[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.03,0:00:01.09,Default,,0000,0000,0000,,- [Instructor] The graph of y
Dialogue: 0,0:00:01.09,0:00:03.00,Default,,0000,0000,0000,,is equal to absolute value of x
Dialogue: 0,0:00:03.00,0:00:04.81,Default,,0000,0000,0000,,is reflected across the x-axis
Dialogue: 0,0:00:04.81,0:00:07.73,Default,,0000,0000,0000,,and then scaled vertically\Nby a factor of seven.
Dialogue: 0,0:00:07.73,0:00:10.60,Default,,0000,0000,0000,,What is the equation of the new graph?
Dialogue: 0,0:00:10.60,0:00:13.07,Default,,0000,0000,0000,,So pause the video and see\Nif you can figure that out.
Dialogue: 0,0:00:13.07,0:00:15.25,Default,,0000,0000,0000,,Alright, let's work\Nthrough it together now.
Dialogue: 0,0:00:15.25,0:00:17.51,Default,,0000,0000,0000,,Now, you might not need\Nto draw it visually
Dialogue: 0,0:00:17.51,0:00:19.88,Default,,0000,0000,0000,,but I will just so that\Nwe can all together
Dialogue: 0,0:00:19.88,0:00:22.38,Default,,0000,0000,0000,,visualize what is going on.
Dialogue: 0,0:00:22.38,0:00:24.88,Default,,0000,0000,0000,,So let's say that's my x-axis
Dialogue: 0,0:00:24.88,0:00:26.71,Default,,0000,0000,0000,,and that is my y-axis.
Dialogue: 0,0:00:29.84,0:00:32.25,Default,,0000,0000,0000,,y equals the absolute value of x.
Dialogue: 0,0:00:32.25,0:00:35.03,Default,,0000,0000,0000,,So for non-negative values of x,
Dialogue: 0,0:00:35.03,0:00:36.55,Default,,0000,0000,0000,,y is going to be equal to x.
Dialogue: 0,0:00:36.55,0:00:37.86,Default,,0000,0000,0000,,Absolute value of zero is zero.
Dialogue: 0,0:00:37.86,0:00:39.02,Default,,0000,0000,0000,,Absolute value of one is one.
Dialogue: 0,0:00:39.02,0:00:40.73,Default,,0000,0000,0000,,Absolute value of two is two.
Dialogue: 0,0:00:40.73,0:00:42.46,Default,,0000,0000,0000,,So it's gonna look like this.
Dialogue: 0,0:00:42.46,0:00:44.92,Default,,0000,0000,0000,,It's gonna have a slope of one
Dialogue: 0,0:00:44.92,0:00:46.33,Default,,0000,0000,0000,,and then for negative values,
Dialogue: 0,0:00:46.33,0:00:47.68,Default,,0000,0000,0000,,when you take the absolute value,
Dialogue: 0,0:00:47.68,0:00:48.58,Default,,0000,0000,0000,,you're gonna take the opposite.
Dialogue: 0,0:00:48.58,0:00:50.06,Default,,0000,0000,0000,,You're gonna get the positive.
Dialogue: 0,0:00:50.06,0:00:52.18,Default,,0000,0000,0000,,So it's gonna look like this.
Dialogue: 0,0:00:52.18,0:00:55.01,Default,,0000,0000,0000,,Let me see if I can draw\Nthat a little bit cleaner.
Dialogue: 0,0:00:55.01,0:00:58.03,Default,,0000,0000,0000,,This is a hand drawn\Nsketch so bear with me
Dialogue: 0,0:00:58.03,0:00:59.08,Default,,0000,0000,0000,,but hopefully this is familiar.
Dialogue: 0,0:00:59.08,0:00:59.91,Default,,0000,0000,0000,,You've seen the graph
Dialogue: 0,0:00:59.91,0:01:04.30,Default,,0000,0000,0000,,of y is equal to absolute\Nvalue of x before.
Dialogue: 0,0:01:04.30,0:01:06.71,Default,,0000,0000,0000,,Now, let's think about the\Ndifferent transformations.
Dialogue: 0,0:01:06.71,0:01:10.88,Default,,0000,0000,0000,,So first, they say is\Nreflected across the x-axis.
