# Graphs of Secant and Cosecant Functions (Including Transformations)

(D4M) — Here is the video transcript for this video.

00:00
in this episode we’ll practice graphing periodic functions that involve secant
and cosecants let’s do some math [Music] all right everyone welcome back and
we’re going to begin by taking a look at
what happened previously on trigonometry
is fun so we went over these uh episodes right here so far
and so we talked about what are trigonometric angles degrees and radians
and we talked about the unit circle special angles on the unit circle we
also talked about reference angles and we talked about the trig functions
we talked about trig functions of acute angles and uh for real numbers
and in the last episode um we worked um quite diligently on graphing sines and
cosines functions that involve sines and cosines we changed the period we worked
with the amplitude we even did some shifts but today we’re going to focus on uh

00:01
secant and cosecant and so let’s see here what the graph of secant looks like
first so let’s start here with secant and then we’ll look at cosecant in a
minute now to graph co to graph secant i’m going to be looking at the function
cosine so remember the relationship between cosine
and secant right whenever we define the cosine function and the secant function
uh secant was the reciprocal of cosine in other words if if we look at this
right here this function right here is one over cosine of x
and so uh i this you know these this right here the shape of this right here
is related to the shape of cosine right here
so um that’s why we’re looking at cosine so i’m going to look at cosine first
here uh before i look at what secant looks like so actually going to make a
good graph right here let’s put it right about here

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and the cosine is going to go up and then come back down and go back up
and come back down and over here also and so yeah we talked about the shape of
cosine in the last episode now because of this relationship right
here this is 1 over cosine of x we have to be worried about where cosine
x is zero because secant is not going to
be defined where cosine is zero so where is this cosine graph right here is zero
so it’s zero right here and right here in here and so on and then it’s also 0
right here and here and so on and so let’s actually put some labels down here
so remember cosine had a period of 2 pi so this is 2 pi right here
and this is pi right here and this is pi over 2 and this is 3 pi over 2.

00:03
so those are those tick marks there so now we’re going to try to sketch the
graph of secant and we’re getting this graph because of this
this relationship right here secant is equal to 1 over cosine x so whenever
x is 0 the cosine will be 0 and this will be 1
over zero right so that’s just not defined right so what we’re gonna have
here is isotopes right here at x equals pi over two
and where else is cosine zero right here so we have another isotope vertical
isotope and this will be three pi over two
and then we have another one right here in fact we have infinitely many of them
and we’d like to figure out how we can write them all down
so this is one pi over two three pi over two
this one will be what five pi over two and also we have a vertical isotope here

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cosine is 0 which means secant is undefined and this will be negative pi over 2
and then we get another one right here and this one will be negative
three pi over two and the pattern continues
and so now we’re ready to look at the graph of secant let’s put that in purple
maybe let’s see how this goes so this is the
highest right here which is uh which is a one right here
for cosine for secant that’ll be um a local minimum or it’ll just be um
it’ll be the minimum between minus pi over two and pi over two
but this this is going to go down here so it won’t be an absolute minimum in
any case we have the graph right here shape like that

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and shape like this right here and we can get a little part of it over
here maybe and we got this part right here so this
purple part here is the secant graph so i’ll put it in purple secant of x
there we go so knowing what the graph of cosine
looks like we can sketch the graph of secant and we can remove the cosine
graph there if we wanted to there’s the graph of secant right there and so it’s
worthwhile to write down the information about it just to make sure we’re clear
so let’s do that over here so we’re going to have the domain what is the domain
of secant and so as you can tell here there’s some
vertical asymptotes so the domain is going to be all the real numbers x
but x cannot be equal to so how can we write down infinitely many values like
there like this so we’re going to kind of need a parameter notice they’re all

00:06
odd so we’re going to say 2n plus 1 times pi over 2 and then we’re going to say
where n is an integer where n is an integer
good there’s the domain that’s set right there
so it’s all real numbers x but x cannot be equal to
an odd multiple of pi over two it cannot
be pi over two three pi over two five pi over two and so on and when
you know n is an integer so it could be negative and so
these negative values here the range here is so what is the range so now we’re
looking along the y-axis right here if i want to label this as the y-axis right
here and so the range here let’s write that one in interval notation perhaps
so it’s going to be from 1 to infinity we’re going to include the 1 it’s going
to hit the 1 right there so 1 to positive infinity union and then minus 1 to

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and we’re including the minus 1 because it touches right there at minus one
so minus one to minus infinity so the union of those two intervals that
is the range right there um some people like to write this one first
but it’s a union so it doesn’t really matter but just because i’ll
write it again so minus 1 to or actually minus infinity will go first
or write like that minus 1 and then union 1 to positive infinity
all right and so either way you want to write it there’s the range
um so notice that the secant here is discontinuous for all values um
right here so let’s write that down discontinuous and discontinuous for
uh these x values right here whenever we’re equal to these x values so we’ll

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say the odd multiples of pi over 2. discontinuous for those also notice that
there’s no x-intercepts no no x-intercepts
right there’s nowhere where it crosses x-axis see it’s just up here it’s down
here and just keeps repeating that pattern right there there’s no
x-intercepts um the period is the same as cosine so period is 2 pi
so for example we can start right here at 0
and go up for this part and then we’re coming up and we’re going down and then
we’re going down and then repeats it’s going up and then
down down and then up and then it’ll repeat again
all right uh so another way you can think about it is
you know this is pi over two to minus pi over two so this is the whole pie right
here in between not including the end points

