The Graphs of Polynomials 5 November 2010 Even

The Graphs of Polynomials 5 November 2010

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

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

Odd l l Degree is an odd number One end of the graph points up and the other end points down l l l 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

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

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

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

Extrema, cont. l l Limited by the degree of a polynomial A polynomial of degree n can have at most n – 1 extrema. l l 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. l l l We can use extrema to figure out the degree of a polynomial # of Extrema = 4 Degree of Polynomial = # of Extrema + 1 5 th Degree Polynomial

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

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

Matching Graphs to Equations: y = 2 x 4 – 5 x 2 +2 or y = -x 6 + 3? l Even or Odd? l l Leading Coefficient Positive or Negative? l l Even Positive # of Extrema = 3 l Degree = 4

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

Multiplicity l l 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.

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

Determining Multiplicity from Graphs (Pg. 265) Zero Multiplicity x-axis -1 Even Touch 2 Odd Cross 3. 5 Odd Cross

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

Hmwk: l Handout, problems 1 -8