1
00:00:01,200 --> 00:00:02,840
NARRATOR: Exercise 10.

2
00:00:02,840 --> 00:00:07,600
Sketch the graph of
y = x to the power of 4 – x squared.

3
00:00:07,600 --> 00:00:10,240
I'm first going to calculate
the intercepts.

4
00:00:10,240 --> 00:00:12,440
So we'll start with the y-intercept,

5
00:00:12,440 --> 00:00:15,840
which is obtained by
making x equal to 0,

6
00:00:15,840 --> 00:00:18,760
so we end up with y = 0.

7
00:00:18,760 --> 00:00:21,120
So we have a y-intercept at (0, 0).

8
00:00:21,120 --> 00:00:24,040
Then we'll consider the x-intercepts.

9
00:00:24,040 --> 00:00:27,680
And we're expecting to get
one x-intercept at 0,

10
00:00:27,680 --> 00:00:30,720
given that we know the graph
goes through (0, 0) already.

11
00:00:30,720 --> 00:00:33,840
So x-intercepts are obtained
by making y equal to 0.

12
00:00:33,840 --> 00:00:37,800
So 0 = x to the power of 4
minus x squared.

13
00:00:37,800 --> 00:00:40,360
x squared is a common factor there

14
00:00:40,360 --> 00:00:44,080
so we have x squared (x squared – 1).

15
00:00:44,080 --> 00:00:49,360
And then using the null factor law,
we know that x squared = 0, so x = 0,

16
00:00:49,360 --> 00:00:53,040
and we also know that
x squared – 1 equals 0

17
00:00:53,040 --> 00:00:55,280
so x squared = 1.

18
00:00:55,280 --> 00:00:59,040
So x = positive or negative
square root of 1,

19
00:00:59,040 --> 00:01:00,840
which is positive or negative 1.

20
00:01:00,840 --> 00:01:03,120
So we know we have x-intercepts at

21
00:01:03,120 --> 00:01:07,760
(0, 0), (1, 0) and (-1, 0).

22
00:01:07,760 --> 00:01:09,680
We also know due to

23
00:01:09,680 --> 00:01:11,760
the repeated factor of x squared

24
00:01:11,760 --> 00:01:13,640
that we have a turning point

25
00:01:13,640 --> 00:01:15,160
at this intercept.

26
00:01:16,960 --> 00:01:20,320
So we're starting to get a sense
of the shape of the graph.

27
00:01:20,320 --> 00:01:22,120
We know that it has x-intercepts

28
00:01:22,120 --> 00:01:25,040
at –1, at 0 and at 1.

29
00:01:25,040 --> 00:01:27,040
We also know that
it's a positive quartic

30
00:01:27,040 --> 00:01:31,200
due to the + x to the power of four
term in the equation,

31
00:01:31,200 --> 00:01:34,920
which means that the graph is going
to start up and also finish up.

32
00:01:34,920 --> 00:01:38,400
So we're getting
this sort of a shape occurring.

33
00:01:44,320 --> 00:01:46,200
So something like that.

34
00:01:46,200 --> 00:01:47,920
So now we need to also determine

35
00:01:47,920 --> 00:01:50,160
the coordinates of these
two stationary points.

36
00:01:50,160 --> 00:01:53,080
We already know that we have the
stationary point here at (0, 0),

37
00:01:53,080 --> 00:01:57,160
so we'll expect to find that, but we
need to find the other two as well.

38
00:01:58,360 --> 00:02:00,200
So for the stationary points,

39
00:02:00,200 --> 00:02:04,320
we know that stationary points occur
where the derivative is 0.

40
00:02:04,320 --> 00:02:07,840
So we first need to find
the derivative of this function.

41
00:02:07,840 --> 00:02:15,600
So dy on dx is equal to
4x cubed – 2x.

42
00:02:15,600 --> 00:02:19,200
And for stationary points, we need
that derivative to equal 0.

