00:00Hi, everybody. So in this
00:02as I'm going to look
00:03at rational functions, now a
00:05rational function is just, well,
00:07it's a function defined as
00:10a function divided by another
00:14function. And in standard level,
00:16we're just going to look
00:17at linear, it's a linear
00:18function divided by another linear
00:20function. Now, they all look
00:25something like this.
00:28So here's my XY grid.
00:30The green line is a
00:31vertical asymptote and the red
00:32line is the horizontal asymptote.
00:35Now what they do is
00:36they come down like this,
00:39cross the x -axis somewhere,
00:41don't touch the, they don't
00:43touch the asymptotes obviously, that's
00:45what an asymptote is and
00:47something like this. So they
00:49all look like that or,
00:52does it look like this?
00:54Or,
00:56Like this. Now this is
01:01one, this is the graph
01:03of one function. It's just,
01:05I haven't drawn two different
01:06functions. This is one function.
01:08But what happens is there's
01:10an asymptote here in the
01:11function doesn't actually exist at
01:13this point. And it's actually
01:14because it's where the denominator
01:16equals zero. So they're either
01:19gonna look like this or
01:20the other way. Now, just
01:23from looking at
01:24So the ABC and the
01:25are just different numbers that
01:26they're the parameters of the
01:28function. So just by looking
01:30at those numbers, it's not
01:31always easy to know which
01:34way it's going to look.
01:35You have to find out
01:37these key features of the
01:39graph to determine exactly how
01:42to draw the graph. And
01:42I'm going to show you
01:43that in an example in
01:44a minute. But first let's
01:45go through these four points.
01:49Now, obviously you can just
01:50learn this when it's in
01:52this
01:52form. This is the line
01:53to set, this is the
01:54acceptor's own lesson, though. This
01:55is the vertical lesson, though.
01:56You can learn that off
01:57by harv and learn it
01:59all you want. But it's
02:02very important that you understand
02:03where these are coming from,
02:05because if the function gets
02:06changed, even slightly. For example,
02:09if I just said it
02:10was this plus two, well,
02:12that changes everything. And then
02:13your horizontal lesson is not
02:16exactly this, well, it's this
02:19plus two, but the
02:20the deeper you understand what's
02:22going on, obviously the more
02:23easily you'll be able to
02:25deal with changes in questions.
02:28So let's begin with the
02:30Y intercept. So the Y
02:31intercept, like for any function,
02:34occurs when X is zero.
02:36That's where the Y intercept
02:37is. It's when X is
02:38zero. So if you make
02:39X zero, what happens? That
02:41disappears. That's zero. That's zero.
02:43And you're left with B
02:44over D. So that's why
02:45Y is B over D
02:46is the Y intercept. Straight
02:47forward.
02:48The x intercept happens when
02:52y is zero or when
02:53f of x is zero.
02:54What happens is you get
02:55a x plus b equals
02:57zero. That's that the only
02:59way this can equal zero
03:00is when the numerator equals
03:01zero. So in that equals
03:03zero rearrange x equals negative
03:08b over a. That's how
03:11we get our x intercept.
03:14Our horizontal asymptote
03:16horizontal asymptote, it's actually the
03:20limit. So it's the limit
03:23as x approaches infinity of
03:27f of x. Now, in
03:29standard level, we don't go
03:30too deeply into limits. But
03:33if you think about this,
03:37think about this graph. I
03:38like to use instead of
03:39infinity. Let's think of a
03:41billion. Now imagine I had
03:44the
03:44function, let's say I had
03:46a function, two x plus
03:51three over x minus four.
03:56Now imagine I got f
03:58of a billion. What is
04:00y when x is a
04:01billion? Well, you end up
04:03with two billion plus three.
04:05Let's write it all out.
04:07I'm going to get two
04:08billion, one, two, three, four,
04:11five, six.
04:12eight and three, two billion
04:14and three, all over, one
04:17billion minus four, which is
04:19basically, all right, let's just
04:20write one billion minus four.
04:26Now, that's pretty much, that's
04:27very, very close to two
04:29billion, and that's very, very
04:30close to one billion. So
04:31what are you left with?
