1
00:00:05,110 --> 00:00:16,770
Is this a quadratic function? Yes. How can we tell? We multiply this out. The

2
00:00:16,770 --> 00:00:20,492
x times the x becomes the x squared. Definitely quadratic. You multiply any two

3
00:00:20,492 --> 00:00:21,010
linear, you

4
00:00:21,010 --> 00:00:30,651
get a quadratic. You multiply three linear, what do you get? So our goal is to graph

5
00:00:30,651 --> 00:00:31,110
this

6
00:00:31,110 --> 00:00:39,386
function just like we were graphing the quadratic functions. This is non-standard

7
00:00:39,386 --> 00:00:40,510
form. Is it

8
00:00:40,510 --> 00:00:51,790
in vertex form? No. Uh-oh. So how are we going to get it in one of those forms? We

9
00:00:51,790 --> 00:00:52,890
just got

10
00:00:52,890 --> 00:01:15,990
Multiply it out: x squared plus three x minus x minus three.

11
00:01:16,350 --> 00:01:22,350
Now it is in standard form. Standard form tells us two things. What are they?

12
00:01:22,350 --> 00:01:35,090
The y-intercept. We definitely know the y-intercept. And whether it

13
00:01:35,090 --> 00:01:45,365
opens up or opens down, which one is this? Open up because what's positive? Leading

14
00:01:45,365 --> 00:01:46,850
coefficient.

15
00:01:47,250 --> 00:01:49,170
Okay, so if we want to find

16
00:01:49,810 --> 00:01:57,960
vertex form, how are we going to do that? You could complete the square. The better

17
00:01:57,960 --> 00:02:00,710
way is to know the vertex is

18
00:02:00,710 --> 00:02:12,965
always at negative b over 2a comma f of negative b over 2a. Negative b over 2a is,

19
00:02:12,965 --> 00:02:17,150
what is b? Is 2. And a is 1.

20
00:02:19,570 --> 00:02:30,590
So that is negative one. That is the x-coordinate of the vertex.

21
00:02:32,450 --> 00:02:36,170
And how do we find the y coordinate? Plug it in to the function.

22
00:02:36,460 --> 00:02:42,903
Oh, no, that's right. I'm sorry. I got confused. Negative 4. We agree? Yes. Okay,

23
00:02:42,903 --> 00:02:43,380
sorry.

24
00:02:45,700 --> 00:02:49,900
So that is the y coordinate of the vertex.

25
00:02:52,840 --> 00:03:04,520
So what is our vertex form? The leading coefficient is one.

26
00:03:05,740 --> 00:03:16,440
Times x minus this number, which is x plus 1 squared minus 4.

27
00:03:20,060 --> 00:03:25,969
So the vertex (negative one, negative four) is right there. To graph it, follow the

28
00:03:25,969 --> 00:03:27,820
pattern: over one, up one;

29
00:03:28,160 --> 00:03:31,200
over two, up four.

30
00:03:31,200 --> 00:03:33,380
I'm just doing that because it's on the wrong side of the pattern.

31
00:03:33,610 --> 00:03:42,510
Over 1 up 1, over 2 up 1, 2, 3, 4. 1, 2, 3, up 9. Up here.

32
00:03:44,270 --> 00:03:47,030
That's going to be the right hand side. And how do we get the left hand side?

33
00:03:47,030 --> 00:03:53,830
On the other side. According to the axis of symmetry, which goes right through the

34
00:03:53,830 --> 00:03:54,410
vertex.

35
00:03:57,430 --> 00:04:00,450
And then we draw a nice curve.

36
00:04:17,030 --> 00:04:18,630
What are the x-intercepts?

37
00:04:20,810 --> 00:04:23,590
Negative three and one.

38
00:04:24,050 --> 00:04:27,390
Going back to the original quadratic.

39
00:04:30,870 --> 00:04:33,810
How can we look at that and tell the x-intercepts?

40
00:04:37,250 --> 00:04:45,050
Uh, we could use a zero product property.

