00:00Hi everybody. So in this
00:02lesson, we're going to look
00:02at transformations of graphs. So
00:05these six transformations here basically
00:07you have to know them.
00:08You have to know how
00:09they work. And what they
00:12are. So a translation is,
00:15well, again, hopefully you do
00:16know what these are. A
00:18translation is, well, a vertical
00:20translation moves the graph, not
00:23that way, either up or
00:24down. And a horizontal translation
00:27moves it either.
00:28the right or left. A
00:29vertical stretch stretches it vertically.
00:34So think about an elastic
00:36band, we stretch it out
00:38this way again vertically or
00:40horizontally this way. And the
00:43negatives reflected in the x
00:45-axis and the y -axis.
00:46Now look, be very careful.
00:47Negative f of x reflects
00:49in the x -axis and
00:50f of negative x reflects
00:51in the y -axis. Also
00:53be careful. The
00:56one's actors you would expect.
00:57So when you add a,
00:58it just moves it up
01:00a. When you subtract it,
01:01moves it down a. But
01:02the horizontal, if you add
01:04a inside the bracket, it
01:06likes to move it left
01:07a. And if you subtract
01:09a, it'll move it right
01:10a. And similarly here, when
01:13you stretch it vertically by
01:15a, it'll stretch with scale
01:17factor a fine. And if
01:19you stretch it horizontally, it
01:22will stretch it with scale
01:23factor one over
01:24over a. So, if a
01:27was to say, it'll actually
01:30squish it, which is definitely
01:31not a technical word, but
01:33we say, the correct word
01:34is actually, you could say
01:36compress, but really the most
01:37technical word is a stretch
01:39of scale factor half. Okay,
01:42now a lot of that
01:42won't make much sense without
01:44actually seeing what's going on.
01:46So, I would strongly advise
01:48you to go into Desmos
01:49and start playing around with
01:50all these different
01:52all these different transformations. What
01:54I have done is I
01:55have created this graph, f
01:58of x. I just like
01:59the look of it because
01:59it's got asymptotes and a
02:02lot of different features, maximum,
02:03minima, things like that. So
02:05this is f of x.
02:07It doesn't really, you haven't
02:08studied, this is a rational
02:10function of a particular type.
02:12You don't have to study
02:12this. Don't worry too much
02:13about it. But note, what
02:17I'm about to do here,
02:17we'll work with any graph.
02:20So F, if I do
02:22F of X, and let's
02:23just say plus two, that
02:25translates it vertically by two.
02:28It moves the graph up
02:29to, if I do minus
02:31two, it'll move it down
02:32to. If I do on
02:34stick with two, because it's
02:35an easy number, if I
02:36do two F of X,
02:37it's vertically stretches it by
02:40two. So this goes from
02:41five to 10. This goes
02:43from one to two. This,
02:48horizontal asymptote which is actually
02:50a two goes up to
02:52four. If I though do
02:56f of two x, see
02:58what I mean by that,
02:59it squishes it. This minimum
03:02which is a negative two
03:05now goes to negative one.
03:08The y intercept says the
03:10same because it was actually
03:11at zero. So what it
03:12does is it halves all
03:14the x coordinates. If you
03:16actually want it to
03:16you'd have to do x,
03:20f of a half x,
03:21or f of x over
03:22two, and that kind of
03:23stretches it out by two.
03:26So this, this minimum is
03:27now at negative four instead
03:29of negative two. And then
03:31finally the, the reflections, if
03:35I do negative f of
03:36x, that reflects it in
03:38the x -axis, and if
03:40I do f of negative
03:42x, that reflects it in
03:44the Y axis. Okay, what
03:48I'm going to do now
03:49is I'm actually going to,
03:52because it's fine, asking Desmos
03:54to do it, because they're
03:56pretty good at drawing graphs.
03:59Let's admit, but when we
04:01actually have to sketch it
04:02ourselves, how do we do
04:03it? So I'm going to
04:04do one of these, this
04:05graph, for all the six
04:07different types, I'm explaining my
04:09thinking as I go along.
04:11What it wants you to
04:12do now, it gives
04:12A and B, so A
04:14is just this kind of
04:15point where the straight line
04:17meets, it's probably like a
04:18straight line parabola kind of
04:20thing. B is the maximum
04:21and these are the intercepts.
