WEBVTT
Kind: captions
Language: en
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like i said if you are not yet signed up for
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the part 1 retake, you need to sign up
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you should have an email from professor sutherland
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you should sign up
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if youre taking it in the testing center this week
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if you dont take it this week you dont take it al all
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and you are stuck with your bad grade
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the midterm is 2 weeks from tomorrow
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the midterm will consist of were not quite sure but
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certainly the stuff we did the last few days
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cognitive division
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stuff with circles and ellipses and then logarithms
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which is coming i know youre
00:00:51.180 --> 00:00:52.300
love logarithms
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so keep in much
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right now were are gonna do a little on rational functions, were not gonna do to much
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so a rational function is a function/function, ratio
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so basically the function became a fraction
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so if you have
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f(x)/g(x)
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that is a rational function
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and of course at out of all these things the bottom cant be zero
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so thats always very important
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if the bottom is 0 you would have a problem
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has everyone gotten some?
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pass around
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are you guys all coming to homecoming on saturday?
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its a big game
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you dont have tickets?
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oh you just go to the athletic center and get tickets in the booth there
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i dont know
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hope a lot
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do you guys know if theres plenty extra tickets
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well you guys can get some, i already got mine
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but i have season tickets
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there you go
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i get took at the back of his head for most of the game
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a lot of too
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you can see me i right by the 45 yard line about five rows up
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near all the other old folks
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right there in the center
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youre suppose to say youre not old
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yea its a little late for that
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so a rational function
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is a function over a function
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you only need a couple of things to worry about
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suppose its something that looks like this
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alright so x-3/x+2
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what do we know about this function?
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we have a problem right there at x-2
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at x =-2 we have 0 in the denominator, which the function is not going to define there
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so at x=-2
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you will get a vertical asymptote
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or more general, what if the bottom is 0?
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and the top is not also 0
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where the bottom is 0 and the top is now also 0
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you will get a vertical asymptope
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so a vertical asymptote means the function will go up
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infinity or down minus infinity
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so if you were graphing it
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you know what graphs look like
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you can have more than one of these of course
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the second possibility, the second thing you have to watch out for are horizontal asymptotes
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as known as end behavior, so im gonna cover this up
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so rational functions use a graph
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they often look something like this
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youll get something that looks like that, see it has vertical asymptotes
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and horizontal asymptotes
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obviously there's many variations of this
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the verticle asymptotes are where the denominator is zero
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horizontal asymptotes has to do with the ratios of the two functions
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say we had
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something that looked like that
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were not gonna ask you to graph it, you can do that in a calculator
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and this will have a vertical asymptote
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2 places it has vertical asymptotes
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where the bottom is 0. so thats 1..
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and x equals -1
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plus or minus 1
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nice and simple, so if you are graphing it
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for example, this is not this
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but if you were graphing it
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[sneeze]
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you are very blessed
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have one at -1
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the curve is either gonna wind up or wind down, it depends on the signs
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so you sorta of have to find out where this is positive and where this is negative
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and the functions also has a horizontal asymptote.
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the horizontal asymptote is the end behavoir
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it means what happens when i say very large
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and there you have to look at how fast the top of the functio is growing
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and how fast the bottom of the function is growing
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so if you put in a number like
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20.. the top is 55 and the bottom is
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21 times 19 so thats a much bigger number
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alright because
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99
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so at a thousand
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the bottom is starting to get much bigger then he top
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and a million the top
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is so much smaller then the bottom that it starts to look like 0
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so this would have a horizontal asymptote at 0
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and thats what they call the end behavior
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this graph has to go out
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along the x axis
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you know to do something like that or come up like this
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id have to think about the graph
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if thats all we know about a horizontal asymptote, there are some rules
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as far as our lost post, help you figure
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end behavior
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so you have a polynomial on top and
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a polynomial on the bottom
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so the ax to the m will be the highest term power on the top
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even if theyre not written in order
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its whatever term on top had the higher power
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similar to the bottom, bx to the n
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whatever term on the bottom has the highest power
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okay we will do actually examples in a moment
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then theres 3 possibilities, either the top is bigger then the bottom
00:07:05.560 --> 00:07:07.560
the bottom bigger than the top
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or there equal
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these are not complicated right?
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bigger. smaller. or the same
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if the top is bigger then the bottom
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there is no horizontal asymptote
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if the top power is bigger than the bottom power, what that means is
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as the numbers start to get big
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the top will be so much bigger than the bottom
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it will dominate the function and the function will go through infinity
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we will do an actually example in a sec
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if the bottom is bigger then the top
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the horizontal asymptote
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is y=0, also known as the x-axis
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so it will go through and slice the x-axis
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and if theyre equal
00:08:05.100 --> 00:08:07.480
the horizontal asymptote is a/b
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hope you guys are following for actual math content
00:08:46.360 --> 00:08:49.420
okay lets do some example, do we have this written down?
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so if we had?