Dialogue: 0,0:01:12.35,0:01:13.60,Default,,0000,0000,0000,,So for example,
Dialogue: 0,0:01:14.56,0:01:18.32,Default,,0000,0000,0000,,if I have some x value right over here,
Dialogue: 0,0:01:18.32,0:01:20.24,Default,,0000,0000,0000,,before, I would take\Nthe absolute value of x
Dialogue: 0,0:01:20.24,0:01:21.88,Default,,0000,0000,0000,,and I would end up there
Dialogue: 0,0:01:21.88,0:01:25.13,Default,,0000,0000,0000,,but now we wanna reflect across the x-axis
Dialogue: 0,0:01:25.13,0:01:28.75,Default,,0000,0000,0000,,so we wanna essentially get\Nthe negative of that value
Dialogue: 0,0:01:28.75,0:01:32.19,Default,,0000,0000,0000,,associated with that corresponding x
Dialogue: 0,0:01:32.19,0:01:33.70,Default,,0000,0000,0000,,and so for example, this x,
Dialogue: 0,0:01:33.70,0:01:35.92,Default,,0000,0000,0000,,before, we would get\Nthe absolute value of x
Dialogue: 0,0:01:35.92,0:01:37.97,Default,,0000,0000,0000,,but now we wanna flip across the x-axis
Dialogue: 0,0:01:37.97,0:01:40.51,Default,,0000,0000,0000,,and we wanna get the negative of it.
Dialogue: 0,0:01:40.51,0:01:42.84,Default,,0000,0000,0000,,So in general, what we are doing
Dialogue: 0,0:01:42.84,0:01:47.00,Default,,0000,0000,0000,,is we are getting the negative\Nof the absolute value of x.
Dialogue: 0,0:01:48.84,0:01:51.74,Default,,0000,0000,0000,,In general, if you're\Nflipping over the x-axis,
Dialogue: 0,0:01:51.74,0:01:53.09,Default,,0000,0000,0000,,you're getting the negative.
Dialogue: 0,0:01:53.09,0:01:57.79,Default,,0000,0000,0000,,You're scaling the expression\Nor the function by a negative.
Dialogue: 0,0:01:57.79,0:02:00.24,Default,,0000,0000,0000,,So this is going to be y\Nis equal to the negative
Dialogue: 0,0:02:00.24,0:02:02.20,Default,,0000,0000,0000,,of the absolute value of x.
Dialogue: 0,0:02:02.20,0:02:03.92,Default,,0000,0000,0000,,Once again, whatever absolute value of x
Dialogue: 0,0:02:03.92,0:02:05.88,Default,,0000,0000,0000,,was giving you before for given x,
Dialogue: 0,0:02:05.88,0:02:08.06,Default,,0000,0000,0000,,we now wanna get the negative of it.
Dialogue: 0,0:02:08.06,0:02:11.33,Default,,0000,0000,0000,,We now wanna get the negative of it.
Dialogue: 0,0:02:11.33,0:02:16.21,Default,,0000,0000,0000,,So that's what reflecting\Nacross the x-axis does for us
Dialogue: 0,0:02:16.21,0:02:20.38,Default,,0000,0000,0000,,but then they say scaled\Nvertically by a factor of seven
Dialogue: 0,0:02:22.92,0:02:26.03,Default,,0000,0000,0000,,and the way I view that is if\Nyou're scaling it vertically
Dialogue: 0,0:02:26.03,0:02:27.42,Default,,0000,0000,0000,,by a factor of seven,
Dialogue: 0,0:02:27.42,0:02:30.17,Default,,0000,0000,0000,,whatever y value you got for given x,
Dialogue: 0,0:02:30.17,0:02:33.36,Default,,0000,0000,0000,,you now wanna get seven times the y value,
Dialogue: 0,0:02:33.36,0:02:35.28,Default,,0000,0000,0000,,seven times the y value
Dialogue: 0,0:02:39.45,0:02:40.62,Default,,0000,0000,0000,,for a given x.
Dialogue: 0,0:02:43.66,0:02:45.78,Default,,0000,0000,0000,,And so if you think\Nabout that algebraically,
Dialogue: 0,0:02:45.78,0:02:48.41,Default,,0000,0000,0000,,well, if I want seven times the y value,
Dialogue: 0,0:02:48.41,0:02:51.51,Default,,0000,0000,0000,,I'd have to multiply this thing by seven.