04:23So it says mark the
04:25new points A and B
04:26and the X and the
04:28X is intercept if possible.
04:30So it won't always be
04:30possible to know what the
04:32intercepts are. Okay, let's do
04:36this. So I need to
04:40f of
04:40plus two, I need to
04:42translate this graph up to.
04:43So every point needs to
04:45go up to. The minus
04:46one one goes to minus
04:48one three, minus one one,
04:51minus one three. So A
04:52is now negative one three.
04:57The two goes up to
04:58four. So the negative two
05:01goes up, or the zero
05:02two goes up to four.
05:04So that's four. The B
05:05goes up to five. So
05:07it's two.
05:085b is a 2, 5
05:12and that's the maximum. Now
05:15this 4, this 4 is
05:180 goes to 4, 2,
05:19but that's not actually a
05:21point that I care about,
05:22but I actually don't know
05:24what the new x intercepts
05:25going to be because it's
05:26basically going to be whatever
05:28is down here. So I
05:29don't know it and that's
05:31why it actually says it
05:31if possible, so I don't,
05:33it's not possible in this
05:34case. So let's try and
05:35draw the graph. I need
05:36to
05:36straight line over to that,
05:39and then it goes up
05:40to here, and then down,
05:43right? Forgive my not perfect
05:46graphs as I do these,
05:48and I'm gonna have to
05:48try and do that one
05:49again. Okay, something like that.
05:53Fine. F of x plus
05:55two. So if it was
05:56F of x minus two,
05:57I just move all the
05:58points down to. F of
06:00x plus two in a
06:01bracket. Now this is a
06:03horizontal translation
06:04to the left, I'm going
06:05to move it left to,
06:09okay? So this moves it
06:10up to, but this one
06:11moves it left to, it's
06:12a bit, well, it is
06:15annoying, but it's different to
06:16what you'd expect. So the
06:18negative one, A negative one
06:20one becomes negative three one,
06:23because I'm going to left,
06:24so A is now negative
06:26three one. This two is
06:30actually moved to the left,
06:32so I don't actually care
06:32about
06:33this point because it's not
06:34no longer an intercept. The
06:36two three is actually going
06:38to become zero three because
06:39look that's moving left two
06:40so that'll become my new
06:42y -intercept. So this is
06:44now two, sorry this is
06:48now zero three so b
06:51is now zero three and
06:53my x -intercept I am
06:55going to know this because
06:56it's four goes left two
06:57so this becomes two.
07:01Let's try and draw the
07:03graph. I'm going to go
07:06straight line, and then I
07:08go up, and then down.
07:14Good enough. That's two. Okay,
07:19next one. Two f of
07:22x, and now I'm going
07:23to stretch it, and I'm
07:24going to stretch it vertically.
07:25So I'm going to multiply
07:26the y coordinates by two.
07:28So this guy goes
07:29from negative one one to
07:30negative one two. So A
07:33is negative one two. This
07:37two goes from zero to
07:39zero four. So this is
07:41now at four. This two
07:44three goes to two six.
07:50This is B two six
07:52because I'm multiplied three by
07:54two. And this is, remember
07:55this is four zero.
07:57I feel like. So if
07:58I multiply 0 by 2,
08:01what do I get? Well,
08:02I still get 0. So
08:03the x intercept isn't going
08:06to change when I stretch
08:07it vertically. So this is
08:09going to stay at 4
08:13like this. OK, let's try
08:16and draw the graph. I'm
08:18going to go across straight
08:21and then up.
08:25Let's stop there and then,
08:28no. This is really difficult
08:32for me to do. There,
08:34something like that. Fine. Now,
08:38it does, yeah, it is,
08:39it's stretched. You can see
08:41the whole, it doesn't look
08:42like this. So these are,
08:43these three should pretty much
08:45look the same. Now they
08:47don't, but that's only because
08:48of me. In the way
08:50they look, this guy doesn't
08:51look the same because it's
08:52been stretched.
08:53So a stretch won't look
08:54the same where a translation
08:56will look the same. Okay,
08:58next one. F of 2x.
