00:00Hi everyone. So I'm going
00:02to do two lessons on
00:03these rational functions with quadratic.
00:05So this is part one
00:06and I'll have to be
00:07part two. Basically, one of
00:10them you have your quadratic
00:11in the denominator and the
00:14other one you have your
00:14quadratic in the numerator and
00:16then the other function is
00:17just a linear. So it's
00:18like a linear function over
00:20a quadratic. So I've listed
00:23five steps of what we
00:25need to do in order
00:26to sketch this. So the
00:28S and it's just basically
00:29sketching it, but obviously once
00:31you sketch it, you can
00:32do all the other things
00:33like solve equations or inequalities
00:39or whatever you need to
00:40do. So this lesson is
00:42just going to focus on
00:43sketching this to kind of
00:44get an idea of what
00:45it looks like. So the
00:46first thing is the x
00:47intercepts. Let's do them step
00:48by step. So the first
00:49thing I'm going to do
00:50here, the first thing I'm
00:52going to do is find
00:53the x intercept. Now the
00:54x intercept
00:56happens when y equals zero.
00:58So this is zero. New
00:59narrator has to be zero.
01:01So literally, all I'm left
01:02with is x minus one
01:03equals zero, x equals one.
01:06So the x intercept is
01:09when x equals one. Second
01:11one. The y intercept with
01:14the y intercept happens when
01:15x equals zero. So x
01:17is zero, zero, zero, zero.
01:18I'm not sure what negative
01:19one over negative six, which
01:22is one over six. So
01:23the y intercept
01:24is y equals one over
01:27six, three. The horizontal asymptote.
01:32Okay, when the quadratic is
01:36in the bottom, the horizontal
01:38asymptote is always y equals
01:42zero. And the reason for
01:44that is the horizontal asymptote
01:46explained this in the previous
01:48rational functions. Listen,
01:52the horizontal lesson is basically
01:54what's happening when x is
01:55really really big or really
01:57really small. So are really
01:59really really large in the
02:00positive direction or the negative
02:02direction. When imagine x is
02:03a million then you're going
02:04to have a million over
02:06a million million which is
02:09basically one over a million
02:10which is very very very
02:11small. So because this the
02:14degree of this polynomial in
02:16the denominator is bigger than
02:17the degree of this linear
02:18function here as x
02:20gets really really large, y
02:22gets really really really small
02:24and it approaches zero. So
02:26you have your horizontal last
02:27note of y equals zero.
02:30For the vertical last note,
02:33so the vertical last note
02:35occur when the denominator equals
02:40zero. I explain that again
02:44in the other lesson, but
02:48it makes sense
02:48And because if you think
02:49as the denominator approaches zero,
02:53the function approaches infinity, because
02:55you're dividing by a really
02:56small number. But you can't
02:58actually divide by zero, so
02:59you get an asymptote. So
03:02this sounds when the denominator
03:04equals zero, x squared minus
03:06x minus six equals zero.
03:08Now we're going to have
03:08to solve this, and we're
03:09going to have two vertical
03:13asymptotes. So I'll just factorize
03:16this quickly.
03:16this is x minus 3x
03:20plus 2x equals 3 or
03:25x equals negative 2. So
03:27these are my two vertical
03:31asymptotes. And then finally what
03:34I'm going to do is
03:35assign diagram right now this
03:39may be your first time
03:41seeing this. So let me
03:44explain it but it's quite
03:47it's quite neat or interesting
03:49or it's gonna help me
03:50sketch this graph definitely. So
03:52what I'm looking for is
03:54this is like my think
03:56of these as my x
03:57values and I'm looking for
03:58my key values so are
04:00the key points the key
04:02points are the x intercept
04:05so this is going to
04:06be one at one so
04:08let's say this is one
04:09and then these are some
04:12total
04:12Let's, our x equals three,
04:15our x equals minus two.
04:16So let's go one, two,
04:18three. This is three, and
04:20I want zero negative one,
04:24negative two. So something like
04:26this. Now, at negative two
04:31and three, we have asymptotes.
04:34So I'm just gonna put
04:35a dot, dot, dot, dot.
