IB · Mathematics Analysis and Approaches (AA)
AASL 2.11.1 Transformations
Learn AASL 2.11.1 Transformations in this free IB Mathematics Analysis and Approaches (AA) video lesson for SL 2.11—Transformation of functions.
Learn AASL 2.11.1 Transformations in this free IB Mathematics Analysis and Approaches (AA) video lesson for SL 2.11—Transformation of functions.
Learn AASL 2.11.1 Transformations in this free IB Mathematics Analysis and Approaches (AA) video lesson for SL 2.11—Transformation of functions.
The video focuses on transformations of graphs, detailing six key types that are essential to understand. These transformations include translations, stretches, and reflections, with specific rules for vertical and horizontal movements, such as:
The video also emphasizes the importance of visualizing these transformations using graphing tools like Desmos and provides examples of how to apply these transformations to a given function. The lesson concludes with a preview of the next topic on composite transformations, which involve applying multiple transformations simultaneously.
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
00:28the 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.