2. Unit 2: Polynomial Functions Graphs of Polynomial Functions 2.2 JMerrill 2005 Revised 2008

3. Learning Goal To find zeros and use transformations to sketch graphs of polynomial functions To use the Leading Coefficient Test to determine end behavior

4. Significant features The graphs of polynomial functions are continuous (no breaks—you draw the entire graph without lifting your pencil). This is opposed to discontinuous functions (remember piecewise functions?). This data is continuous as opposed to discrete.

5. Examples of Polynomials DegreeNameExample 0Constant5 1Linear3x+2 2QuadraticX 2 – 4 3CubicX 3 + 3x + 1 4Quartic-3x 4 + 4 5QuinticX 5 + 5x 4 - 7

6. Significant features The graph of a polynomial function has only smooth turns. A function of degree n has at most n – 1 turns. −A−A 2 nd degree polynomial has 1 turn −A−A 3 rd degree polynomial has 2 turns −A−A 5 th degree polynomial has…

7. Cubic Parent Function Draw the parent functions on the graphs. f(x) = x 3 XY -3-27 -2-8 00 11 28 327

8. XY -381 -216 1 00 11 216 381 Quartic Parent Function Draw the parent functions on the graphs. f(x) = x 4

9. Graph and Translate Start with the graph of y = x 3. Stretch it by a factor of 2 in the y direction. Translate it 3 units to the right. XY -3-27 -2-8 00 11 28 327 XY 0-54 1-16 2-2 30 42 516 654

10. XY -381 -216 1 00 11 216 381 Graph and Translate Start with the graph of y = x 4. Reflect it across the x-axis. Translate it 2 units down. XY -3-83 -2-18 -3 0-2 1-3 2-18 3-83 XY -3-81 -2-16 00 1 2-16 3-81

12. Max/Min A parabola has a maximum or a minimum Any other polynomial function has a local max or a local min. (extrema) Local max Local min max min

13. Polynomial Quick Graphs From yesterday’s activity: f(x) = x 2 + 2x f(x) =

14. Look at the root where the graph of f(x) crossed the x-axis. What was the power of the factor? A.3 B.2 C.1

15. Look at each root where the graph of a function“wiggled at” the x-axis. Were the powers even or odd? A.Even B.Odd

16. Look at each root where the graph of a function was tangent to the x-axis. What was the power of the factor? A.4 B.3 C.2 D.1

17. Describe the end behavior of a function if a > 0 and n is even. A.Rise left, rise right B.Fall left, fall right C.Rise left, fall right D.Fall left, rise right

18. Describe the end behavior of a function if a < 0 and n is even. A.Rise left, rise right B.Fall left, fall right C.Rise left, fall right D.Fall left, rise right

19. Describe the end behavior of a function if a > 0 and n is odd. A.Rise left, rise right B.Fall left, fall right C.Rise left, fall right D.Fall left, rise right

20. Describe the end behavior of a function if a < 0 and n is odd. A.Rise left, rise right B.Fall left, fall right C.Rise left, fall right D.Fall left, rise right

21. Leading Coefficient Test As x moves without bound to the left or right, the graph of a polynomial function eventually rises or falls like this: In an odd degree polynomial: −I−If the leading coefficient is positive, the graph falls to the left and rises on the right −I−If the leading coefficient is negative, the graph rises to the left and falls on the right In an even degree polynomial: −I−If the leading coefficient is positive, the graph rises on the left and right −I−If the leading coefficient is negative, the graph falls to the left and right

22. End Behavior If the leading coefficient of a polynomial function is positive, the graph rises to the right. y = x 2 y = x 3 + … y = x 5 + …

23. Finding Zeros of a Function If f is a polynomial function and a is a real number, the following statements are equivalent: x = a is a zero of the function x = a is a solution of the polynomial equation f(x)=0 (x - a) is a factor of f(x) (a, 0) is an x-intercept of f

24. Example Find all zeros of f(x) = x 3 – x 2 – 2x Set function = 00 = x 3 – x 2 – 2x Factor0 = x(x 2 – x – 2) Factor completely0 = x(x – 2)(x + 1) Set each factor = 0, solve0 = x 0 = x – 2; so x = 2 0 = x +1; so x = -1

25. You Do Find all zeros of f(x) = - 2x 4 + 2x 2 X = 0, 1, -1

26. Multiplicity (repeated zeros) How many roots?How many roots? 3 roots; x = 1, 3, 3. 4 roots; x = 1, 3, 3, 4. 3 is a double root. It is tangent to the x- axis

27. Roots of Polynomials How many roots? How many roots? 5 roots: x = 0, 0, 1, 3, 3. 0 and 3 are double roots 3 roots; x = 2, 2, 2 Double roots Double roots (tangent ) Triple root – lies flat then crosses axis (wiggles)

28. Given Roots, Find a Polynomial Function There are many correct solutions. Our solutions will be based only on the factors of the given roots: Ex: Find a polynomial function with roots 2, 3, 3 Turn roots into factors: f(x) = (x – 2)(x – 3)(x – 3) Multiply factors: f(x) = (x – 2)(x 2 – 6x + 9) Finish multiplying: f(x) = x 3 – 8x 2 + 21x -18

29. You Do Find a polynomial with roots – ½, 3, 3 One answer might be: f(x) = 2x 3 – 11x 2 + 12x +9

30. Sketch graph f(x) = (x - 4)(x - 1)(x + 2) Step 1: Find zeros. Step 2: Mark the zeros on a number line. Step 3: Determine end behavior Step 4: Sketch the graph Fall left, rise right

31. Sketch graph f(x)= -(x-4)(x-1)(x+2)

32. You Do f(x) = (x+1) 2 (x-2)

33. You Do f(x) = - (x-4) 3

34. Sketch graph. f(x) = (x-2) 2 (x+3)(x+2) roots: -3, -2 and 2 Rise left, rise right

35. Roots: -3, 2 and 6 Factors: (x+3), (x-2) and (x-6) Factored Form: f(x) = (x+3)(x-2)(x-6) Write an equation. Polynomial Form: f(x) = (x+3)(x 2 – 8x + 12) = x3 x3 – 5x 2 – 12x + 36

36. Write equation. Zeros: -2, -1, 3 and 5 Factors: (x+2), (x+1), (x-3) and (x-5) Factored Form: f(x) = (x + 2)(x + 1)(x – 3)(x – 5) Polynomial Form:

37. Gateway Problem Sketch the graph of f(x) = x 2 (x – 4)(x + 3) 3 Double root at x = 0 Root at x = 4 Triple root at x = -3 Roots? Degree of polynomial? 6 End Behavior? Rise left Rise right