04:37That is my vertical asymptote,
04:40dot dot dot. And at
04:42one, I have zero. So
04:45what I'm kind of, what
04:46I'm doing with a sine
04:47diagram is I'm writing is
04:51the function positive or negative
04:53or zero or an asymptote.
04:55So here it's an asymptote
04:57zero and asymptote. I'm more
04:59interested though, is it positive
05:01or negative? Because I've already
05:02got these. So when the
05:05function is less than negative
05:07two,
05:08Is this function positive or
05:14negative? Now you can think
05:15of it as well, let's
05:18think of a value and
05:18you could choose any value
05:20less than negative 2 and
05:21it will work, but you
05:22might as well if it's
05:23saying any value less than
05:24negative 2 and it's not
05:25going to change. It's not
05:26because there's no other kind
05:27of key point here. It's
05:28not going to change anywhere
05:30below negative 2. So I
05:32might as well pick negative
05:33100 or even negative 1
05:34million, but let's go with
05:35negative 100. If I have
05:36of negative 100. So if
05:42I sub negative 100 in
05:43here, I'm going to have
05:44the numerator is going to
05:45be a negative value. And
05:47then the denominator is going
05:49to be negative 100 squared,
05:51which is 10 ,000, and
05:52then plus 100. Well, but
05:54I don't even have to
05:54do the calculation. I know
05:55that the numerator is going
05:56to be negative, and the
05:58denominator is going to be
05:59positive. So if I get
06:00a negative over a positive,
06:03I get a negative. So
06:04everything
06:04less the function for x
06:08values less than negative 2
06:10is going to be negative
06:11and I put a minus
06:13sign or a negative sign.
06:16Between negative 2 and 1,
06:20what's it going to be?
06:21Well, the easiest thing to
06:22put in here is 0
06:23because 0 is pretty negative
06:252 and 1. If I
06:27put in 0, I get
06:281 sixth. So here it's
06:30going to be positive
06:33of between 1 and 3.
06:37What's it going to be?
06:38Well, let's put in 2.
06:422 is value to put
06:43in there. If I put
06:45in 2, I get a
06:48positive on the numerator. 2
06:50minus 1 is 1. So,
06:51a positive on the numerator.
06:53And then 2 squared is
06:544, 4 minus 2 is
06:552 minus 4 is negative
06:574, negative. So, I have
06:58a positive over negative. So,
07:00this is negative.
07:01and then if I put
07:02in a number bigger than
07:04three, let's go with again
07:05a hundred. I'm gonna have
07:07a positive over a positive,
07:10definitely, so this is going
07:11to be positive. So that's
07:14my signed diagram. Now you're
07:16gonna see where this is
07:16very, very useful as I
07:19draw my graph. Okay, so
07:23let's get out. Let's get
07:27out a set of
07:29axes here. Okay, now I
07:37know I have, there's some
07:42key points here happening at
07:43negative two, one, three. So
07:45let's put in my asymptotes
07:47first. So I'm gonna put
07:49in, I'm gonna put in,
07:51I'm gonna use different colors
07:53obviously, you don't need them,
07:54I just use dotted lines
07:55if you want, but for
07:56Nate I'm gonna put
07:57negative two, a vertical asymptote
08:01there. And the three, a
08:04vertical asymptote. So that's negative
08:06two, one, two, three. Let's
08:09go here. Doesn't have to
08:11be perfectly accurate, but this
08:14is negative two, and this
08:17is three. And at one
08:20here, it's gonna cross the
08:25the x -axis. So let's
08:29try and figure out what's
08:30going on here. So we
08:33know there is a horizontal
08:37asymptote at zero, so here.
08:40And we know there's a
08:41vertical asymptote at negative two,
08:43and we know that the
08:44function is always negative anywhere
08:46less than negative two. So
08:48what's gonna happen is, look,
08:50I have a horizontal asymptote
08:51here.
08:53And a vertical asymptote here.
08:56So this is going to
08:57approach this, approach this, and
08:59it's always negative. Fine. Next
09:02bit. Between negative two and
09:05one. So between negative two
09:07and one, it is positive.