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so we have something like this
00:10:10.400 --> 00:10:13.360
well it says the second but ill make it the 3rd
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it doesnt matter okay, the only thing that matters is
00:10:16.700 --> 00:10:20.860
it doesnt matter about the cubed or squared root it only matters the highest power of each one
00:10:21.580 --> 00:10:24.880
because when you start to get to a really big number, like a million
00:10:24.880 --> 00:10:28.760
a million to fourth is a lot bigger than a million to the third
00:10:28.760 --> 00:10:31.600
you guys who have taken chemistry you know your scientific notation
00:10:31.920 --> 00:10:33.200
a million to the 4th
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is 10 to the 24th
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a million to the 3rd is only 10 to the 8th
00:10:37.740 --> 00:10:39.740
big difference, theres a million concepts
00:10:41.920 --> 00:10:46.060
similarly you only care about the highest term on the bottom so you only care about these two terms
00:10:47.840 --> 00:10:52.500
now since the term on top is bigger than the term on the bottom
00:10:52.880 --> 00:10:57.040
no horizontal asymptote
00:11:02.880 --> 00:11:04.740
we wouldnt ask you to graph this, its messy
00:11:08.560 --> 00:11:09.520
what if it said
00:11:20.200 --> 00:11:23.160
now i have the same power on the top and bottom
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so even though this is cubed and this has an x, it doesnt matter
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only thing that matters is the highest powers
00:11:33.740 --> 00:11:37.180
all the terms start to not matter when its very large
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you can use your calculator to see if you want
00:11:42.260 --> 00:11:45.700
so this will look like, 3x to the 4th over 5 x to the 4th
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so the horizontal asymptote will be 305
00:12:10.100 --> 00:12:12.420
alright and one other possibility
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now i have 5x to the 5th on the bottom
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for those of you who cant read my hand writting
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so again it doesnt matter whats going on on top
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if the bottom power is bigger than the top power
00:12:39.760 --> 00:12:42.800
so now the horizontal asymptote is the x-axis
00:12:44.260 --> 00:12:44.760
y=0
00:12:55.620 --> 00:12:56.740
that make sense?
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were actually gonna learn how to draw one of these
00:13:01.860 --> 00:13:02.660
get excited
00:13:03.980 --> 00:13:04.860
very excited
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so far so good? alright im going to cover this up
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no
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you guys are going slowly over ther
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alright?
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i cant wait any longer im not getting any youger
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lets do one of these
00:14:10.200 --> 00:14:12.040
suppose i want to graph that
00:14:15.160 --> 00:14:18.440
see you very nervous and youre like ugh seriously
00:14:20.100 --> 00:14:20.740
graphing
00:14:31.360 --> 00:14:34.720
look ill need very good doctors when the time comes
00:14:34.720 --> 00:14:36.920
i want to know where my a students are
00:14:36.920 --> 00:14:39.480
dont want any of the b students doing any surgeries
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just a students alright
00:14:43.860 --> 00:14:46.780
so we can figure out a few things right away from looking at it
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first of all we notice the vertical asymptotes
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vertical asymptotes where the bottom is 0
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bigger power on the bottom the horizontal asymptote is 0
00:15:50.600 --> 00:15:52.120
this curve eventually
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go out to the x axis
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what else do i know?
00:15:58.280 --> 00:16:00.520
when x=2, what does this equal to?
00:16:01.340 --> 00:16:03.020
when x equals 2 the top is 0
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bottom is not 0
00:16:05.460 --> 00:16:09.060
any fraction with a zero on top and not on the bottom is 0
00:16:10.180 --> 00:16:13.780
okay so im gonna cross the x axis and have an xintercept
00:16:17.240 --> 00:16:18.640
x=2
00:16:19.040 --> 00:16:21.520
so again if you plug in 2 on top you get 0
00:16:21.520 --> 00:16:25.380
you plus in 2 on the bottom and you get something else and dont really care
00:16:26.060 --> 00:16:27.580
0 over numbers equals 0
00:16:28.160 --> 00:16:29.760
have you realized that?
00:16:31.560 --> 00:16:33.480
this is going to go right here
00:16:33.520 --> 00:16:34.480
gonna go from 2
00:16:37.900 --> 00:16:40.140
other what happens when i plug in 0
00:16:40.140 --> 00:16:41.280
thats the y intercept right?
00:16:46.900 --> 00:16:49.780
well lets see when i plug in 0 i get negative 2
00:16:50.420 --> 00:16:53.060
over 3 times 4 over negative 12
00:16:53.620 --> 00:16:56.020
so negative 2 over negative 12 is 1/6
00:16:58.700 --> 00:17:00.140
hat would be this spot
00:17:06.100 --> 00:17:09.620
so now i have a pretty good idea what this graph looks like, believe it or not
00:17:10.100 --> 00:17:10.980
in the middle
00:17:12.960 --> 00:17:13.680
this graph
00:17:14.100 --> 00:17:16.740
has to look like that, how do i know that?
00:17:18.740 --> 00:17:21.140
well lets see it has to go through 1/6
00:17:21.720 --> 00:17:22.920
has to go through 2
00:17:22.920 --> 00:17:27.360
these are vertical asymptotes so when it gets close to the asymptote you have to go to infinity
00:17:27.820 --> 00:17:30.380
how do i know it goes up here rather down
00:17:30.380 --> 00:17:31.600
because if it went down
00:17:31.600 --> 00:17:33.620
i would have another x intercept
00:17:35.780 --> 00:17:38.260
so if it went f=down through here it would cut through the x axis
00:17:38.260 --> 00:17:40.960
and it doesnt cut through the x axis it has to go up
00:17:41.840 --> 00:17:44.240
here i guess it could bounce and go up
00:17:44.240 --> 00:17:46.460
and just sort of touch the x axis and go up
00:17:46.740 --> 00:17:47.620
its possible
00:17:47.620 --> 00:17:50.640
pick a number through 3 and 4 and see what happens
00:17:51.260 --> 00:17:53.020
you plug in 3 you get 1 on top
00:17:53.740 --> 00:17:56.780
you 6 times -1, you get a negative on the bottom
00:17:57.500 --> 00:17:59.520
you get a negative number so you must be going down
00:18:01.760 --> 00:18:02.480
do it again
00:18:02.480 --> 00:18:06.380
just sign test it, you only care if its positive or negative
00:18:06.920 --> 00:18:09.480
take a number between 2 and 4 and thats 3
00:18:10.000 --> 00:18:11.760
and you plug it in each part
00:18:11.760 --> 00:18:14.080
you plug in 3 here and you get a positive number
00:18:14.080 --> 00:18:18.380
plug in 3 here you get a positive number, and plug in 3 here you get a negative number
00:18:18.380 --> 00:18:21.960
positive, positive, negative will make the whole thing negative
00:18:23.100 --> 00:18:25.380
you guys get that? no?
