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.

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

Finding values of a polynomial: Substitute values of x into polynomial and simplify: Find each value for 1. 2. 3. 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)

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

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:

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

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

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 )

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:

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

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

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

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

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.

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