1
00:00:00,000 --> 00:00:05,080
WOMAN: In this interactive, we will
explore graphs of quartic functions,

2
00:00:05,080 --> 00:00:08,080
that is, quartic functions
with equations of the form

3
00:00:08,080 --> 00:00:12,480
y = ax to the power of four
+ bx cubed

4
00:00:12,480 --> 00:00:16,120
+ cx squared + dx + e.

5
00:00:16,120 --> 00:00:19,680
With five unknowns,
'a', 'b', 'c', 'd' and 'e',

6
00:00:19,680 --> 00:00:23,720
there are numerous different
combinations that we could create

7
00:00:23,720 --> 00:00:26,640
and so we won't attempt to explore
every possible combination

8
00:00:26,640 --> 00:00:29,040
of 'a', 'b', 'c', 'd' and 'e',

9
00:00:29,040 --> 00:00:31,880
but we will try to look
at the different graph shapes

10
00:00:31,880 --> 00:00:36,640
that we can obtain
when graphing quartic functions.

11
00:00:36,640 --> 00:00:40,600
So currently we have
'b', 'c', 'd' and 'e' set to zero

12
00:00:40,600 --> 00:00:42,240
and 'a' set to one.

13
00:00:42,240 --> 00:00:45,520
So we're looking at the graph
of y = x to the power of 4,

14
00:00:45,520 --> 00:00:47,480
the most basic quartic function,

15
00:00:47,480 --> 00:00:51,920
and we see this
is a slightly parabolic shape

16
00:00:51,920 --> 00:00:54,280
but with a flatter base
and steeper sides.

17
00:00:54,280 --> 00:00:59,200
And we can alter this basic shape
by simple dilation,

18
00:00:59,200 --> 00:01:01,160
so that is by changing 'a',

19
00:01:01,160 --> 00:01:05,520
making it steeper
or flatter or reflecting it.

20
00:01:05,520 --> 00:01:09,760
But this essentially forms
one type of quartic function.

21
00:01:09,760 --> 00:01:12,360
We'll note that
it's a symmetric shape

22
00:01:12,360 --> 00:01:15,480
and that's to do with
the even power, obviously.

23
00:01:15,480 --> 00:01:18,600
Similarly, we can create
other symmetric quartics

24
00:01:18,600 --> 00:01:23,360
by just introducing the x squared
term into the equation as well.

25
00:01:23,360 --> 00:01:28,280
So if we, for example,
look here at the graph

26
00:01:28,280 --> 00:01:31,920
of y = x to the power
of 4 – 3x squared,

27
00:01:31,920 --> 00:01:36,640
we see a nice symmetric shape
with three turning points

28
00:01:36,640 --> 00:01:39,000
and three x-intercepts.

29
00:01:39,000 --> 00:01:44,120
Obviously, altering 'e' would simply
translate the graph up and down.

30
00:01:44,120 --> 00:01:47,360
So we could take this same shape
and translate it up or down,

31
00:01:47,360 --> 00:01:51,000
hence also being able
to create a quartic shape

32
00:01:51,000 --> 00:01:54,120
with three turning points
and four x-intercepts.

33
00:01:54,120 --> 00:01:56,440
Similarly, three turning points
and no x-intercepts

34
00:01:56,440 --> 00:01:58,960
is quite possible as well.

35
00:01:58,960 --> 00:02:04,320
Making 'a' negative here would
obviously reflect this shape.

36
00:02:04,320 --> 00:02:09,760
We would need to also make 'c'
positive there and make 'a' negative

37
00:02:09,760 --> 00:02:15,200
to get the three turning points
occurring.

38
00:02:15,200 --> 00:02:18,520
Alright, so putting this back to one.

39
00:02:25,480 --> 00:02:28,600
And then we can have many
different quartic functions

40
00:02:28,600 --> 00:02:30,520
which aren't symmetric.