00:18:25.680 --> 00:18:26.400
do it again
00:18:27.380 --> 00:18:29.720
when you plug in 3, this is 3-2
00:18:30.400 --> 00:18:31.120
this is 3+3
00:18:32.340 --> 00:18:33.300
and this is 3+4
00:18:33.740 --> 00:18:34.780
thats equal to 1
00:18:34.880 --> 00:18:37.200
6 times negative 1, thats negative
00:18:37.780 --> 00:18:40.260
you dont care what the actual value is
00:18:40.400 --> 00:18:41.520
in the sketching
00:18:41.520 --> 00:18:43.920
so thats how i know it goes down here rather than up
00:18:45.340 --> 00:18:50.940
so dont worry i dont expect you guys to be able to do this instantly, just follow along
00:18:51.300 --> 00:18:53.860
now what happens when x is bigger then 5
00:18:54.440 --> 00:18:57.000
on either, i got it here on the asymptote
00:18:57.000 --> 00:18:59.420
and later i have to get this asymptote so i either
00:18:59.420 --> 00:19:03.440
i either go like this or go like that and i got to figure out which one
00:19:03.820 --> 00:19:06.860
if you just take any number bigger than 4 like 5
00:19:06.860 --> 00:19:09.760
and you plug it in and see if you get a positive or negative
00:19:09.840 --> 00:19:11.200
if you get a positive
00:19:11.280 --> 00:19:13.680
it must be this and if i get a negative
00:19:13.980 --> 00:19:14.940
it must be that
00:19:15.580 --> 00:19:17.020
that make sense? Yes?
00:19:20.240 --> 00:19:23.520
the y intercept is when y equals 0, you plug in 0 and get negative 2
00:19:24.820 --> 00:19:25.320
3
00:19:25.760 --> 00:19:26.260
-4
00:19:26.260 --> 00:19:27.940
you multiply it out you get
00:19:29.080 --> 00:19:29.580
1/6
00:19:32.540 --> 00:19:35.380
so what you are doing is you just want to find the signs of things
00:19:36.460 --> 00:19:37.420
so if i say x is 5
00:19:37.420 --> 00:19:42.120
if i get a positive value ill be up here, if i get a negative value ill be down there
00:19:42.440 --> 00:19:44.620
and you know it has to have one of those two shapes
00:19:44.700 --> 00:19:48.180
it has to have this asymptote and this asymptote
00:19:48.400 --> 00:19:50.160
so you put in a number like 5
00:19:51.620 --> 00:19:54.040
5 minus 2 on top is positive
00:19:55.160 --> 00:19:56.440
5 plus 3 is positive
00:19:57.460 --> 00:20:00.820
5 minus 4 is positive, the whole this is positive so
00:20:02.760 --> 00:20:03.880
its gotta be that
00:20:06.140 --> 00:20:08.540
well practice more than one of these
00:20:10.140 --> 00:20:12.200
and now i take a number to the left of negative 3
00:20:12.200 --> 00:20:15.820
to figure out if its doing this or doing that
00:20:17.100 --> 00:20:18.700
so i pick a number like -4
00:20:19.140 --> 00:20:20.020
i go to the top
00:20:21.560 --> 00:20:23.560
-4-2 thats -6, thats negative
00:20:24.540 --> 00:20:27.140
-4+3 thats -1 so thats negative
00:20:27.860 --> 00:20:30.420
-4-4, tis negative 8 thats negative
00:20:30.420 --> 00:20:32.400
so negative negative negative
00:20:32.580 --> 00:20:34.980
3 negatives means negatives right?
00:20:34.980 --> 00:20:37.880
because 2 negatives is positive so one negative makes it negative again
00:20:40.820 --> 00:20:44.740
and youre done, you dont have to be anymore precise than tha
00:20:45.660 --> 00:20:49.040
okay so what did we do.... we figured it out where it crosses the axis
00:20:49.800 --> 00:20:53.460
we figured out the asymptotes and then we just figure out if its positive or negative
00:20:53.780 --> 00:20:57.540
you all long a bit stunned so well do alittle more of these
00:21:05.100 --> 00:21:07.100
can slightly use the other one
00:21:20.420 --> 00:21:22.260
this will be nice and simple like that
00:21:23.640 --> 00:21:24.520
simple for me
00:21:24.580 --> 00:21:26.420
now you guys are doing it for
00:21:40.820 --> 00:21:43.060
so were gonna find some key things
00:21:44.040 --> 00:21:47.240
first well find where the vertical asymptote is
00:21:47.240 --> 00:21:50.660
the vertical asymptote is where the bottom is equal to 0
00:21:52.200 --> 00:21:54.200
so wheres the bottom equal to 0
00:21:56.600 --> 00:21:58.080
x equals -2
00:22:10.200 --> 00:22:11.720
then i look and say okay
00:22:11.720 --> 00:22:14.780
how about a horizontal asymptote, well the power of the top is x
00:22:15.040 --> 00:22:16.800
the power of the bottom is x
00:22:17.200 --> 00:22:20.160
they are the same but look at the coefficents
00:22:21.500 --> 00:22:22.700
1/1 so its just y=1
00:22:26.400 --> 00:22:30.040
this is x to the 1 on top and this is x to the one on the bottom so same pwero
00:22:30.220 --> 00:22:35.420
we just care what the coefficient of this is and that is. its y=1 so that tells me
00:22:37.700 --> 00:22:40.580
when they go out to infinity, im gonna be at1
00:22:42.800 --> 00:22:43.840
y=1 again, sure
00:22:44.120 --> 00:22:45.720
okay so the power of this
00:22:47.180 --> 00:22:49.980
is x to the 1 and the power of this is x to the 1
00:22:51.080 --> 00:22:53.060
remember the rule, when the powers are the same
00:22:53.060 --> 00:22:55.800
its just whatever that coefficient is, a.