09:11It's also, I should have
09:12put in, it's going to
09:12cross the y -axis at
09:15one sixth. These don't have
09:17to be the x and
09:19y axis kind of
09:21different scales but whatever. So
09:23this is going to be
09:24this is one sixth. So
09:27it's positive actually I don't
09:30think that's going to work
09:31too nicely. Let's put that
09:33in after right but it's
09:36between negative two and one
09:36it's positive and it's going
09:38to touch the x axis
09:40at one and then it's
09:42going to go negative and
09:44it has a an asymptote
09:46here. So what's actually happening
09:47is it's positive so it
09:49has to
09:49approach this asymptote. And this
09:52is not drawn very well.
09:54It has to approach that
09:55asymptote there. And then it
09:58has to do this. Sorry
10:00guys, this I just cannot
10:02accept that. Okay, so something
10:06like this. Now here's my
10:08asymptote. Here is my asymptote
10:12there, whatever. It's going to
10:15touch go through and here's
10:17my one sixth goes through
10:19one sixth it goes through
10:21one and then from one
10:23to three it's negative but
10:25again I have this vertical
10:27as much as gonna have
10:28to come down like this
10:29and do something like this
10:33select that is not a
10:34great drawing but whatever you
10:35get the point just come
10:36down here through one down
10:39like that so here it's
10:41negative fine and then after
10:43three it's positive and again
10:45And I have my vertical
10:47estimate out here and my
10:48horizontal estimate out here is
10:49just gonna look a bit
10:50like this, but in the
10:51positives it's gonna look like
10:53this. Okay, that's it drawn.
11:00That is this function, x
11:02minus one over x squared
11:04minus x minus six. Now,
11:08if, I mean, depending on
11:10the question, if he says
11:12indicate
11:13equations of vertical horizontal asymptotes
11:17will then write down here
11:18at the side what your
11:19horizontal asymptotes are. Or in
11:21your vertical asymptotes, or you
11:22can write here x equals
11:24negative two, x equals three,
11:27you can even put here
11:28y equals zero. And again,
11:30like you've shown, I mean
11:32you will show your working
11:35here. If I hadn't put
11:36that I'd have put number
11:37three horizontal asymptote by equals
11:39zero. Okay, I want
11:41want to just show you
11:45on Desmos what's going on
11:48here. So this is basically
11:51what I've drawn. We have
11:53our vertical estimate of negative
11:54two and the three. It's
11:57come down here through a
11:59sixth. That will be one
12:01sixth and goes into the
12:03negative. So it's obviously a
12:05better version of what I
12:06drew. There are different, this
12:09can look
12:09different ways. So like here
12:13for example it went from
12:14the it went from the
12:16positive into the negative but
12:17it's possible for that to
12:20look these generally look depending
12:23on how things depending on
12:24what these these letters are
12:27and I've added these sliders
12:28here just to change things
12:29and show you the transformations
12:30but I mean this doesn't
12:33have to go back into
12:34the negative so depending on
12:36I mean obviously
12:38calculating those equations of asymptotes
12:40and things change as you
12:41go through here, but these
12:43graphs can look slightly different.
12:46Something like this, it doesn't
12:47have a vertical asymptote because,
12:49well, if you did the
12:52math, the denominator doesn't have,
12:55there are no solutions for
12:56this to equal zero. So
12:58let's put that back to,
13:01sorry, this was one. So
13:03you can change, I recommend
13:05you actually
13:06going around and playing with
13:07these just to see the
13:10different types that you can
13:12get. But the two, the
13:13two usual ones are the
13:14ones that I drew and
13:16the ones like this where
13:18instead of going down into
13:19the negative, it comes back
13:20up. Now exactly once this
13:21minimum point, well, until you
13:24study calculus, you won't be
13:26able to get that, unless
13:28obviously you have a calculator.
13:29And that's the final thing
13:31I want to say is
13:32obviously if this is a
13:34paper
13:34per 2 and you have
13:34a calculator, you can just
13:37sketch it or you can
13:39just draw it yourself. What
13:41the calculator are used the
13:42calculator to solve any questions
13:46or solve any equations. Okay,
13:49that's it in the next
13:51lesson. I'll do the next
13:53type where I have to
13:54quadratic in the numerator and
13:55the linear function on as
13:57the denominator.