41
00:04:47,450 --> 00:04:52,886
But it is simply the opposite of that number and the opposite of that number. Why is

42
00:04:52,886 --> 00:04:53,210
that?

43
00:04:56,670 --> 00:04:59,910
Yeah, right. We plug in the opposite of negative 1, which is 1.

44
00:05:01,450 --> 00:05:04,070
And you get 0 times whatever. It doesn't matter. It's 0.

45
00:05:07,050 --> 00:05:15,124
And the opposite of 3. So we plug in negative 3 plus 3. We get 0 times whatever.

46
00:05:15,124 --> 00:05:15,830
It's 0.

47
00:05:16,930 --> 00:05:22,630
Okay. So this is a special form.

48
00:05:25,330 --> 00:05:31,910
And it tells us something right away. What form is this? Oh, can anyone take this?

49
00:05:31,910 --> 00:05:36,470
Factored form, because the polynomial is factored.

50
00:05:38,030 --> 00:05:39,970
Okay. So this is called factored form.

51
00:05:45,770 --> 00:05:55,910
Okay. And what does it tell you? Those are the x-intercepts.

52
00:05:59,230 --> 00:06:03,490
Also known as zeros. I'll define that for you in a minute.

53
00:06:09,990 --> 00:06:15,190
Okay. If we have the x-intercepts

54
00:06:20,230 --> 00:06:25,908
without doing this negative b over 2a, how can we use the x-intercepts to figure out

55
00:06:25,908 --> 00:06:26,990
the x-coordinate

56
00:06:26,990 --> 00:06:28,390
of the vertex?

57
00:06:28,610 --> 00:06:30,510
We find the mean of these two?

58
00:06:30,930 --> 00:06:36,761
We find the mean, the average of the two, exactly halfway. Okay. Why is it exactly

59
00:06:36,761 --> 00:06:37,330
halfway?

60
00:06:41,390 --> 00:06:46,070
Because they are split about the axis of symmetry.

61
00:06:46,070 --> 00:06:52,750
Any two points on this parabola on the same horizontal line are symmetric.

62
00:06:52,750 --> 00:07:00,470
Symmetric ordered pairs. So if you find two such points,

63
00:07:03,230 --> 00:07:07,770
especially when they, like let's say they landed right here and here or something,

64
00:07:08,470 --> 00:07:14,592
where we couldn't know the zeros, we'll just go halfway. Exactly as the axis of

65
00:07:14,592 --> 00:07:15,290
symmetry,

66
00:07:15,750 --> 00:07:19,430
which is the x-coordinate of the vertex. So...

67
00:07:22,610 --> 00:07:24,810
So that's an alternative way,

68
00:07:24,960 --> 00:07:31,180
To find this value. And once we have that value, we just plug it into this function

69
00:07:31,180 --> 00:07:36,600
or this function to figure it out, the y-coordinate.

70
00:07:38,980 --> 00:07:42,180
So, right, if we plug in what's halfway? Negative one.

71
00:07:43,200 --> 00:07:48,760
Negative one minus one times negative one plus three, which is

72
00:07:53,840 --> 00:07:58,540
negative two times two, which is our negative four, our y-coordinate of the vertex.

73
00:07:58,540 --> 00:08:06,860
Okay, so that is factored form. So factored form

74
00:08:17,420 --> 00:08:23,573
Factored form is f of x equals the leading coefficient a, the same a as in vertex

75
00:08:23,573 --> 00:08:26,080
and standard forms, times x minus

76
00:08:26,960 --> 00:08:34,360
z one, times x minus z two, where

77
00:08:36,960 --> 00:08:47,360
z one and z two are the zeros, the x-coordinates of the intercepts.

78
00:08:51,620 --> 00:08:54,720
Okay, let me define the word zero for it.

79
00:08:54,910 --> 00:08:55,850
a zero

80
00:09:00,330 --> 00:09:01,550
of a function

81
00:09:05,130 --> 00:09:08,230
is an x value

82
00:09:11,690 --> 00:09:13,790
that makes

83
00:09:16,230 --> 00:09:23,070
the y-value, or function value,

84
00:09:31,150 --> 00:09:36,370
equal to zero, the number zero.

