00:00Hi guys. So in this
00:02video we are going to
00:02graph modulus functions. In this
00:04two types we need to
00:05graph it's f of mod
00:07f of x and f,
00:09well y equals f of
00:11mod x. So notice the
00:12difference there's mod f of
00:13x and f of mod
00:14x. Now mod is obviously
00:15short for modulus. Now what
00:18does modulus mean? Well hopefully
00:20you've seen this before but
00:22the mod of negative 5
00:24or the modulus of negative
00:255 is sometimes called the
00:27absolute value of negative
00:28and mod of negative 5
00:30is 5. It's kind of
00:31like the size of it.
00:34So the size of this
00:35is negative. The size of
00:38negative 5 is 5. Think
00:39of it like your speed.
00:42If you are reversing, your
00:44velocity is negative, whatever, 50
00:47kilometers per hour. But your
00:49speed is 50 kilometers per
00:51hour, so that the modulus
00:52of the velocity anyway.
00:56going to draw these graphs.
00:58Now first I'll do it
00:59without a calculator and then
01:00we'll do it with a
01:00calculator. I'm going to use
01:02this example, this quadratic that
01:04hopefully works somewhat familiar with
01:06and we can draw, we're
01:10going to use this as
01:11f of x and then
01:11we'll draw these two. So
01:14firstly, I should have put
01:17my y here and x
01:20here, I guess I should
01:21do it for them all,
01:22x and y.
01:24x and y. Now, how
01:28do I draw this function?
01:29Well, this is not the
01:30main part of the lesson.
01:31Hopefully, you know how to
01:31do this. The factors of
01:33this are x minus two
01:35times x plus one, which
01:37means it has roots at
01:40negative one, negative one, and
01:45two. And a y -intercept
01:47at negative two. So this
01:49comes down here. Now, the
01:51y -intercept
01:52is not the minimum, just
01:54be careful. The minimum would
01:56be at 1, 0 .5,
01:58half way between the two,
01:59but that is negative two.
02:02Okay, so if we want
02:04to draw the mod of
02:05f of x, the modulus
02:06of this, well what happens
02:07is the, what this means
02:11is there cannot be negative
02:14y values because we're in
02:17this mod sign. So whatever
02:18we put into f,
02:20the value we get out
02:21has to be positive. So
02:23these negative values, which are
02:24the values underneath the x
02:25-axis become positive. The negative
02:28two becomes two. Whatever this
02:32value here is, its negative
02:33becomes positive. This minimum becomes
02:38positive here, and now hopefully
02:39you'll see what's happening is
02:41it's actually a reflection. The
02:45negative part gets reflected in
02:48the x -axis. So that's
02:49the first kind of key
02:52point that I want you
02:53to learn from this lesson.
02:55Now, the positive part just
02:57stays the same. So this
02:58is still, it's still going
02:59to be a root negative
03:00one. It's going to be
03:01another root of two. So
03:03these positive bits stay the
03:05same. It still comes down
03:07here. It still goes up
03:08there. But now instead of
03:11going underneath the line, these
03:13become positive. It gets reflected
03:15in
03:16the x -axis. So I'm
03:21actually going to write that,
03:22oh, this is now two.
03:27I'm going to write that
03:28down. I'm going to say
03:28reflect, reflect, negative y values
03:36in x -axis. OK.
03:44fine. Next part is slightly
03:48more tricky. It's f of
03:50mod x. Now what that
03:51means is what's going to
03:54happen is, well firstly, all
03:57the positive values. So imagine
03:59I do f of, if
04:02I do f of mod
04:065. Well that's just the
04:08same as f, 5. So
04:11all the positive
04:12of x values that go
04:14from here to here, they're
04:15going to be exactly the
04:16same. So I'm still going
04:18to root at 2 here.
04:21This is still going to
04:23come down like this. This
04:26is still going to turn,
04:28I'm going to turn, which
04:30I'll try this properly. It's
04:32still going to turn here
04:33and go up there to
04:352. So this is at
04:372 and it would normally
04:38just continue on to
04:40one. This isn't a perfect
04:43drawing, but whatever. So let's
04:45say this is negative two.
04:49Okay, but what happens to
04:52these negative values? Well, what
04:57happens is if I put
04:58in, let's say, well, let's
05:06do negative, let's do negative
05:08five,
05:08put in negative five into
05:11f of mod x. What
05:12I get is the same
05:14value as f of five.
05:17So all these negative x's
05:20behave exactly like the positive
05:23x's. So you end up
05:26with instead of getting, instead
05:28of getting f of negative
05:30one, you get a y
05:32value the same as f
05:33of one. Instead of getting
05:34f of negative five, you
05:36get a
05:36y value of f of
05:385 and what happens is
05:40it becomes a ref... these
05:44negative x's behave the exact
05:48same as the positive x's
05:49and it ends up being
05:50a reflection. Let me try
05:52and draw that. Okay that's
05:56not too bad. It becomes
05:57a reflection in the y
06:01axis. So
06:04Again, what I'm gonna write
06:06here, draw, so reflect the
06:10negative one of those next
06:11axis here, draw, draw, graph
06:22for positive x's, for positive
06:29x's, and then
06:33and reflect, reflect in y
06:43axis. Okay, so hopefully that
06:45makes sense. There's basically two
06:46different ones, and this is
06:48what you do. For the
06:51mod of f of x,
06:52you get the negative y
06:54values, and you reflect them
06:55in the x -axis like
06:56this. For the f of
06:59mod x, you draw the
07:00positive
07:01sort of x first or
07:02anything to the right of
07:03the y -axis because they
07:04behave normally and then you
07:06get that and just whatever
07:08you have reflected in the
07:10y -axis like this. Okay,
07:13hopefully that makes sense. And
07:15the next lesson I am
07:16going to be using these
07:18graphs to solve equations with
07:21mod functions in them. Okay,
07:27I almost forgot I wanted
07:28to
07:29graph these with the calculator.
07:32So let's bring this here.
07:35Let's go into our graph.
07:38Now, and at this point
07:39here, I am going to
07:42put it. So let's first
07:43do mod f of x.
07:47So I need to get
07:48my modulus signs, which are
07:49here. Click this button beside
07:51the nine. I have them
07:54right there. Be careful.
07:57So, I have seen, see
07:59this little line here. This
08:02is not a mod sign,
08:04so just be careful there,
08:06right? We need to use
08:07this one here. And then
08:11inside it, we're gonna put
08:12x squared minus x minus
08:14two. x squared minus x
08:22minus two. There you go.
08:25So that is the graph
08:27that I drew here. Not
08:31quite as well as a
08:32calculator, but close enough. And
08:35the second one, the second
08:38one, let's do this press
08:41tab. We can do it
08:42on the same set of
08:44axes. I want to do
08:46f of mod x. Okay,
08:48so now I need to
08:49just put the mod in
08:50the axis. So what that
08:51actually looks like is this
08:53idea.
08:53mod x. So it's mod
08:58x squared. The squared is
09:01outside the modulus. Mod x
09:03squared minus mod x and
09:10then minus 2. And then
09:17I press enter and this,
09:21yeah, look, this
09:21This is not exactly the
09:25right way one, but again,
09:27close enough. We have the
09:30same values for the positive,
09:35the positive x values, because
09:37if I drew, let me
09:38draw the other one. Let
09:40me draw the original x,
09:46yeah.
09:49squared, do that again, x
09:52squared minus x minus two.
09:56So you see we have
09:57the red graph follows the
10:02graph for the positive values
10:04and then it just kind
10:06of, well it reflects in
10:08the y axis. Okay, yeah,
10:12next doesn't what we solving
10:13equations.