00:22:55.800 --> 00:22:58.360
over this coefficient b. well theyre both 1
00:22:59.920 --> 00:23:01.120
right? 1x to the 1x
00:23:01.620 --> 00:23:02.800
so 1/1 is 1
00:23:04.300 --> 00:23:05.340
okay, you sure?
00:23:08.620 --> 00:23:10.660
so now we know when i go to infinity
00:23:10.660 --> 00:23:13.620
im getting to 1, so the question is just whether im gonna go above
00:23:14.080 --> 00:23:15.280
or below to get to 1
00:23:15.280 --> 00:23:17.000
well lets see whats the x intercept
00:23:17.320 --> 00:23:18.920
another when the top is 0
00:23:20.080 --> 00:23:21.880
top is 0 and x=4
00:23:23.220 --> 00:23:26.180
when i get x=4, i get 0 on top and 6 on the bottom
00:23:27.380 --> 00:23:28.620
0/6 is 0
00:23:29.420 --> 00:23:30.140
so when x=4
00:23:32.140 --> 00:23:33.500
i go through the x axis
00:23:40.340 --> 00:23:43.360
now with the y intercept, with the y intercept i plug in x=0
00:23:46.100 --> 00:23:47.540
so i get 0 minus 4 on top
00:23:48.980 --> 00:23:50.100
0+2 on the bottom
00:23:50.960 --> 00:23:51.760
so minus 4/2
00:23:52.700 --> 00:23:53.200
is -2
00:24:00.920 --> 00:24:02.280
now im gonna get y=-2
00:24:02.360 --> 00:24:04.600
which is.. that looks about right
00:24:05.600 --> 00:24:07.960
now we just have to sorta figure out what happens
00:24:08.540 --> 00:24:10.700
okay i have avertical asymptote
00:24:10.700 --> 00:24:14.360
and i have a horizontal asymptote and i have to go from -2,4
00:24:15.040 --> 00:24:17.440
so i probably do something like thae
00:24:20.260 --> 00:24:22.340
the only place i cross the x axis
00:24:22.340 --> 00:24:24.900
is here the only place i cross the y axis is here
00:24:24.900 --> 00:24:27.720
and i have to get down there some how and there somehow
00:24:29.640 --> 00:24:32.760
so on the other side im either going to have this
00:24:32.760 --> 00:24:33.820
or im gonna have this
00:24:34.760 --> 00:24:35.640
how do i know?
00:24:36.220 --> 00:24:40.400
well you have two possibilities, one you can take the number to the left of -2, like -3
00:24:40.600 --> 00:24:42.840
and see if you are above 1 or below 1
00:24:43.280 --> 00:24:44.960
but i also know something
00:24:45.720 --> 00:24:50.020
if the bottom curve is right there, were gonna have an x intercept somewhere around there
00:24:50.020 --> 00:24:53.200
i dont have anymore x intercepts, i only have the one'
00:24:53.540 --> 00:24:55.380
so its gotta be the top arch
00:24:58.260 --> 00:25:00.260
thats it thats the whole graph
00:25:04.220 --> 00:25:04.880
how we feeling about these?
00:25:13.940 --> 00:25:15.960
want to practice some more, lets practice some
00:25:20.320 --> 00:25:22.080
okay any you guys want to do
00:25:24.940 --> 00:25:25.980
nice simple one
00:25:41.360 --> 00:25:44.240
how bout that, lets see if you can graph that
00:25:44.240 --> 00:25:49.460
so find the vertical asymptote, the horizontal asymptote, the x's, the y intercept and sketch the graph
00:28:07.140 --> 00:28:11.060
okay to find the x intercept you set the numerator equal to 0
00:28:11.580 --> 00:28:14.360
to find the y intercept you let x=0
00:30:11.580 --> 00:30:13.220
do we need onemore minute or are we ready
00:30:52.760 --> 00:30:54.160
alright thats probably long enough
00:31:17.060 --> 00:31:18.920
okay so va stands for vertical asymptote
00:31:22.340 --> 00:31:25.420
whats the vertical asymptote, when you take the bottom and set it equal to 0
00:31:25.800 --> 00:31:26.980
you get x=-1
00:31:30.260 --> 00:31:32.020
thats where the bottom is 0
00:31:37.900 --> 00:31:38.620
thats easy
00:31:39.480 --> 00:31:41.000
horizontal asymptote
00:31:41.000 --> 00:31:43.180
well now we look at the power of the top
00:31:43.480 --> 00:31:46.200
which is 3x and the power of the bottom is x
00:31:46.740 --> 00:31:48.180
so we just do the ratio
00:31:48.900 --> 00:31:51.800
get 3 and 1, so its 3/1 is 3
00:31:54.900 --> 00:31:55.700
and thats a 3
00:31:56.420 --> 00:31:57.380
so far so good?
00:31:59.020 --> 00:32:02.460
so the power of the top is 3x so the highest power is 3x
00:32:02.460 --> 00:32:04.940
the highest term, the highest term on the bottom is x
00:32:06.200 --> 00:32:07.960
so we look at the ratio, 3/1
00:32:08.320 --> 00:32:09.040
include 1x
00:32:09.760 --> 00:32:10.640
so 3/1 is 3
00:32:12.000 --> 00:32:14.320
we have a horizontal asymptote of 3
00:32:17.500 --> 00:32:19.660
we have a vertical asympote at -1
00:32:20.240 --> 00:32:22.320
okay lets find some intercepts
00:32:22.320 --> 00:32:24.720
the x intercept is when the top is equal to
00:32:24.720 --> 00:32:26.440
to 0. when the numerator is 0
00:32:27.140 --> 00:32:29.360
which is x=, sorry
00:32:32.920 --> 00:32:35.400
x=5/3
00:32:35.800 --> 00:32:36.600
cause again
00:32:36.920 --> 00:32:38.220
when x=5/3
00:32:38.560 --> 00:32:39.280
the top is 0
00:32:39.700 --> 00:32:42.740
0 over anything other than 0 is 0.