85
00:09:44,630 --> 00:09:49,630
It is the same as the x-coordinate of an x-intercept. Sometimes

86
00:09:54,350 --> 00:10:01,785
we could say the x-intercepts are negative three and one, or you can actually talk

87
00:10:01,785 --> 00:10:03,870
about the ordered pair.

88
00:10:13,790 --> 00:10:20,333
Typically, the x-intercept is actually an ordered pair, but we often just refer to

89
00:10:20,333 --> 00:10:21,530
it as a number.

90
00:10:25,430 --> 00:10:30,667
But these would be the zeros of the function. Zeros are numbers, x values that make

91
00:10:30,667 --> 00:10:31,550
the y value or

92
00:10:31,750 --> 00:10:32,490
the function value zero.

93
00:10:32,700 --> 00:10:37,380
What form is this in? Factored form.

94
00:10:46,700 --> 00:10:49,260
Okay, so what do we instantly know?

95
00:10:49,360 --> 00:10:51,360
What do we know from this form?

96
00:10:52,960 --> 00:10:53,520
The zeros.

97
00:10:53,820 --> 00:10:54,480
Zeros.

98
00:10:55,980 --> 00:11:00,240
Okay, also known as the x-intercepts.

99
00:11:03,620 --> 00:11:09,634
The zeros are the opposite of that number and the opposite of that number, two,

100
00:11:09,634 --> 00:11:10,700
negative four.

101
00:11:16,520 --> 00:11:18,200
Two, negative four.

102
00:11:20,360 --> 00:11:21,600
Okay.

103
00:11:23,160 --> 00:11:28,540
And then the vertex, the x-coordinate of the vertex,

104
00:11:34,240 --> 00:11:38,160
of the vertex is...

105
00:11:42,520 --> 00:11:50,403
Right, it's z one plus z two divided by two, which is two plus negative four divided

106
00:11:50,403 --> 00:11:51,060
by two.

107
00:11:51,860 --> 00:11:57,940
Here that is negative two over two, which is negative one.

108
00:12:01,140 --> 00:12:04,260
So we know this is the axis of symmetry.

109
00:12:11,320 --> 00:12:16,980
And then how would we find the y-coordinate of the vertex?

110
00:12:17,640 --> 00:12:22,880
Plug it in. We don't even need to multiply this out.

111
00:12:24,100 --> 00:12:32,307
So f of negative one is two times (negative one minus two) times (negative one plus

112
00:12:32,307 --> 00:12:32,900
four),

113
00:12:32,900 --> 00:12:39,460
which is two times negative three times three, which is

114
00:12:44,040 --> 00:12:49,860
negative eighteen. Uh-oh. So the only problem with this is way off my graph.

115
00:12:52,280 --> 00:13:00,800
Way down here at negative eighteen. Sorry, negative one comma negative eighteen.

116
00:13:07,960 --> 00:13:14,200
It's going to be a little harder to graph because I don't have a graph paper right

117
00:13:14,200 --> 00:13:14,580
here.

118
00:13:19,180 --> 00:13:25,655
But we would follow our pattern over... We're not going to go over one up one. Where

119
00:13:25,655 --> 00:13:27,120
are we going to go?

120
00:13:28,340 --> 00:13:38,760
Over one, up two. The parent offsets are (0,0), (1,1), (2,4), and (3,9).

121
00:13:42,260 --> 00:13:49,700
Our new pattern, instead of x and x squared, uses x and two x squared.

122
00:13:51,000 --> 00:13:56,040
Multiply those vertical offsets by two.

123
00:14:01,140 --> 00:14:04,640
Two, eight, eighteen.

124
00:14:07,680 --> 00:14:10,040
So I go over one up two.

125
00:14:13,860 --> 00:14:18,140
Over three up... Sorry, over two up eight.

