1. Using Multiplicity and Graphs to Determine Equations 8 November 2010

2. Multiplicity Review 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. y = (x + 1) 3 (x – 1)(x – 3) 2 RootLinear FactorMultiplicityX-Axis

4. Determining Multiplicity from Graphs The graph above is a 5 th degree polynomial. RootLinear Factor MultiplicityX-Axis

5. Your Turn: For problems 15-18 on page 270 of your Precalculus textbook, complete a “multiplicity table”. RootLinear FactorMultiplicityX-Axis

6. Determining Equations of Graphs Using Multiplicity We can use multiplicity to determine the equations of graphs. We need: a graph of the polynomial the degree of the polynomial

7. Determining Equations of Graphs Using Multiplicity, cont. Step 1: Complete a “multiplicity table” for the graph. Step 2: Write the polynomial as the product of linear factors. Step 3: Determine the exponent for each factor using the degree of the polynomial and the “multiplicity table”. Step 3: Expand the polynomial. Step 4: Check the degree.

8. Determining Multiplicity from Graphs The graph above is a 5 th degree polynomial. RootLinear Factor MultiplicityX-Axis (x + 1)EvenTouch 1(x – 1)OddCross 2(x – 2)EvenTouch

9. Determining Equations of Graphs Using Multiplicity, cont. Remember, the graph is a 5 th degree polynomial. RootLinear Factor MultiplicityX- Axis (x + 1)EvenTouch 1(x – 1)OddCross 2(x – 2)EvenTouch

10. Determining Equations of Graphs Using Multiplicity, cont. (x + 1) 2 (x – 1)(x – 3) 2 (x + 1) 2 = (x + 1)(x + 1) (x – 3) 2 = (x – 3)(x – 3)

11. Determining Equations of Graphs Using Multiplicity, cont. (x + 1) 2 (x – 1)(x – 3) 2 (x 2 + 2x + 1)(x – 1) * (x 2 – 6x + 9)

12. Determining Equations of Graphs Using Multiplicity, cont. The graph above is a 5 th degree polynomial.

13. Your Turn: For problems 15-18 on page 270 of your Precalculus textbook, use your multiplicity tables to determine the expanded (or standard) form of the polynomial. 15. 3 rd degree16. 5 th degree 17. 4 th degree18. 4 th degree

14. Homework In Precalculus textbook, pg. 249: 51 – 54 Pg. 272: 43 – 44 Plus – Determine the equation of the polynomial using your answer from part d.