00:32:43.520 --> 00:32:45.860
okay that means as long as you have 0
00:32:48.520 --> 00:32:51.200
the whole thing is 0. okay? thats the x intercept
00:32:52.840 --> 00:32:53.880
the y intercept
00:32:55.080 --> 00:32:56.040
if you plug in 0
00:32:56.900 --> 00:32:58.880
you plug in 0 on top you get -5
00:32:59.500 --> 00:33:00.820
you plug in 0 on the bottom you get 1
00:33:01.620 --> 00:33:04.180
the y intercept, is y=-5
00:33:05.460 --> 00:33:06.580
itd be down there
00:33:07.920 --> 00:33:09.280
that means the graph
00:33:11.280 --> 00:33:13.540
up. and go something like that on the right side
00:33:15.020 --> 00:33:16.620
okay how we doing so far?
00:33:17.520 --> 00:33:18.020
happy
00:33:18.660 --> 00:33:19.300
love life
00:33:21.980 --> 00:33:24.220
you only learn so much in math class
00:33:26.260 --> 00:33:28.320
now we can figure out the left side well..
00:33:28.800 --> 00:33:30.960
the left side cant look like this
00:33:31.540 --> 00:33:35.220
because if it looks like that we have an x intercept here
00:33:35.720 --> 00:33:36.360
so it must
00:33:37.960 --> 00:33:38.920
look like that
00:33:40.440 --> 00:33:42.440
the other way you can test that
00:33:42.920 --> 00:33:43.960
is take a number
00:33:44.280 --> 00:33:46.220
to the left of -1, like 2
00:33:46.540 --> 00:33:48.240
and we plug in -2 here
00:33:48.240 --> 00:33:52.460
you get a number above the horizontal asymptote, a number bigger than 3
00:33:52.460 --> 00:33:55.880
than its this branch, if you get a number less than 3 you get this branch
00:33:58.580 --> 00:33:59.700
that make sense?
00:34:00.160 --> 00:34:01.440
how we do on this one
00:34:02.780 --> 00:34:03.280
good
00:34:05.820 --> 00:34:06.320
yes
00:34:07.560 --> 00:34:09.720
howd i get which one? negative 5?
00:34:10.400 --> 00:34:12.240
when i plug in 0 on top i get -5
00:34:12.340 --> 00:34:14.580
when i plugin 0 on the bottom i get 1
00:34:15.140 --> 00:34:16.640
so -5/1 is -5
00:34:20.460 --> 00:34:21.500
oh right good so
00:34:21.500 --> 00:34:22.640
nick raised a good point
00:34:23.040 --> 00:34:24.320
the previous graph
00:34:24.540 --> 00:34:25.900
or couple graphs ago
00:34:25.900 --> 00:34:28.000
i crossed the horizontal asymptotes
00:34:28.000 --> 00:34:32.780
you can cross horizontal asymptotes but you cannot cross vertical asymptotes
00:34:32.780 --> 00:34:37.260
a horizontal asymptote is really just a description on what is going on outside of infinity
00:34:37.820 --> 00:34:41.500
but you can cross, you can cross as many times as you want
00:34:49.140 --> 00:34:51.860
if i plug in the number -2 it will be above 3
00:34:53.260 --> 00:34:54.860
the other way you know is
00:34:54.860 --> 00:34:58.020
so theres two ways to tell what the left side looks like
00:34:58.260 --> 00:35:01.520
one is pick a number to the left of -2, -1
00:35:02.100 --> 00:35:02.600
see what happens
00:35:03.120 --> 00:35:03.920
the other is
00:35:04.980 --> 00:35:07.860
just remember if the number to the left of -1
00:35:07.860 --> 00:35:09.500
where the graph looks like this
00:35:09.860 --> 00:35:11.620
you have to cross the x axis
00:35:12.040 --> 00:35:14.040
which means you have another x
00:35:14.040 --> 00:35:16.740
intercept which you dont, you only have 1 intercept
00:35:20.220 --> 00:35:21.340
lets do another 1
00:35:22.900 --> 00:35:23.840
slightly more dificult
00:35:32.680 --> 00:35:34.120
-5 once you plug in x=0
00:35:35.720 --> 00:35:37.720
you plug in x=0 on top and get -5
00:35:38.540 --> 00:35:40.780
you plug in 0 on the bottom you get 1
00:35:40.780 --> 00:35:42.600
-5/1 is -5
00:35:50.720 --> 00:35:51.520
so plug in -2
00:35:51.900 --> 00:35:53.940
you get a=-6-5
00:35:54.120 --> 00:35:55.580
over -2+1
00:35:56.540 --> 00:35:59.300
-11/1=11
00:36:00.120 --> 00:36:01.860
so when x is -2, y is 11
00:36:02.980 --> 00:36:06.700
which is above 3, so that means you must be above the curve not below the curve
00:36:19.800 --> 00:36:21.820
alright lets do one thats slightly more difficult
00:36:25.900 --> 00:36:29.660
i will work you guys up to a hard one and then we will be very impressed
00:36:43.120 --> 00:36:44.080
how bout that?
