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 1)Write the polynomial in descending order 2)Identify the DEGREE and LEADING COEFFICIENT of the polynomial

3. Evaluating a Polynomial: Substitute values of x into polynomial and simplify: Find each value for 1. 2. 3. 4.

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

5. OBSERVATIONS of Polynomial Graphs: 1)DEGREE and ROOT (x-intercept or Zero) Observations: How does the degree of a polynomial function relate the number of roots (zeroes) of the graph? 2) DEGREE and SHAPE OBSERVATIONS How EVEN versus ODD degree graphs start and end? How are the number of direction changes (up, down) related to the degree?

6. LEADING COEFFICIENT AFFECTS SHAPE Numerical Value of Degree NEGATIVE Leading Coefficient POSITIVE Leading Coefficient: Describe possible shape of the following based on the degree and leading coefficient: How does the graph start and end? How many changes in direction?

7. Degree Practice with Polynomial Functions Identify the degree as odd or even and state possible degree value. Identify leading coefficient as positive or negative. Degree: Odd or Even Possible Value: ________ LC: Pos or Neg

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

9. RANGE of POLYNOMIAL FUNCTIONS Describes the possible y-values of the function. Is there a highest or lowest value? ODD DEGREE EVEN DEGREE (3, -9) (-2, 5) (-6, 15) (8, 11) (1, -8)

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

11. 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. Another way of saying it starts and ends going UP or DOWN Determine the end behavior for each of the given graphs Decreasing to the Right Negative: “Down” Decreasing to the Left Right: “Ends” Negative: “Down”Left: “Starts”

12. END BEHAVIOR of a polynomial: Continued Decreasing to the Right Increasing to the Left Negative: “Down” Right: “Ends”Positive: “Up” Left: “Starts” Use Range Graphs #1 – 2 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 Degree: ODD or EVEN LC: POS or NEG GRAPH #2

13. Use Range Graphs #3 – 6 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 #3 GRAPH #4 GRAPH #5 GRAPH #6

14. 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: Odd or Even Leading Coefficient: Pos or Neg END BEHAVIOR Degree: Odd or Even Leading Coefficient: Pos or Neg END BEHAVIOR

15. Degree: Odd or Even Leading Coefficient: Pos or Neg END BEHAVIOR Degree: Odd or Even Leading Coefficient: Pos or Neg END BEHAVIOR

16. Degree: Odd or Even Leading Coefficient: Pos or Neg END BEHAVIOR Degree: Odd or Even Leading Coefficient: Pos or Neg END BEHAVIOR

17. 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 EXTREMA : MAXIMUM and MINIMUM points are the highest and lowest points on the graph.

18. Identify ALL Maximum or Minimum Points Distinguish if each is RELATIVE or ABSOLUTE (-6, -9) (4, -15) (-2, 8) (0, 11) (13, 9) (7, -2) (-17, -10) (-3, 3) (-5, -4) (1, -3) (2, 2) (4, -5) (1, 4) (-5, -9) Graph #1 Graph #2 Graph #3 Graph #4

19. Identify ALL Maximum or Minimum Points Distinguish if each is RELATIVE or ABSOLUTE (1, -9) (6, 11) (-3,12) (-2, 22) (6, 3) Graph #5 Graph #6 (1, -27) (-4,19) (-7, 1.3) (8, -2.5) Graph #7 Graph #8 (-7.5, 6) (10, -4.5) (-17, -1.1)

20. 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 Y 1 = function and Y 2 = 0 [2 nd ]  [Calc]  [Intersect] To find EXTEREMA (maximums and minimums): Input [ Y=] as Y 1 = function [2 nd ]  [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. CALCULATOR COMMANDS for POLYNOMIAL FUNCTIONS