09:01So this is what I
09:02mean. We are going to
09:04horizontally stretch it, scale factor
09:07a half. So I'm actually
09:08going to write here just
09:09to remind us scale, factor
09:11is a half. So we're
09:12going to half all the
09:15x coordinates right now. This
09:17guy is a negative 1,
09:201. So the new coordinate
09:21for A is actually going
09:24to be negative 1. So
09:32I have half that. The
09:342 is actually going to
09:36stay the same. So the
09:382 stays the same because
09:41what's half of this is
09:430, 2 if you like.
09:45And what's half of 0?
09:47Well, it's 0. So that
09:49doesn't change.
09:49when I'm stretching it horizontally,
09:52the Y -intercept bone change.
09:54This two, three, will become
09:56one, three, because a half,
09:59two. So this is going
10:00to be one, three. Now,
10:04again, certainly not perfectly to
10:07scale. But what's important is
10:08that I put in, and
10:09that's why this question says,
10:11label the new points, because
10:12you want to make sure
10:12you know how to stretch
10:14it. So A is half,
10:16this is half,
10:17which is to zero, this
10:18is half to one, and
10:19now my four has three
10:20half to two. So this
10:23is gonna become two. Okay,
10:28let's try and draw that.
10:31So I do my straight
10:32line here, and there we
10:38go. So, as I said,
10:42the not technical word here
10:44is, well,
10:45really not technically word is
10:47squished it's been squished horizontally
10:50pushed in towards that what
10:54the y axis the the
10:58more technical is compressed and
11:01the really technical way of
11:02saying it is it has
11:03been stretched by scale factor
11:05one half okay this is
11:08now okay let's see this
11:10is now going to be
11:11the reflection so negative f
11:13of x is a reference
11:13the reflection in the x
11:15axis. So what we're going
11:16to do is we were,
11:17well look, all the y
11:19coordinates become negative. So negative
11:211, 1 becomes negative 1,
11:24negative 1. So this goes
11:25down here. This is my
11:26a. It is a negative
11:311, negative 1. And then
11:35my 0, 2 becomes 0,
11:39negative 2. So this is
11:41going to be
11:41negative 2. And my 2
11:453, 2 3 becomes 2,
11:48negative 3. This is B,
11:512, negative 3. And my
11:544 0 will stay at
11:564 0 because it becomes
11:57negative 0 if you want
11:58to think of it like
11:59that. But that's obviously still
12:01going to be 4. So
12:02you'll see now I reflect
12:04and I like to always
12:05do the points first and
12:06then draw the graph. So
12:08it's going to come down
12:09like this.
12:09Turn there and then up
12:15like that. That's not a
12:16bad effort. So you can
12:17see that is clearly, this
12:19has been reflected in the
12:21x axis. Okay, last one,
12:24I now need to reflect
12:25in the y axis. So
12:28negative one one has to
12:31turn into one one. So
12:34I'm going to multiply now
12:35the x coordinates by negative
12:37one. So I'm going to
12:37So negative 1, 1 becomes
12:411, 1. So A is
12:45now 1, 1. The 2
12:49is going to stay the
12:50same because it is 0,
12:552. So multiply 0 by
12:572. And multiply 0 by
13:00negative 1. And you still
13:01get 0. The 2, 3
13:04is going to become 1.
13:06negative two three. So this
13:08is going to be at
13:09negative two three something like
13:12this. B is negative two
13:14three and the four zero
13:16is going to become negative
13:18four zero. So this will
13:20be at negative four. So
13:22I do my straight line
13:25is going to come from
13:26this side. We're going to
13:28go up like this max
13:31out there and then down.
13:34like that. I'm clearly looking,
13:36if you reflect that in
13:37the way axis, you would
13:38get this. Okay, that's the
13:41lesson. Hopefully, it's clear I'm
13:44aware that I went through
13:45that quite quickly. As always,
13:47you have the opportunity to
13:50pause the video or rewatch
13:51particular particular parts that you
13:53want to see. But the
13:54main thing you absolutely have
13:57to be able to do
13:58or recognize are these six
14:01transformations.
14:02In the next lesson, I'm
14:03going to go over composite
14:05transformations, which is where we
14:07do more than one transformation
14:09in one go, which you
14:10don't love.