00:36:48.040 --> 00:36:51.080
so a couple of rules for everybody to remember
00:37:10.440 --> 00:37:13.800
alright if you want to find the vertical asymptote
00:37:13.940 --> 00:37:16.180
you set the denominator equal to 0
00:37:18.360 --> 00:37:20.520
you want to find the x intercepts
00:37:26.720 --> 00:37:28.800
you set the numerator equal to 0
00:37:38.180 --> 00:37:39.860
so to find the y intercept
00:37:50.320 --> 00:37:51.920
we plug in 0
00:37:53.760 --> 00:37:54.260
for x
00:37:57.000 --> 00:37:59.640
okay to find the horizontal asymptotes
00:37:59.700 --> 00:38:00.820
you need our rule
00:38:05.620 --> 00:38:08.420
you need the rule i gave you before
00:38:08.420 --> 00:38:10.080
and then to figure anything else out
00:38:10.380 --> 00:38:11.780
you do whats called sine test
00:38:12.300 --> 00:38:14.540
figure out where its positive and where its negative
00:38:14.660 --> 00:38:15.780
and thats enough
00:38:16.160 --> 00:38:18.880
to come up with a basic sketch of the graph
00:38:18.880 --> 00:38:21.440
doesnt have to be perfect you just have to know those things
00:38:21.440 --> 00:38:23.800
can you guys see that on the right side of the room?
00:38:25.240 --> 00:38:25.740
sorta?
00:38:29.980 --> 00:38:32.220
so lets figure this out, lets find
00:38:33.260 --> 00:38:34.300
vertical asymptote
00:38:36.340 --> 00:38:40.100
set the bottom equal to 0, its equal to 0 in two places. x=4
00:38:42.100 --> 00:38:42.800
and x=-4
00:38:54.700 --> 00:38:56.860
find the horizontal asymptotes
00:38:57.140 --> 00:38:58.740
the highest power on top
00:39:01.700 --> 00:39:02.820
is just 2x, just 1
00:39:03.100 --> 00:39:05.420
and the highest power on the bottom
00:39:05.420 --> 00:39:07.880
well its gonna be x squared multiplied out
00:39:07.880 --> 00:39:10.960
so youll get an x on top and a x squared on the bottom
00:39:11.860 --> 00:39:13.220
so thats gonna be y=0
00:39:14.920 --> 00:39:15.700
or the x axis
00:39:16.340 --> 00:39:17.720
did we find the asymptotes okay?
00:39:19.200 --> 00:39:20.160
so far so good?
00:39:21.260 --> 00:39:21.820
yes what
00:39:23.780 --> 00:39:25.440
i have 2 possible vertical asymptotes
00:39:25.580 --> 00:39:28.300
the bottom is 0 in the denominator is 0 at 4
00:39:28.300 --> 00:39:30.180
the denominator is 0 at negative 4
00:39:30.180 --> 00:39:32.620
there are two places where this gonna be
00:39:33.640 --> 00:39:34.840
asymptotes, yes?
00:39:43.600 --> 00:39:45.600
the top is greater than the bottom
00:39:45.800 --> 00:39:47.160
if the top is less than the bottom
00:39:48.340 --> 00:39:49.620
im not really gonna do to many of those
00:39:50.500 --> 00:39:51.060
alright
00:39:51.400 --> 00:39:55.240
now i want to find the x intercept, so i set the top equal to 0
00:40:00.000 --> 00:40:01.860
and thats what 2x+1=0
00:40:02.100 --> 00:40:02.600
or x is
00:40:04.720 --> 00:40:05.220
-1/2
00:40:12.240 --> 00:40:14.000
and to find the y intercept
00:40:18.440 --> 00:40:23.460
to find the x intercept, take 2x+1 set it equal to 0 and get x=-1/2
00:40:23.680 --> 00:40:25.920
find the y intercept set x=0
00:40:26.720 --> 00:40:27.680
and get 1 on top
00:40:28.620 --> 00:40:31.420
-4 and 4 on the bottom which is 16
00:40:31.900 --> 00:40:34.740
so y=-1/16
00:40:36.520 --> 00:40:38.120
okay lets see i have -1/2
00:40:41.600 --> 00:40:42.100
-1/16
00:40:45.260 --> 00:40:46.460
so my guess is this
00:40:48.120 --> 00:40:49.020
looks like thay
00:40:49.380 --> 00:40:50.440
how you doing so far?
00:40:51.380 --> 00:40:52.180
doing good?
00:40:55.960 --> 00:40:59.960
now lets try and figure out what the left side of -4 looks like
00:41:00.820 --> 00:41:03.100
you pick a number to the left of -4
00:41:04.580 --> 00:41:05.080
like -5
00:41:06.520 --> 00:41:08.420
plug it in the top and you get -11
00:41:08.940 --> 00:41:09.440
-9
00:41:10.900 --> 00:41:13.700
you plug it in the bottom you get -9 times -1
00:41:13.840 --> 00:41:16.160
so a negative, negative, negative
00:41:16.300 --> 00:41:18.380
so 3 negatives equal a negative
00:41:20.300 --> 00:41:22.220
the graph goes down like that
00:41:23.340 --> 00:41:26.300
you got to pick a number to the right of 4 like 5
00:41:26.980 --> 00:41:32.300
the top would be positive, this will be positive, this will be positive, so the whole thing will be positive
00:41:34.300 --> 00:41:35.980
look something like that
00:41:35.980 --> 00:41:38.300
anybody able to get the right graph?
00:41:40.360 --> 00:41:41.480
some of you? yay!
00:41:43.220 --> 00:41:47.860
not with this variation, depends on the minus signs and the plus signs
00:41:49.020 --> 00:41:52.540
notice we cross the horizontal asymptote right here
00:41:52.900 --> 00:41:54.920
thats okay alright
00:41:54.920 --> 00:41:57.380
you just cant cross the horizontal asymptote
00:41:57.640 --> 00:41:58.520
way out there
00:41:58.520 --> 00:42:03.480
now ill do one more to make sure everyone has the idea and ten we will do something else
00:42:05.980 --> 00:42:07.820
how we feeling about these?
00:42:13.160 --> 00:42:14.120
i plug in 0 for x
00:42:14.120 --> 00:42:15.460
to get the y intercept
00:42:16.660 --> 00:42:17.540
so the top is 1
00:42:18.480 --> 00:42:18.980
0 plus 1
00:42:19.680 --> 00:42:22.640
-4 and positive 4 is -16
00:42:23.400 --> 00:42:25.240
we feel good about this one?