126
00:14:20,060 --> 00:14:23,540
It's going to be right here at ten. I didn't draw this very well.

127
00:14:25,000 --> 00:14:28,880
Negative ten. And over three, up eighteen reaches the zeros right there.

128
00:14:30,460 --> 00:14:32,200
And then we got some symmetric points.

129
00:14:37,360 --> 00:14:39,020
We get a nice smooth curve.

130
00:14:52,860 --> 00:14:53,680
Okay.

131
00:14:53,680 --> 00:14:57,580
Okay. So what is the...

132
00:15:01,420 --> 00:15:02,800
What is the vertex form?

133
00:15:09,640 --> 00:15:11,040
Two...

134
00:15:11,040 --> 00:15:17,280
What was the vertex?

135
00:15:20,120 --> 00:15:23,360
Was it negative one...

136
00:15:25,240 --> 00:15:25,740
Negative eighteen.

137
00:15:26,460 --> 00:15:35,160
It is two times (x plus one) squared, minus eighteen.

138
00:15:37,320 --> 00:15:42,720
Okay. And then how would we get standard forms?

139
00:15:45,140 --> 00:15:49,523
We could either multiply this out, multiply it by two, and then subtract 18, or it's

140
00:15:49,523 --> 00:15:49,940
probably

141
00:15:49,940 --> 00:16:01,420
easier right here: two times (x squared plus four x minus two x minus eight).

142
00:16:01,820 --> 00:16:05,440
Which form should we... Let's go with the factored form first.

143
00:16:15,720 --> 00:16:22,720
Factored form is the leading coefficient times x minus one zero, times

144
00:16:22,720 --> 00:16:28,640
x minus the other zero. What are the zeros?

145
00:16:30,680 --> 00:16:32,280
Negative one, negative three.

146
00:16:32,280 --> 00:16:39,140
Negative one and negative three. So we have x minus negative one, or x plus one,

147
00:16:39,560 --> 00:16:43,338
and x minus negative three, or x plus three. How do we determine the leading

148
00:16:43,338 --> 00:16:44,680
coefficient from the graph?

149
00:16:49,900 --> 00:16:53,040
You start at the vertex. We go one to the right.

150
00:16:55,140 --> 00:17:00,740
And since we go down two, we know the leading coefficient is negative two.

151
00:17:04,860 --> 00:17:08,760
We can go right to the vertex form.

152
00:17:15,160 --> 00:17:21,080
The leading coefficient is negative two, times

153
00:17:24,440 --> 00:17:30,080
(x plus two) squared, plus two. Lots of twos.

154
00:17:33,480 --> 00:17:39,080
And then standard form. We know it's going to be

155
00:17:40,860 --> 00:17:44,420
a coefficient times x squared, plus or minus a coefficient times x.

156
00:17:45,160 --> 00:17:47,640
We do know the y-intercept, which is what? Negative six.

157
00:17:50,320 --> 00:17:53,720
But you need to multiply this out. So we got x squared

158
00:17:57,440 --> 00:18:06,320
negative two times x squared plus three x plus x plus three.

159
00:18:09,460 --> 00:18:14,260
Three x plus x is four x. Distributing negative two gives negative eight x and

160
00:18:14,260 --> 00:18:15,060
negative six.

161
00:18:19,380 --> 00:18:20,180
And that is standard form.

162
00:18:21,150 --> 00:18:23,190
What two things do we know?

163
00:18:29,530 --> 00:18:35,133
It opens upward because the leading coefficient is positive, and its y-intercept is

164
00:18:35,133 --> 00:18:36,010
negative one.

165
00:18:40,590 --> 00:18:46,910
How can we get this into factored form? One option is to factor it.

166
00:18:50,470 --> 00:18:51,450
Does it factor?

167
00:18:54,150 --> 00:18:59,070
The integer factors are one and negative one; those do not add to two.

168
00:19:00,550 --> 00:19:02,590
So how are we going to factor it?

