1. Graphing Polynomial Functions We will now use the strategies from the previous sections to sketch the graphs of higher order polynomials.

2. Graphing Polynomial Functions We will now use the strategies from the previous sections to sketch the graphs of higher order polynomials. Follow these steps and the process is quite easy…

3. Graphing Polynomial Functions We will now use the strategies from the previous sections to sketch the graphs of higher order polynomials. Follow these steps and the process is quite easy… 1. Using the Remainder / Factor theorem and synthetic division, factor the function completely. This will give you the roots of the function where it crosses the x – axis. It will also set up intervals in which we will test points to get a general picture of the function.

4. Graphing Polynomial Functions We will now use the strategies from the previous sections to sketch the graphs of higher order polynomials. Follow these steps and the process is quite easy… 1.Using the Remainder / Factor theorem and synthetic division, factor the function completely. This will give you the roots of the function where it crosses the x – axis. It will also set up intervals in which we will test points to get a general picture of the function. 2.Plot the roots on the graph

5. Graphing Polynomial Functions We will now use the strategies from the previous sections to sketch the graphs of higher order polynomials. Follow these steps and the process is quite easy… 1.Using the Remainder / Factor theorem and synthetic division, factor the function completely. This will give you the roots of the function where it crosses the x – axis. It will also set up intervals in which we will test points to get a general picture of the function. 2.Plot the roots on the graph 3.Create a test table using the roots of the function

6. Graphing Polynomial Functions We will now use the strategies from the previous sections to sketch the graphs of higher order polynomials. Follow these steps and the process is quite easy… 1.Using the Remainder / Factor theorem and synthetic division, factor the function completely. This will give you the roots of the function where it crosses the x – axis. It will also set up intervals in which we will test points to get a general picture of the function. 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points.

7. Graphing Polynomial Functions We will now use the strategies from the previous sections to sketch the graphs of higher order polynomials. Follow these steps and the process is quite easy… 1.Using the Remainder / Factor theorem and synthetic division, factor the function completely. This will give you the roots of the function where it crosses the x – axis. It will also set up intervals in which we will test points to get a general picture of the function. 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function

8. Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function

9. Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function

10. Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function

11. Graphing Polynomial Functions Then, using synthetic division… 1 1 6 3 - 10 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function

12. Graphing Polynomial Functions Then, using synthetic division… 1 1 6 3 - 10 + 1 1 7 7 10 0 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function

13. Graphing Polynomial Functions Then, using synthetic division… 1 1 6 3 - 10 + 1 1 7 7 10 0 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function

14. Graphing Polynomial Functions The roots of the function are the x – values that create a zero… 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function

15. Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function

16. Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function

17. Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function

18. Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function Interval Test Value ( x ) Graph - The intervals are set by the roots

19. Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function Interval Test Value ( x ) Graph - The test value will be some “x” within the interval or close to the root…

20. Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function Interval Test Value ( x ) Graph - The test value will be some “x” within the interval or close to the root…

21. Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function Interval Test Value ( x ) Graph - Now plug each test value into the function…

22. Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function Interval Test Value ( x ) negative Graph Below x - axis

23. Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function Interval Test Value ( x ) negative Graph Below x - axis - The point is off the graph, no need to plot extreme values…we will use arrows to show the “ends” of the graph

24. Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function Interval Test Value ( x ) negativepositive Graph Below x – axis Above x – axis

25. Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function Interval Test Value ( x ) negativepositive Graph Below x – axis Above x – axis Plot the point…

26. Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function Interval Test Value ( x ) negativepositivenegative Graph Below x – axis Above x – axis Below x – axis

27. Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function Interval Test Value ( x ) negativepositivenegative Graph Below x – axis Above x – axis Below x – axis Plot the point…

28. Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function Interval Test Value ( x ) negativepositivenegativepositive Graph Below x – axis Above x – axis Below x – axis Above x _ axis

29. Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function - The point is off the graph, no need to plot extreme values…we will use arrows to show the “ends” of the graph Interval Test Value ( x ) negativepositivenegativepositive Graph Below x – axis Above x – axis Below x – axis Above x _ axis

30. Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function Now sketch in the rest of the graph using smooth curves…

31. Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function

32. Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function

33. Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function

34. Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function Then use synthetic division… 1-313

35. Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function Then use synthetic division… 1-313 + 1 1 -2 -3 0

36. Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function Then use synthetic division… 1-313 + 1 1 -2 -3 0

37. Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function The roots of the function are the x – values that create a zero…

38. Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function The roots of the function are the x – values that create a zero…

39. Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function The roots of the function are the x – values that create a zero…

40. Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function Interval Test Value ( x ) Graph - The intervals are set by the roots

41. Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function Interval Test Value ( x ) Graph - The test value will be some “x” within the interval or close to the root…

42. Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function Interval Test Value ( x ) Graph - The test value will be some “x” within the interval or close to the root…

43. Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function Interval Test Value ( x ) Graph - Now plug each test value into the function…

44. Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function Interval Test Value ( x ) negative Graph Below x - axis

45. Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function Interval Test Value ( x ) negative Graph Below x - axis - The point is off the graph, no need to plot extreme values…we will use arrows to show the “ends” of the graph

46. Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function Interval Test Value ( x ) negativepositive Graph Below x - axis Above x – axis

47. Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function Plot the point… Interval Test Value ( x ) negativepositive Graph Below x - axis Above x – axis

48. Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function Interval Test Value ( x ) negativepositiveNegative Graph Below x - axis Above x – axis Below x – axis

49. Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function Interval Test Value ( x ) negativepositiveNegative Graph Below x - axis Above x – axis Below x – axis Plot the point…

50. Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function Interval Test Value ( x ) negativepositiveNegativepositive Graph Below x - axis Above x – axis Below x – axis Above x – axis

51. Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function Interval Test Value ( x ) negativepositiveNegativepositive Graph Below x - axis Above x – axis Below x – axis Above x – axis - The point is off the graph, no need to plot extreme values…we will use arrows to show the “ends” of the graph

52. Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function Now sketch in the rest of the graph using smooth curves…

53. Graphing Polynomial Functions One note here… I am choosing values in between the roots that is half way. The sketch then shows the resulting point as what would be called a relative maximum or minimum. There are functions where this is not the case. Some values to the left or right of the middle of two roots could be higher or lower.

54. Graphing Polynomial Functions One note here… I am choosing values in between the roots that is half way. The sketch then shows the resulting point as what would be called a relative maximum or minimum. There are functions where this is not the case. Some values to the left or right of the middle of two roots could be higher or lower.

55. Graphing Polynomial Functions One note here… I am choosing values in between the roots that is half way. The sketch then shows the resulting point as what would be called a relative maximum or minimum. There are functions where this is not the case. Some values to the left or right of the middle of two roots could be higher or lower.

56. Graphing Polynomial Functions One note here… I am choosing values in between the roots that is half way. The sketch then shows the resulting point as what would be called a relative maximum or minimum. There are functions where this is not the case. Some values to the left or right of the middle of two roots could be higher or lower.

57. Graphing Polynomial Functions One note here… I am choosing values in between the roots that is half way. The sketch then shows the resulting point as what would be called a relative maximum or minimum. There are functions where this is not the case. Some values to the left or right of the middle of two roots could be higher or lower.

58. Graphing Polynomial Functions One note here… I am choosing values in between the roots that is half way. The sketch then shows the resulting point as what would be called a relative maximum or minimum. There are functions where this is not the case. Some values to the left or right of the middle of two roots could be higher or lower.