00:42:25.240 --> 00:42:27.360
well do one more, lets make it a little messier
00:42:30.220 --> 00:42:31.820
see how good you guys are
00:42:53.340 --> 00:42:54.700
i factored it for you
00:42:58.360 --> 00:43:00.360
still some shakiness thats good
00:43:01.680 --> 00:43:04.160
this is al the energy we are putting in for rational functions
00:43:05.220 --> 00:43:07.040
not more than this
00:43:07.880 --> 00:43:10.760
so first lets find the vertical asymptotes
00:43:10.760 --> 00:43:13.840
vertical asymptotes, where the bottom is equal to 0
00:43:14.860 --> 00:43:16.060
its equal to 0 at -4
00:43:16.140 --> 00:43:17.900
its equal to 0 at positive 3
00:43:22.420 --> 00:43:23.940
so wehen graphing this
00:43:24.860 --> 00:43:27.100
we have a verticle asymptote at -4
00:43:28.400 --> 00:43:31.200
we have a vertical asymptote at positive 3
00:43:32.420 --> 00:43:33.380
so far so good?
00:43:33.520 --> 00:43:34.400
that was easy
00:43:34.420 --> 00:43:35.620
thats the fun part
00:43:36.500 --> 00:43:38.260
now i got to find horizontal asymptote
00:43:41.660 --> 00:43:44.540
okay you can have many vertical asymptotes
00:43:44.540 --> 00:43:47.740
you can have no more than 1 horizontal asymptote
00:43:47.740 --> 00:43:49.500
unless we give you a really bazar function
00:43:49.500 --> 00:43:51.380
but were not gonna do it okay?
00:43:51.780 --> 00:43:53.980
so horizontal asymptotes, well if you multiply out the top
00:43:54.600 --> 00:43:56.680
you get x squared and something
00:43:56.680 --> 00:43:59.880
and you multiply out the bottom you get x squared something
00:44:00.320 --> 00:44:02.200
so x squared over x squared is 1
00:44:02.200 --> 00:44:04.080
so the horizontal asymptote is
00:44:04.860 --> 00:44:06.480
is y=1, so agian
00:44:06.920 --> 00:44:08.760
x times x gives you x squared
00:44:08.760 --> 00:44:10.560
x times x gives you x squared
00:44:10.920 --> 00:44:12.200
so this is x squared
00:44:12.560 --> 00:44:14.320
and the bottom is x squared
00:44:15.560 --> 00:44:16.060
1/1
00:44:18.700 --> 00:44:19.900
its not 2x squared
00:44:20.160 --> 00:44:21.120
its not x cubed
00:44:21.120 --> 00:44:22.560
its just x squared and x squared
00:44:28.360 --> 00:44:29.480
okay x intercept
00:44:32.580 --> 00:44:33.940
theres 2 x intercept
00:44:33.940 --> 00:44:36.360
there are two places where the numorator is 0
00:44:36.560 --> 00:44:37.620
at x=1
00:44:38.740 --> 00:44:40.120
and x=-2
00:44:47.140 --> 00:44:50.660
agin you can have multiple places where x intercepts
00:44:50.720 --> 00:44:52.480
you only have 1 y intercept
00:44:54.200 --> 00:44:58.280
you have multiple intercepts the graph goes through -2 and -1
00:44:59.160 --> 00:44:59.960
y intercept
00:45:00.800 --> 00:45:03.300
you get by plugging in 0
00:45:05.240 --> 00:45:08.360
so you plug in 0 on top what do you get, you get 0-1
00:45:09.520 --> 00:45:10.820
times 0+2
00:45:11.940 --> 00:45:13.960
the bottom you get 0+4
00:45:15.260 --> 00:45:16.500
times 0-3
00:45:20.040 --> 00:45:21.940
alright thats plugging in 0 for all the x's
00:45:22.960 --> 00:45:23.520
so thats
00:45:24.760 --> 00:45:25.960
-1 times 2 thats -2
00:45:26.440 --> 00:45:28.820
-4 times 3 thats -12
00:45:30.020 --> 00:45:30.820
so thats 1/6
00:45:32.100 --> 00:45:34.340
that looks like a good spot for 1/6
00:45:37.780 --> 00:45:38.740
so far so good?
00:45:40.940 --> 00:45:41.820
so this graph
00:45:42.360 --> 00:45:43.880
gotta go through there
00:45:45.280 --> 00:45:47.160
probably look something like that in the middle
00:45:50.220 --> 00:45:52.620
how do i know it doesnt go up and down?
00:45:53.300 --> 00:45:56.020
you can try some values if you are not sure also
00:45:56.440 --> 00:45:59.160
ill give you a clue if this is gonna bounce
00:45:59.160 --> 00:46:02.260
one of these terms are gonna be squared so its gonna go like
00:46:03.140 --> 00:46:04.340
up like that, yes?
00:46:07.560 --> 00:46:08.600
horizontal, oh yes i didnt draw that sorry
00:46:10.740 --> 00:46:13.300
thats the horizontal asymptote okay?
00:46:15.980 --> 00:46:16.940
so far so good?