Dialogue: 0,0:02:51.51,0:02:55.52,Default,,0000,0000,0000,,So I would get y is\Nequal to negative seven
Dialogue: 0,0:02:55.52,0:02:58.29,Default,,0000,0000,0000,,times the absolute value of x
Dialogue: 0,0:02:58.29,0:02:59.98,Default,,0000,0000,0000,,and that's essentially\Nwhat they're asking,
Dialogue: 0,0:02:59.98,0:03:01.74,Default,,0000,0000,0000,,what is the equation of the new graph,
Dialogue: 0,0:03:01.74,0:03:03.18,Default,,0000,0000,0000,,and so that's what it would be.
Dialogue: 0,0:03:03.18,0:03:05.80,Default,,0000,0000,0000,,The negative flips us over the x-axis
Dialogue: 0,0:03:05.80,0:03:09.03,Default,,0000,0000,0000,,and then the seven scales\Nvertically by a factor of seven
Dialogue: 0,0:03:09.03,0:03:11.52,Default,,0000,0000,0000,,but just to understand\Nwhat this would look like,
Dialogue: 0,0:03:11.52,0:03:13.70,Default,,0000,0000,0000,,well, you multiply zero times seven,
Dialogue: 0,0:03:13.70,0:03:15.26,Default,,0000,0000,0000,,it doesn't change anything
Dialogue: 0,0:03:15.26,0:03:17.84,Default,,0000,0000,0000,,but whatever x this is,
Dialogue: 0,0:03:17.84,0:03:20.05,Default,,0000,0000,0000,,this was equal to negative x
Dialogue: 0,0:03:20.05,0:03:23.08,Default,,0000,0000,0000,,but now we're gonna get\Nto negative seven x.
Dialogue: 0,0:03:23.08,0:03:27.76,Default,,0000,0000,0000,,So let's see, two, three,\Nfour, five, six, seven
Dialogue: 0,0:03:27.76,0:03:29.97,Default,,0000,0000,0000,,so it'd put it something around that.
Dialogue: 0,0:03:29.97,0:03:32.54,Default,,0000,0000,0000,,So our graph is now going to look,
Dialogue: 0,0:03:32.54,0:03:34.62,Default,,0000,0000,0000,,is now going to look like this.
Dialogue: 0,0:03:34.62,0:03:39.33,Default,,0000,0000,0000,,It's going to be stretched\Nalong the vertical axis.
Dialogue: 0,0:03:39.33,0:03:41.70,Default,,0000,0000,0000,,If we were scaling vertically
Dialogue: 0,0:03:41.70,0:03:44.65,Default,,0000,0000,0000,,by something that had an\Nabsolute value less than one
Dialogue: 0,0:03:44.65,0:03:46.70,Default,,0000,0000,0000,,then it would make the graph less tall.
Dialogue: 0,0:03:46.70,0:03:47.80,Default,,0000,0000,0000,,It would make it look,
Dialogue: 0,0:03:47.80,0:03:49.55,Default,,0000,0000,0000,,it would make it look wider.
Dialogue: 0,0:03:49.55,0:03:52.24,Default,,0000,0000,0000,,Let me make it at least look\Na little bit more symmetric.
Dialogue: 0,0:03:52.24,0:03:54.67,Default,,0000,0000,0000,,So it's gonna look something,
Dialogue: 0,0:03:54.67,0:03:57.05,Default,,0000,0000,0000,,something like that but the key issue
Dialogue: 0,0:03:57.05,0:03:58.88,Default,,0000,0000,0000,,and the reason why I'm\Ndrawing is so you can see
Dialogue: 0,0:03:58.88,0:04:00.71,Default,,0000,0000,0000,,that it looks like it's\Nbeing scaled vertically.
Dialogue: 0,0:04:00.71,0:04:02.67,Default,,0000,0000,0000,,It's being stretched in\Nthe vertical direction
Dialogue: 0,0:04:02.67,0:04:06.23,Default,,0000,0000,0000,,by a factor of seven and the\Nway we do that algebraically
Dialogue: 0,0:04:06.23,0:04:07.42,Default,,0000,0000,0000,,is we multiply by seven
Dialogue: 0,0:04:07.42,0:04:11.58,Default,,0000,0000,0000,,and the negative here is what\Nflipped us over the x-axis.