The Graphs of Polynomials 5 November 2010

Even vs. Odd All polynomials are either even or odd Describes the shape of a graph Good for generalizing graphs Degree determines if polynomial is even or odd

Even Degree is an even number Roughly U-Shaped Both ends of the graph are either increasing or decreasing Can open either up or down Depends on leading coefficient Positive leading coefficient = opens up Negative leading coefficient = opens down

Leading Coefficient Positive Leading Coefficient Negative Even, cont. Opens Up Opens Down Leading Coefficient Positive Leading Coefficient Negative

Odd Degree is an odd number One end of the graph points up and the other end points down Depends on leading coefficient Positive leading coefficient = left end points down, right end points up Negative leading coefficient = left end points up, right end points down

Leading Coefficient Positive Leading Coefficient Negative Odd, cont. Left Down, Right Up Left Up, Right Down Leading Coefficient Positive Leading Coefficient Negative

Your Turn: Decide if the graphs for problems 7-8, 10-12 on page 269-70 in your Precalculus textbook are even or odd.

Extrema The vertex points of polynomials Extrema (maximum) the maximum or minimum points peaks and valleys Extrema (maximum) Extrema (minimum)

Extrema, cont. Limited by the degree of a polynomial A polynomial of degree n can have at most n – 1 extrema. Translation: The greatest number of extrema a polynomial can have is one less than the degree of the polynomial Or: The degree of the polynomial is 1 greater than the number of extrema

Extrema, cont. 5th Degree Polynomial We can use extrema to figure out the degree of a polynomial # of Extrema = 4 Degree of Polynomial = # of Extrema + 1

Your Turn: Determine the degree of each polynomial for problems 7-8, 10-12 on page 269-70 in your Precalculus textbook.

Matching Equations to Graphs Step 1: Determine if the equation is even or odd. Step 2: Check if the leading coefficient is positive or negative. Step 3: Determine the maximum number of extrema for the equation. Step 4: Match a graph to the characteristics determined in steps 1-3.

Matching Equations to Graphs: y = x5 + 12x Even or Odd? Odd Leading Coefficient Positive or Negative? Positive Degree = 5 Max Extrema = 4

Matching Graphs to Equations: y = 2x4 – 5x2 +2 or y = -x6 + 3? Even or Odd? Even Leading Coefficient Positive or Negative? Positive # of Extrema = 3 Degree = 4

Your Turn: Answer questions 19-24 on pages 270-71 in your Precalculus textbook.

Multiplicity Describes how often a root or factor occurs in an equation or graph Either even or odd If a root occurs an even number of times, then the graph touches the x-axis at the root. If a root occurs an odd number of times, then the graph crosses the x-axis at the root.

-3 2 Touch -1 1 Cross 3 y = (x + 3)2(x + 1)(x – 1)3 Zero Multiplicity x-axis -3 2 Touch -1 1 Cross 3

-1 3 Cross 1 2 Touch y = (x + 1)3(x – 1)(x – 3)2 Zero Multiplicity x-axis -1 3 Cross 1 2 Touch

Determining Multiplicity from Graphs (Pg. 265) Zero Multiplicity x-axis -1 Even Touch 2 Odd Cross 3.5

Your Turn: Complete problems 15-18 on page 270 of your Precalculus textbook.

Hmwk: Handout, problems 1-8