Unit 2: Polynomial Functions Graphs of Polynomial Functions 2.2

1. Unit 2: Polynomial FunctionsGraphs of Polynomial Functions2.2 JMerrill 2005 Revised 2008

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

3. 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.

4. Examples of Polynomials

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

6. Cubic Parent Function Draw the parent functions on the graphs. f(x) = x3

7. Quartic Parent Function Draw the parent functions on the graphs. f(x) = x4

8. Graph and Translate Start with the graph of y = x3. Stretch it by a factor of 2 in the y direction. Translate it 3 units to the right.

9. Graph and Translate Start with the graph of y = x4. Reflect it across the x-axis. Translate it 2 units down.

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

12. f(x) = f(x) = Polynomial Quick Graphs • From yesterday’s activity: • f(x) = x2 + 2x

13. Look at the root where the graph of f(x) crossed the x-axis. What was the power of the factor? • 3 • 2 • 1

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

15. Look at each root where the graph of a function was tangent to the x-axis. What was the power of the factor? • 4 • 3 • 2 • 1

16. Describe the end behavior of a function if a > 0 and n is even. • Rise left, rise right • Fall left, fall right • Rise left, fall right • Fall left, rise right

17. Describe the end behavior of a function if a < 0 and n is even. • Rise left, rise right • Fall left, fall right • Rise left, fall right • Fall left, rise right

18. Describe the end behavior of a function if a > 0 and n is odd. • Rise left, rise right • Fall left, fall right • Rise left, fall right • Fall left, rise right

19. Describe the end behavior of a function if a < 0 and n is odd. • Rise left, rise right • Fall left, fall right • Rise left, fall right • Fall left, rise right

20. 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: • If the leading coefficient is positive, the graph falls to the left and rises on the right • If the leading coefficient is negative, the graph rises to the left and falls on the right • In an even degree polynomial: • If the leading coefficient is positive, the graph rises on the left and right • If the leading coefficient is negative, the graph falls to the left and right

21. End Behavior • If the leading coefficient of a polynomial function is positive, the graph rises to the right. y = x3 + … y = x5 + … y = x2

22. 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

23. Example • Find all zeros of f(x) = x3 – x2 – 2x • Set function = 0 0 = x3 – x2 – 2x • Factor 0 = x(x2 – x – 2) • Factor completely 0 = x(x – 2)(x + 1) • Set each factor = 0, solve 0 = x 0 = x – 2; so x = 2 0 = x +1; so x = -1

24. You Do • Find all zeros of f(x) = - 2x4 + 2x2 • X = 0, 1, -1

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

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

27. 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)(x2 – 6x + 9) • Finish multiplying: f(x) = x3 – 8x2 + 21x -18

28. You Do • Find a polynomial with roots – ½, 3, 3 • One answer might be: f(x) = 2x3 – 11x2 + 12x +9

29. 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

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

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

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

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

34. Write an equation. Roots: -3, 2 and 6 Factors: (x+3), (x-2) and (x-6) Factored Form: f(x) = (x+3)(x-2)(x-6) Polynomial Form: f(x) = (x+3)(x2 – 8x + 12) = x3 – 5x2 – 12x + 36

35. 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:

36. Gateway Problem • Sketch the graph of f(x) = x2(x – 4)(x + 3)3 Roots? Double root at x = 0 Root at x = 4 Triple root at x = -3 Degree of polynomial? 6 Rise left Rise right End Behavior?