41
00:02:30,520 --> 00:02:35,680
So we might, for example ... Let's
move that back to zero as well.

42
00:02:35,680 --> 00:02:40,960
We might, for example,
have something like this,

43
00:02:40,960 --> 00:02:42,840
where, in fact,
we have a quartic function

44
00:02:42,840 --> 00:02:48,520
still with only one turning point
but with two x-intercepts,

45
00:02:48,520 --> 00:02:53,160
and this sort of slightly strange
asymmetric shape.

46
00:02:53,160 --> 00:02:55,600
And that can obviously
go the other way as well.

47
00:02:55,600 --> 00:02:57,960
We can shift it up and down

48
00:02:57,960 --> 00:03:03,400
and we could reflect it and still
maintain that same kind of shape.

49
00:03:06,920 --> 00:03:10,160
Sticking with the asymmetric vein,

50
00:03:10,160 --> 00:03:14,120
we could also create
an asymmetric shape

51
00:03:14,120 --> 00:03:18,240
simply by adding in an x cubed term,

52
00:03:18,240 --> 00:03:21,920
and we see here we get
only two stationary points,

53
00:03:21,920 --> 00:03:24,120
one turning point
and one point of inflection,

54
00:03:24,120 --> 00:03:27,760
and also only two x-intercepts.

55
00:03:27,760 --> 00:03:30,440
But again, translation up or down.

56
00:03:30,440 --> 00:03:33,840
We can affect what's going on

57
00:03:33,840 --> 00:03:38,000
and maintain that same shape,
but in a different position.

58
00:03:38,000 --> 00:03:40,880
And the last kind of shape
we might get

59
00:03:40,880 --> 00:03:43,240
would be returning
to the 'W' sort of shape,

60
00:03:43,240 --> 00:03:46,680
but with some asymmetry involved.

61
00:03:46,680 --> 00:03:50,840
So if I also add in a 'c' term here,
for example ...

62
00:03:52,160 --> 00:03:53,760
... you'll see -

63
00:03:53,760 --> 00:03:57,840
let me try making 'a' bigger as well
so we can still see that there -

64
00:03:57,840 --> 00:04:00,360
you'll see we're starting to get
an extra actual turning point

65
00:04:00,360 --> 00:04:06,120
come out of that point of inflection
in our previous example,

66
00:04:06,120 --> 00:04:08,960
so we now have three turning points
occurring there

67
00:04:08,960 --> 00:04:10,800
and hence three x-intercepts.

68
00:04:10,800 --> 00:04:14,760
Similarly, a shift up or down
could make that four x-intercepts.

69
00:04:14,760 --> 00:04:17,320
And so we get lots of different
shapes that we could have.

70
00:04:17,320 --> 00:04:19,120
We can get symmetric
sort of bowl shapes

71
00:04:19,120 --> 00:04:21,200
with just one stationary point,

72
00:04:21,200 --> 00:04:24,720
we could get symmetric 'W' shapes
with three stationary points,

73
00:04:24,720 --> 00:04:28,520
we could get asymmetric shapes

74
00:04:28,520 --> 00:04:32,760
with just the one turning point,
one stationary point,

75
00:04:32,760 --> 00:04:34,560
we could get asymmetric 'W's,

76
00:04:34,560 --> 00:04:37,560
we could get asymmetric shapes
with one turning point

77
00:04:37,560 --> 00:04:39,880
and one point of inflection ...

78
00:04:39,880 --> 00:04:42,280
So there are many different
combinations that could occur

79
00:04:42,280 --> 00:04:46,240
and 'a', 'b', 'c', 'd' and 'e' here
are all quite entwined,

80
00:04:46,240 --> 00:04:50,080
although we can continue to link 'a'
with the dilations and reflections

81
00:04:50,080 --> 00:04:53,000
and 'e' with the translations
up and down.

82
00:04:53,000 --> 00:04:58,840
So there's a snapshot of what
can occur with quartic functions.