00:46:16.940 --> 00:46:20.600
its hard to sketch these, its easy to find points but hard to sketch them
00:46:21.780 --> 00:46:23.280
alright now
00:46:23.460 --> 00:46:25.540
whats happened to the left of -4
00:46:25.680 --> 00:46:28.320
are we above the x axis or below the x axis
00:46:28.900 --> 00:46:30.260
take anumber like -5
00:46:30.740 --> 00:46:31.860
this is negative
00:46:32.760 --> 00:46:33.880
this is negative
00:46:35.160 --> 00:46:35.800
negative
00:46:36.120 --> 00:46:37.240
this is negative
00:46:37.240 --> 00:46:40.700
4 negatives, an even number of negatives can make something positive
00:46:40.700 --> 00:46:44.160
negative times negative times negative times negative so
00:46:45.120 --> 00:46:46.480
gonna look like that
00:46:49.220 --> 00:46:51.980
okay take a number to the right of 3 like
00:46:52.280 --> 00:46:52.780
4
00:46:54.020 --> 00:46:55.780
positive, 4-1 is positive
00:46:56.360 --> 00:46:58.120
positive, 4+2 is positive
00:46:59.120 --> 00:47:00.160
4+4 is positive
00:47:00.820 --> 00:47:01.860
4-3 is positive
00:47:01.860 --> 00:47:04.840
everything is positive so the whole thing is positive, so again
00:47:05.640 --> 00:47:06.840
it looks like that
00:47:10.240 --> 00:47:11.440
these are painful
00:47:11.440 --> 00:47:13.920
i dont think we will give you antything this hard
00:47:14.700 --> 00:47:17.700
we might not do graphs at all we might just say find the points
00:47:18.040 --> 00:47:21.080
okay we may ask you the end behavior im not sure
00:47:23.040 --> 00:47:24.160
questions, yes?
00:47:27.580 --> 00:47:29.740
how do i know this goes down here?
00:47:29.740 --> 00:47:31.880
how do i know it doesnt go up like this?
00:47:33.800 --> 00:47:34.920
well first of all
00:47:35.580 --> 00:47:37.020
i know i love graphing
00:47:38.920 --> 00:47:41.160
pick a number between 1 and 3 like 2
00:47:41.440 --> 00:47:44.800
and see if you are above the x axis or below the x axis
00:47:46.040 --> 00:47:49.740
also take a number between -2 and -4 like -3
00:47:50.760 --> 00:47:54.680
see if you go up or you go down and the clue is when you get this
00:47:54.840 --> 00:47:57.320
when you bounce off the axis like that
00:47:57.720 --> 00:48:00.120
one of these factors will be squared
00:48:00.900 --> 00:48:02.100
or to an even power
00:48:02.100 --> 00:48:04.580
thats whats causing, whats happening is
00:48:04.580 --> 00:48:06.180
were getting sign changes
00:48:06.560 --> 00:48:10.420
every time, change. every time you get between the 0's
00:48:10.480 --> 00:48:12.320
the sign of this is changing
00:48:12.980 --> 00:48:15.500
okay as the sign changes
00:48:16.340 --> 00:48:17.940
notice this is positive
00:48:17.940 --> 00:48:20.840
then its negative and positive and negative and positive
00:48:20.840 --> 00:48:21.960
see you figure out
00:48:21.960 --> 00:48:24.380
when its above the axis and when its below the axis
00:48:36.480 --> 00:48:38.400
wanna take a number left of -4
00:48:39.280 --> 00:48:41.480
-5, -10, negative billion
00:48:43.820 --> 00:48:47.180
what you can do is make a sine chart if youre not sure
00:48:48.060 --> 00:48:49.340
one thing you can do
00:48:55.980 --> 00:48:59.100
you take the 0's of this and put them on the x axis
00:49:00.440 --> 00:49:02.100
not on an axis
00:49:02.100 --> 00:49:06.480
-4,2,1,3
00:49:07.120 --> 00:49:10.400
and then you can try and number each of these zones
00:49:10.400 --> 00:49:13.500
and see whether this expression is positive or negative
00:49:15.480 --> 00:49:18.360
its gonna do that when you try numbers okay?
00:49:18.720 --> 00:49:22.960
where do i get those numbers, i take a number less than -4, like -5
00:49:23.000 --> 00:49:26.280
and i plug it in and find the whole this is positive
00:49:26.600 --> 00:49:29.160
i find a number between -4 and -2 like -3
00:49:29.160 --> 00:49:31.620
and i plug it in and find it is negative
00:49:31.620 --> 00:49:33.400
and that will help you figure out
00:49:33.400 --> 00:49:35.420
when you go above and below the x axis
00:49:36.840 --> 00:49:39.400
this is called sin chart or sin testing
00:49:39.400 --> 00:49:43.660
sine testing is often a very good way to find out whats going on with a function
00:50:14.700 --> 00:50:16.140
how did we do on these?
00:50:16.760 --> 00:50:20.120
okay now were gonna lead to exponential functions
00:50:20.740 --> 00:50:23.540
so exponential functions and logarithms
00:50:23.680 --> 00:50:25.040
are closely related
00:50:25.040 --> 00:50:27.220
so lets give a few minutes to do exponential functions
00:50:27.840 --> 00:50:29.120
heres what happens
00:50:30.140 --> 00:50:32.540
so far we have been doing x to a number
00:50:32.920 --> 00:50:35.160
now we are gonna do a number to the x
00:50:35.160 --> 00:50:37.040
okay thats an exponential function
00:50:37.720 --> 00:50:39.560
3 to the x, 2 to the x, 5 to the x
00:50:40.800 --> 00:50:43.360
they exhibit a specific kind of growth
00:50:43.360 --> 00:50:46.020
the type you see in biology and chemistry
00:50:47.920 --> 00:50:53.200
biology you see a lot of exponential gorwth, for physics you see radio activity
00:50:53.820 --> 00:50:56.780
you see all sorts of things, you see the sound
00:50:57.100 --> 00:51:01.460
so when you ring a bell the sound dies off exponentially
00:51:01.460 --> 00:51:07.220
all sorts of things and in order to solve exponential equations we are gonna use our favorite thing from math
00:51:07.300 --> 00:51:08.100
logarithms
00:51:08.100 --> 00:51:11.860
okay if get stuck on them youll get an extra point on the tes
00:51:19.420 --> 00:51:22.700
so thats youre homework ,learn about logartihms
00:51:22.700 --> 00:51:25.120
wednesday were gonna do exponential functions