1. POLYNOMIALS REVIEW
The DEGREE of a polynomial is the largest degree of any single term in the polynomial
(Polynomials are often written in descending order of the degree of its terms)
COEFFICIENTS are the numerical value of each term in the polynomial
The LEADING COEFFICIENT is the numerical value of the term with the HIGHEST DEGREE.

2. Polynomials Review Practice
For each polynomial
Write the polynomial in descending order
Identify the DEGREE and LEADING COEFFICIENT of the polynomial

3. Finding values of a polynomial: Substitute values of x into polynomial and simplify:
Find each value for

4. Graphs of Polynomial Functions:  
  Constant Function Linear Function Quadratic Function
(degree = 0) (degree = 1) (degree = 2)
Cubic Function Quartic Function Quintic Function
(deg. = 3) (deg. = 4) (deg. = 5)

5. OBSERVATIONS of Polynomial Graphs:
1) How does the degree of a polynomial function relate the number of roots of the graph?
The degree is the maximum number of zeros or roots that a graph can have.
2) Is there any relationship between the degree of the polynomial function and the shape of the graph?
Number of Changes in DIRECTION OF THE GRAPH = DEGREE
EVEN DEGREES: Start and End both going UP or DOWN
ODD DEGREES: Start and End as opposites  UP and DOWM

6. OBSERVATIONS of Polynomial Graphs:
3) What additional information (value) related the degree of the polynomial may affect the shape of its graph?
LEADING COEFFICIENT
Numerical Value of Degree
EVEN DEGREE:
POSITIVE Leading Coefficient
= UP
NEGATIVE Leading Coefficient
= DOWN
ODD DEGREE:
POSITIVE Leading Coefficient = START Down and END Up
NEGATIVE Leading Coefficient = START Up and END Down
Describe possible shape of the following based on the degree and leading coefficient:

7. Degree Practice with Polynomial Functions
Identify the degree as odd or even and state the assumed degree.
Identify leading coefficient as positive or negative.

8. Draw a graph for each descriptions:
Degree = 4
Leading Coefficient = 2
Description #2:
Degree = 6
Leading Coefficient = -3
Description #3:
Degree = 3
Leading Coefficient = 1
Description #5:
Degree = 5
Leading Coefficient = -4
Description #4:
Degree = 8
Leading Coefficient = -2

9. Graphs # 1 – 6 Identify RANGE: Interval or Inequality Notation
(-2, 8)
(0, 11)
(1, 4)
(13, 9)
(7, -2)
(-6, -9)
(-17, -10)
(-5, -9)
(4, -15)
Range, y: (-∞, ∞ )
Range, y: (-15, ∞ )
Range, y,: (-∞, ∞ )
Graph #4
Graph #5
(-5,17)
Graph #6
(-3,12)
(6, 11)
(1, 12)
(-3, 3)
(4, 8)
(2, 2)
(-2, 6)
(3, 2)
(1, -3)
(-5, -4)
(1, -9)
(4, -5)
Range, y: (-5, ∞ )
Range, y: (-∞, 12 )
Range, y,: (-∞, 17 )

10. Right: Right: Left: Left:
The END BEHAVIOR of a polynomial describes the RANGE, f(x), as the DOMAIN, x, moves LEFT (as x approaches negative infinity: x → - ∞) and RIGHT (as x approaches positive infinity : x → ∞) on the graph. Determine the end behavior for each of the given graphs
Increasing to the Left
Decreasing to the
Left
Decreasing to the Right
Decreasing to the Right
Right:
Left:
Right:
Left:

11. Use Graphs #1 – 6 from the previous Slide
Describe the END BEHAVIOR of each graph
Identify if the degree is EVEN or ODD for the graph
Identify if the leading coefficient is POSITIVE or NEGATIVE
GRAPH #1
GRAPH #2
GRAPH #3
Degree: EVEN
LC: POS
Degree: ODD
LC: NEG
Degree: ODD
LC: NEG
GRAPH #4
GRAPH #5
GRAPH #6
Degree: EVEN
LC: POS
Degree: EVEN
LC: NEG
Degree: EVEN
LC: NEG

12. Describing Polynomial Graphs Based on the Equation
Based on the given polynomial function:
Identify the Leading Coefficient and Degree.
Sketch possible graph (Hint: How many direction changes possible?)
Identify the END BEHAVIOR
Degree: 5  Odd
LC: 2  Pos
Start Down, End Up
Degree: 4  Even
LC: -1  Neg
Start Down, End Down
Degree: 3  Odd
LC: -2  Neg
Start Down, End Up
Degree: 6  Even
LC: 1  Pos
Start Up, End Up

13. EXTREMA: MAXIMUM and MINIMUM points are the highest and lowest points on the graph.
Point A is a Relative Maximum because it is the highest point in the immediate area (not the highest point on the entire graph).  
Point B is a Relative Minimum because it is the lowest point in the immediate area (not the lowest point on the entire graph).
Point C is the Absolute Maximum because it is the highest point on the entire graph.
There is no Absolute Minimum on this graph because the end behavior is:
(there is no bottom point) A B C

14. Identify ALL Maximum or Minimum Points Distinguish if each is RELATIVE or ABSOLUTE
Graph #3
(1, 4)
(-5, -9)
Graph #1
Graph #2
R: Max
R: Max
R: Max
(-2, 8)
(0, 11)
(13, 9)
R: Max
(7, -2)
(-6, -9)
R: Min
(-17, -10)
(4, -15)
R: Min
R: Min
R: Min
A: Min
Graph #4
Graph #5
Graph #6
R: Max
R: Max
(-3,12)
(6, 11)
R: Max
(-2, 22)
(-3, 3)
(2, 2)
A: Max
R: Max
(6, 3)
R: Min
(1, -3)
(1, -9)
(-5, -4)
(4, -5)
R: Min
R: Min
R: Min
A: Min

15. CALCULATOR COMMANDS for POLYNOMIAL FUNCTIONS
The WINDOW needs to be large enough to see graph!
The ZEROES/ ROOTS of a polynomial function are the x-intercepts of the graph.
Input [ Y=] as Y1 = function and Y2 = 0
[2nd ] [Calc]  [Intersect]
To find EXTEREMA (maximums and minimums):
Input [ Y=] as Y1 = function
[2nd ][Calc]  [3: Min] or [4: Max]
LEFT and RIGHT bound tells the calculator where on the domain to search for the min or max.
y-value of the point is the min/max value.

16. Using your calculator: GRAPH the each polynomial function and IDENTIFY the ZEROES, EXTREMA, and END BEHAVIOR.
[1]
[2]
[3]
[4]