43
00:02:19,200 --> 00:02:21,360
So we have a common factor of 2 here

44
00:02:21,360 --> 00:02:23,360
that we're going to
divide through by,

45
00:02:25,440 --> 00:02:28,760
And then we have a common factor
of x that we can factor out.

46
00:02:31,080 --> 00:02:32,880
So now using the null factor law,

47
00:02:32,880 --> 00:02:38,480
we know that either x = 0
or 2x squared – 1 = 0.

48
00:02:38,480 --> 00:02:41,600
So 2x squared = 1.

49
00:02:41,600 --> 00:02:43,880
x squared equals a half.

50
00:02:43,880 --> 00:02:46,240
So we know that x is equal to

51
00:02:46,240 --> 00:02:48,840
the positive or negative
square root of a half,

52
00:02:48,840 --> 00:02:52,320
which is positive or negative 1
on root 2,

53
00:02:52,320 --> 00:02:56,880
which is equivalent to positive
or negative root 2 on 2.

54
00:02:56,880 --> 00:03:00,200
So we need to work out
the corresponding y-coordinates.

55
00:03:00,200 --> 00:03:04,520
We already know that
the turning point at 0

56
00:03:04,520 --> 00:03:06,600
has coordinates of (0, 0),

57
00:03:06,600 --> 00:03:09,080
but let's think about
these other two.

58
00:03:09,080 --> 00:03:12,040
So we know that when x =...

59
00:03:12,040 --> 00:03:14,360
And I'm gonna substitute
square root of a half

60
00:03:14,360 --> 00:03:16,600
because that's gonna make
the simplification easier.

61
00:03:16,600 --> 00:03:18,720
When x = square root of a half,

62
00:03:18,720 --> 00:03:22,640
y is equal to square root
of half to the power of four

63
00:03:22,640 --> 00:03:26,320
minus square root of half squared.

64
00:03:27,440 --> 00:03:32,000
So that's half squared minus half.

65
00:03:33,880 --> 00:03:36,600
So a quarter minus a half,

66
00:03:36,600 --> 00:03:38,960
which means we have
negative a quarter.

67
00:03:38,960 --> 00:03:42,000
Which matches the picture
that we saw earlier as well.

68
00:03:42,000 --> 00:03:44,400
So we also know
we need to think about

69
00:03:44,400 --> 00:03:47,080
substituting
negative square root of a half,

70
00:03:47,080 --> 00:03:49,640
but we're not actually going to get
any different result.

71
00:03:49,640 --> 00:03:52,360
So I know if I had
negative square root of a half here,

72
00:03:52,360 --> 00:03:53,360
that would give me

73
00:03:53,360 --> 00:03:56,360
negative square root of a half and
negative square root of a half there.

74
00:03:56,360 --> 00:03:59,880
And then because of these even
powers - power of 4 and power of 2 -

75
00:03:59,880 --> 00:04:02,040
those negatives aren't
going to make any difference

76
00:04:02,040 --> 00:04:04,960
and we'll still end up with
a y-coordinate at negative a quarter.

77
00:04:04,960 --> 00:04:08,920
So that means that we have
turning points with coordinates

78
00:04:08,920 --> 00:04:11,560
(root 2 on 2, negative a quarter)

79
00:04:11,560 --> 00:04:17,240
and also (negative root 2 on 2,
negative one quarter).

80
00:04:18,760 --> 00:04:21,000
So we have three stationary points.

81
00:04:21,000 --> 00:04:22,920
So now we can finalise our graph.

82
00:04:22,920 --> 00:04:27,880
So here's our graph with our
intercepts at –1, 0 and 1.

83
00:04:27,880 --> 00:04:32,040
We now also have coordinates
for these two stationary points.

84
00:04:32,040 --> 00:04:36,160
This is (root 2 on 2,
negative one quarter).

85
00:04:36,160 --> 00:04:41,520
And this is (negative root 2 on 2,
negative one quarter).

86
00:04:41,520 --> 00:04:45,880
And we can fill this one in as well
with coordinates of (0, 0).

87
00:04:45,880 --> 00:04:51,280
And there is the graph of
y = x to the power of 4 – x squared.