169
00:19:03,790 --> 00:19:06,870
Find the zeros. We need to determine the zeros.

170
00:19:11,150 --> 00:19:13,030
How do we figure out the zeros

171
00:19:18,730 --> 00:19:21,710
Okay, we could do that, but I have a feeling

172
00:19:26,770 --> 00:19:31,390
They will not be obvious integers. This is just a rough sketch.

173
00:19:33,150 --> 00:19:35,490
Okay to find any zeros of a function

174
00:19:37,230 --> 00:19:40,210
You set the function equal to zero

175
00:19:43,030 --> 00:19:48,010
So we go zero equals x squared plus 2x minus 1

176
00:19:54,410 --> 00:19:57,670
And now we have a quadratic equation in one variable

177
00:20:01,450 --> 00:20:04,290
It cannot be solved by integer factoring. What is the next option?

178
00:20:05,430 --> 00:20:06,510
Use the quadratic formula.

179
00:20:07,210 --> 00:20:10,570
We sing it all together. Yes, we ready. Yes

180
00:20:13,030 --> 00:20:13,990
Negative b,

181
00:20:16,030 --> 00:20:20,750
the square root of b squared minus four a c.

182
00:20:20,750 --> 00:20:31,990
Negative two plus or minus the square root of two squared minus four times one times

183
00:20:32,910 --> 00:20:34,390
negative 1

184
00:20:35,430 --> 00:20:36,970
All over two.

185
00:20:52,890 --> 00:20:53,930
And

186
00:20:53,930 --> 00:20:58,770
The square root of eight is two square roots of two.

187
00:21:05,290 --> 00:21:09,770
Either factor out a two, or

188
00:21:10,770 --> 00:21:16,090
divide negative two by two and divide plus or minus two square roots of two by two.

189
00:21:27,250 --> 00:21:31,490
Those are the zeros, the x-coordinates of the intercepts.

190
00:21:41,110 --> 00:21:41,950
Oh

191
00:21:41,950 --> 00:21:43,730
So how do we put it in factored form?

192
00:21:44,770 --> 00:21:47,650
Well, do we know the leading coefficient which is

193
00:21:48,290 --> 00:21:48,830
one

194
00:21:49,810 --> 00:21:52,310
These are two numbers: negative one plus the square root of two

195
00:21:53,850 --> 00:21:54,690
and

196
00:21:54,690 --> 00:21:56,830
and negative one minus the square root of two.

197
00:21:58,370 --> 00:22:00,330
Factored form is

198
00:22:01,130 --> 00:22:03,530
One times x minus

199
00:22:04,450 --> 00:22:07,570
(negative one plus the square root of two),

200
00:22:08,570 --> 00:22:13,770
times x minus (negative one minus the square root of two).

201
00:22:17,250 --> 00:22:19,250
Distribute the signs to simplify.

202
00:22:20,370 --> 00:22:22,590
Get x

203
00:22:23,570 --> 00:22:26,070
plus one minus the square root of two;

204
00:22:27,250 --> 00:22:29,370
X plus 1

205
00:22:30,370 --> 00:22:31,070
Plus

206
00:22:33,130 --> 00:22:34,230
the square root of two.

207
00:22:44,370 --> 00:22:44,850
So

208
00:22:45,330 --> 00:22:48,410
So where the heck are those two numbers on the graph

209
00:22:51,630 --> 00:22:53,090
We'll get there in a second

210
00:22:55,230 --> 00:22:57,510
What is the square root of two approximately?

211
00:23:00,610 --> 00:23:01,630
One point something

212
00:23:03,070 --> 00:23:04,930
Let us use about

213
00:23:05,770 --> 00:23:07,310
One point four ish

214
00:23:07,310 --> 00:23:11,730
so negative one plus one point four and

215
00:23:12,290 --> 00:23:15,130
negative one minus one point four is

216
00:23:16,350 --> 00:23:20,530
Point four and negative two point four

217
00:23:23,490 --> 00:23:29,150
So point four and negative two point four

218
00:23:37,150 --> 00:23:40,670
Okay, the axis of symmetry is halfway

219
00:23:43,070 --> 00:23:46,290
Okay, which is the average of these two numbers

220
00:23:48,950 --> 00:23:55,550
Negative one plus the square root of two, plus negative one minus

221
00:23:56,330 --> 00:23:59,270
the square root of two, all over two.