04:32Well, two billion over one
04:34billion is two, or A
04:36over C, and that's where
04:37we get the A over
04:38C from. And finally, the
04:40vertical
04:40I mentioned this earlier, it's
04:42when this vertical asymptote is,
04:44when this function can't exist,
04:47or when the denominator equals
04:49zero, what happens when the
04:51denominator equals zero? CX plus
04:53D equals zero, I'm not
04:57for CX equals negative D,
05:00and X equals negative D
05:03over C, and that's for
05:04this control. So those are
05:06four, let's say,
05:08key features for these rational
05:10functions. Now, let's, we're going
05:11to do an example, we're
05:12going to try and draw
05:13the rational function graph. So,
05:17sketch the graph of this.
05:19So, I'm going to start,
05:20well, I'm going to get
05:20those four different features. So,
05:24let's say x intercept. The
05:29x intercept will be x
05:33equals, well, it's, it's going
05:36to be
05:36When the x intercept is
05:38when y equals zero, so
05:41let's do it is, well
05:43let's write it out properly.
05:44It's when y is zero
05:45or when 2x minus 9
05:48equals zero. 2x equals 9x
05:57equals 9 over 2. That
05:59is my x intercept. Why
06:02intercept?
06:04My y intercept, just do
06:08that. My y intercept is
06:09when x is zero, when
06:10x is zero, and I've
06:11got negative nine over negative
06:14three, which is just three.
06:17My horizontal asymptote, my horizontal
06:21asymptote is going to be
06:24y equals two x over
06:26x, which is just going
06:27to be two, or two
06:29over one. And my vertical
06:31asymptote, let's write it here,
06:33My vertical asymptote is when
06:36the denominator equals zero or
06:40x equals three. Okay, so
06:44what we're gonna do, we're
06:45gonna draw, we're gonna draw
06:48our x and we're gonna
06:49draw our, sorry, we're gonna
06:53draw our y, we're gonna
06:54draw our x, this is
06:55x and this is y.
06:57Now let's put the asymptotes
06:58first. The horizontal asymptote
07:01is y equals two. So
07:03let's go with, I think
07:04I chose red, let's go
07:07red, y equals two, let's
07:09say that is, let's say
07:12that is here, this is
07:14y equals two. And the
07:18vertical asymptote is when x
07:21equals three. So that would
07:24be x equals, let's say
07:25that's two, let's say x
07:26equals three is here.
07:29This is x equals three.
07:32And this is y equals
07:36two. Now the question actually
07:38says show clearly any x's
07:40intercepts and asymptotes. So here's
07:42a clear vertical asymptote. Here's
07:44a clear horizontal asymptote. Now
07:47I just need, and this
07:49would, I don't know yet
07:50I'm not going to draw
07:50it here and here or
07:52here and here, but my
07:54intercepts is going to, are
07:56going to tell me.
07:57So, my x intercept is
08:029 over 2. So, 9
08:04over 2, this is 3,
08:069 over 2 is, let's
08:07just write that, it's 4
08:08.5. So, it's going to
08:10be somewhere to the right
08:11of this, it's going to
08:12be here. And my y
08:14intercept will be at 3.
08:17So, this is 2, this
08:18is 3. So, clearly now
08:20it's going to be here
08:21and here, because these are
08:23asymptotes and I'm not able
08:24to go past the
08:25and tots is going to
08:26look like this. They don't
08:27have to be perfectly accurate.
08:28Just make sure you're approaching
08:30the asymptotes. You're going through
08:32the intercept and I'm going
08:34to actually label that 4
08:35.5. And I'm going through
08:38this intercept there. This is
08:463. And we're done. And
08:48that's the sketch. Now let's
08:50just, this
08:53This might well be a
08:54paper one, but I just
08:55want to draw it with
08:56my calculator just to be
08:58sure I know how to
08:59do that as well. I'm
09:01just make sure I'm correct.
09:02So I'm going to press
09:03Ctrl. I need to press
09:06Tab first. Then in here
09:08I'm going to press Ctrl
09:09and divide to give me
09:10this. I'm going to write
09:112x minus 9 over x
09:16minus 3. And here I
09:21get
09:21I go off that if
09:23I zoom out and let's
09:29go with trace just to
09:33make sure my so here
09:35I have my Y -inches
09:36up that 0 3 I'm
09:38going to go along here
09:39you see my my asymptote
09:43is at 3 is as
09:443 undefined because the function
09:46is undefined there but it's
09:48at 3 and if I
09:49keep
09:49moving along my 0 or
09:53x intercept is at 4
09:54.5 and that looks like
09:56that so I know I
09:57am correct. Okay, so that
09:59is how we draw rational
10:01functions. Find the x intercept,
10:03find the y intercept, find
10:06the horizontal asymptote and the
10:07vertical asymptote and there you
10:10have it.