and
x-, y

30. Factored form & Standard form Sign of Leading Coefficient
Quartic Polynomials
The following chart shows the properties of the graphs on the left.
Equation
Factored form & Standard form
X-Intercepts
Sign of Leading Coefficient
End Behaviour
Domain and Range
Factored
y=(x-3)(x-1)(x+1)(x+2)
Standard
y=x4-x3-7x2+x+6
-2,-1,1,3
Positive
As x, y and x-, y
Domain
{x| x Є R}
Range
{y| y Є R,
y ≥ }
y=-(x-3)(x-1)(x+1)(x+2)
y=-x4+x3+7x2-x-6
Negative
As x, y- and
x-, y-
y ≤ 12.95}

31. Factored form & Standard form Sign of Leading Coefficient
Quartic Polynomials
The following chart shows the properties of the graphs on the left.
Equation
Factored form & Standard form
X-Intercepts
Sign of Leading Coefficient
End Behaviour
Domain and Range
Factored
y=(x-4)2(x-1)(x+1)
Standard
y=x4-8x3+15x2+8x-16
-1,1,4
Positive
As x, y and x-, y
Domain
{x| x Є R}
Range
{y| y Є R,
y ≥ }
y=-(x-4)2(x-1)(x+1)
y=-x4+8x3-15x2-8x+16
Negative
As x, y- and
x-, y-
y ≤ 16.95}

32. Factored form & Standard form Sign of Leading Coefficient
Quartic Polynomials
The following chart shows the properties of the graphs on the left.
Equation
Factored form & Standard form
X-Intercepts
Sign of Leading Coefficient
End Behaviour
Domain and Range
Factored
y=(x+2)3(x-1)
Standard
y=x4+5x3+6x2-4x-8
-2,1
Positive
As x, y and x-, y
Domain
{x| x Є R}
Range
{y| y Є R,
y ≥ -8.54}
y=-(x+2)3(x-1)
y=-x4-5x3-6x2+4x+8
Negative
As x, y- and
x-, y-
y ≤ 8.54}

33. Factored form & Standard form Sign of Leading Coefficient
Quartic Polynomials
The following chart shows the properties of the graphs on the left.
Equation
Factored form & Standard form
X-Intercepts
Sign of Leading Coefficient
End Behaviour
Domain and Range
Factored
y=(x-3)4
Standard
y=x4-12x3+54x2-108x+81 3
Positive
As x, y and x-, y
Domain
{x| x Є R}
Range
{y| y Є R,
y ≥ 0}
y=-(x-3)4
y=-x4+12x3-54x2+108x-81
Negative
As x, y- and
x-, y-
y ≤ 0}