222
00:24:00,910 --> 00:24:04,370
The two radical terms cancel to zero.

223
00:24:05,410 --> 00:24:07,610
So we get negative 2 over 2

224
00:24:08,390 --> 00:24:09,410
negative 1

225
00:24:10,830 --> 00:24:13,810
Okay, this is our axis of symmetry

226
00:24:16,070 --> 00:24:18,370
To get the y-coordinate of the vertex,

227
00:24:19,790 --> 00:24:21,410
evaluate f at negative one.

228
00:24:23,130 --> 00:24:29,390
F of negative one is (negative one) squared plus two times

229
00:24:29,390 --> 00:24:30,150
negative one,

230
00:24:31,030 --> 00:24:31,870
minus one,

231
00:24:33,030 --> 00:24:35,030
which is

232
00:24:39,110 --> 00:24:39,950
negative two.

233
00:24:42,580 --> 00:24:44,580
Follow our pattern

234
00:24:58,530 --> 00:25:00,550
Let's put it in vertex form

235
00:25:03,370 --> 00:25:07,330
The leading coefficient is one. What was the

236
00:25:08,930 --> 00:25:15,750
x-coordinate of the vertex? Negative one, so use (x plus one) squared,

237
00:25:17,270 --> 00:25:19,890
minus two.

238
00:25:24,810 --> 00:25:25,290
Okay

239
00:25:26,670 --> 00:25:27,150
So

240
00:25:29,650 --> 00:25:30,610
Almost every

241
00:25:33,190 --> 00:25:36,911
quadratic we have discussed can be written in all three forms. Each form tells us

242
00:25:36,911 --> 00:25:37,830
something different.

243
00:25:40,410 --> 00:25:42,730
We can convert to all the forms

244
00:25:44,910 --> 00:25:46,170
The only time

245
00:25:47,690 --> 00:25:49,810
We're not going to be able to write in factored form

246
00:26:03,710 --> 00:26:04,210
Okay

247
00:26:05,850 --> 00:26:07,070
Okay is this

248
00:26:10,350 --> 00:26:12,570
Why can we not write this in factored form

249
00:26:14,790 --> 00:26:17,090
If there are no x intercepts

250
00:26:17,770 --> 00:26:19,830
It cannot be factored into real linear factors.

251
00:26:21,170 --> 00:26:23,290
We know this equation, right?

252
00:26:24,710 --> 00:26:26,690
Or this function would be

253
00:26:27,690 --> 00:26:34,410
(x minus three) squared plus two.

254
00:26:37,330 --> 00:26:38,870
So how can you tell

255
00:26:48,390 --> 00:26:50,710
Yeah, if you have the graph it's obvious

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if the

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00:26:53,290 --> 00:26:55,850
It opens up and above the x-axis

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It has no x-intercepts. Likewise, a downward-opening parabola with a negative

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00:27:00,402 --> 00:27:01,230
maximum has no

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x-intercepts.

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What if we only have standard form?

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This is x squared minus 6x

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plus

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11

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If we substitute into the

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00:27:20,290 --> 00:27:21,270
Quadratic formula

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00:27:21,270 --> 00:27:27,030
You're gonna get negative negative 6 plus or minus

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36 minus 4 times 1 times 11

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Thirty-six minus forty-four. What is the radicand?

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00:27:43,450 --> 00:27:45,470
Sorry, what's the discriminant?

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00:27:46,410 --> 00:27:46,670
negative

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00:27:46,670 --> 00:27:51,310
Negative eight, which means there are no real

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solutions to f of x equals zero: no real zeros and no x-intercepts.

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It cannot be written as a product of real linear factors.