Turning Points and X Intercepts of a Polynomial Function — Transcript

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1. 0:00Welcome to a lesson on the intercepts and  turning points of polynomial functions.
2. 0:06Sometimes these topics are referred to as  short-run behavior of polynomial functions.
3. 0:12We are given a polynomial function of degree  n. This gives us information about the graph
4. 0:20of the polynomial function. The function has at  most n X intercepts, or horizontal intercepts,
5. 0:27which means it also has at most n real zeros.  The function has at most n minus one turns.
6. 0:35Let's take a look at some examples. Here  we have a degree four polynomial function,
7. 0:42which means the graph will have at  most four X intercepts. In this case,
8. 0:47notice how the graph does have exactly one,  two, three, and four X intercepts. And again,
9. 0:55because the degree is four, the function will have  at most n minus one, or in this case, three turns.
10. 1:02Again, looking at the graph of our function,  notice how the function turns here, it turns here,
11. 1:10and it turns here. So we do have exactly n  minus one turns for this polynomial function.
12. 1:18Notice that when the graph turns, we either  have a high point or a low point on the graph,
13. 1:24which will always give us a maximum or  minimum function value at that location.
14. 1:30Let's take a look at another degree  four polynomial function. Again,
15. 1:34the degree is four, but notice in this case that  even though it is a degree four function, we
16. 1:38only have one and two X intercepts. This is  still acceptable. Remember, it says at most
17. 1:45four X intercepts, so having two intercepts  still satisfies the conditions listed above.
18. 1:51Also, notice this degree four polynomial  function only has one turn here. So it
19. 1:57doesn't have exactly n minus one turns; it  has at most n minus one turns. Therefore,
20. 2:02anything less than three turns and the polynomial  function could still be of degree four.
21. 2:08One last thing to mention here about the end  behavior: notice how the leading coefficient
22. 2:13in this first example is positive one.  Because we have an even degree and a
23. 2:18positive leading coefficient, as X moves to  the right, or approaches positive infinity,
24. 2:24or as X moves to the left and approaches  negative infinity, notice how in both cases
25. 2:29the graph is moving up. Therefore, in both  cases, F of X approaches positive infinity.
26. 2:35However, in the second case, when the degree is  even and the leading coefficient is negative,
27. 2:43as X approaches the right or approaches positive
28. 2:45infinity, and as X approaches the left or  approaches negative infinity, notice in both
29. 2:50cases the graph is going down. Therefore, F of X  is approaching negative infinity in both cases.
30. 2:57Now, let's take a look at two examples  of a polynomial function that has an odd
31. 3:01degree. Here we have a degree five polynomial  function, which means we can have at most five
32. 3:08X intercepts. Let's see how many we have. We  have one, two, three, four, and five, so we
33. 3:17have exactly five X intercepts, which is the most  we can have for a degree five polynomial function.
34. 3:23Now let's see how many turns we have. We have a  turn here, one, two, three, and four. Remember,
35. 3:32we can have at most n minus one turns. Five  minus one is equal to four. So in this case,
36. 3:38we have the most possible turns for  a degree five polynomial function.
37. 3:42For a second example of a degree five polynomial  function, again we can have at most five X
38. 3:49intercepts and four turns. Notice in this case  we only have one, two, and three X intercepts
39. 3:56and one, two turns. So if we didn't have the  equation for this polynomial function, we
40. 4:03might think this function would only be a degree  three polynomial function, which is possible. A
41. 4:08degree three polynomial function could have the  same short-run behavior as this function here.
42. 4:14However, because we have the equation, we  know it is a degree five polynomial function.
43. 4:18Let's finish by talking about the end behavior  of these odd degree polynomial functions. Notice
44. 4:24how for this first example, the leading  coefficient is negative one. Whenever
45. 4:28we have an odd degree polynomial function  and the leading coefficient is negative,
46. 4:33as X moves to the right or as  X approaches positive infinity,
47. 4:37notice how the function is going down. F of X  is always going to approach negative infinity.
48. 4:43As X approaches the left or approaches  negative infinity, notice how the graph
49. 4:48goes up. So as X approaches negative infinity,  F of X will always approach positive infinity.
50. 4:54It's just the opposite when the degree  is odd and the leading coefficient
51. 4:58is positive. As X approaches positive  infinity, notice how the graph goes up,
52. 5:04so F of X approaches positive infinity. As X  approaches negative infinity, or as X moves left,
53. 5:11notice how the function goes down. So  F of X approaches negative infinity.
54. 5:17We had a whole other lesson on end  behavior, or long-run behavior,
55. 5:21but I wanted to include this as a review.
56. 5:24Okay, I hope you found this lesson helpful.