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so there’s one branch and then here’s another branch right here so you can
think about it as the period is two pi right there
also notice that there’s no max and no min no max no min
and by that i mean no absolute maximum no absolute minimum um
okay so there’s secret so i feel like there’s
you know some basic information that everyone needs to know about secant in
particular to sketch it what i’m always going to do is i’m going to
sketch first the corresponding secant graph or cosine graph and then i’ll
purple in the actual graph that i’m looking for which will be the secant
graph right there all right so now before we go on
let’s look at the cosecant graph here so let me erase this real quick
so as you might guess the cosecant graph is going to be based on
the sine function so this right here and that’s because

00:10
this right here is 1 over sine of x and so we’re looking at the reciprocal of
a known function we know what sine looks like and we’re looking for the
reciprocal of that so let’s draw a sketch of the sine graph first
and so let’s see here we go now sign starts here so i’m going to go up
and then down and then up and then down and i’ll try to do the same thing over
here and so on all right that’s good enough so this is the sine graph
just y equals sine x now of course this continues and that
and it continues over there um but to make the
cosecant graph well we need to worry about where sine is zero now because
that’ll be one over zero that’ll be undefined so where is sine zero it’s

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zero here and here and here and here and so on and also here and here
and so wherever sine is zero we’re going to have vertical isotopes
so let’s do do these vertical isotopes right here so the y-axis is going to be
a vertical isotope right there for the cosecant graph and we’re going
to have a another vertical isotope right here and actually let’s go ahead and
make a period so this is 2 pi and then this is pi and this is pi over 2
where we hit our height right here of one and this will be three pi over two
where we hit our minimum right there at minus one
and so we have another vertical isotope where sine is zero
another vertical isotope goes sine of zero again and again
and so this is x equals pi x equals two pi and then this will be three pi

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and this would be 4 pi and this is the period twice so go
through it once we go through it again and now we can go through it this way
and so this will be minus pi minus pi and then this right here will be -2 pi
and so on and so there we have the vertical
isotopes and that will help us draw the sketch of the secant graph
all right so here we go these are vertical isotopes so let’s
draw them like we’re getting close to them
some people like to put arrows on them so people don’t
and then let’s put them right here like this and like this and
hopefully you can tell that purple from the black
and then just to make sure that they’re all touching

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let’s put some points on them right there
there’s there we go there’s a height of one and they’re touching and they’re
touching and here’s a height of -1 oh and i forgot this little branch over here
something like that any case there’s a couple of branches
of the sine graph right there so i mean the cosecant so the label down
here so this is y equals cosecant x is the one in purple the one in black was
just to help us define it and the red lines the red dashed lines are the
vertical isotopes they’re not part of the graph either but just to make clear
let’s write down over here what the domain and the range is
so the domain let’s let’s put that up here let’s put it over here i guess so
the don’t let’s put it over here the domain
is the all the real numbers such that so um you know when we’re looking at

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cosecant x uh it’s almost defined everywhere except
for these isotopes here so we’re going to say all real numbers and then we’re
going to say the exceptions there are infinitely many exceptions
so x is not going to be equal to and so what are the values we’re going
to roll out here so in this case right here we’re going to say n times pi
and where n is an integer this will allow us to make infinitely
many values here where n is any integer all right so the domain is that set
right there and the range here we can write it out
in interval notation because it’s just got two pieces to it it’s going to be
one to infinity right here it’s going to be minus infinity to minus one so let’s
say minus infinity to minus one and then oh it includes minus one here so union

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and then one to positive infinity right there all right so there’s the range
so let’s notice where it’s discontinuous so where do you have to pick up your
pencil right so i’m drawing i’m drawing cosecant i’m drawing i’m drawing i’m
drawing i’m drawing and now we got to pick up our pencil to get over here
right so discontinuous so discontinuous mean
as long as you’re defined right so it’s discontinuous where x is equal to the n
pi where n is an integer that’s where it’s discontinuous
so the domain is where you can plug in numbers and you can plug in anything
where it’s not equal because where it is equal well it’s not defined it’s
actually discontinuous there the period is also 2 pi
we can see that by looking at the graph here’s one branch and here’s pi and
here’s another branch and here’s also pi and then it starts to repeat itself so

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the period is 2 pi and again there’s um no maximums no minimums no max
why is there’s no max well this part right here for example
just keeps going up and up there’s no maximum and this branch right here just
keeps going down there’s no minimum um yeah and so
now we’re going to start looking at the um you know
functions that trig functions or periodic functions that involve
secant and cosecant like for example you know what if we change the period
and things like that so let’s look at some examples
all right so here we go let’s get rid of this real quick
all right so let me outline the strategy for you that i’m going to use here so
let’s look at the y equals a and then we’re going to have cosecant bx

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and y equals a secant bx and b is going to be greater than zero for right now
all right and so how do we sketch these graphs here
right so we have a b and if you remember the last episode the b was changing the
period and the a was the the absolute value of the a
was the amplitude right and so what we’re going to do is we’re first going
to sketch the graph of a and then this one will correspond to sine and we’ll
first graph this right here and this will correspond to cosine
and so we’ll first graph these and we graph these right here in the previous
episode so again the previous episode here’s the trailer for it
trigonometry is fun step-by-step tutorials for beginners if you haven’t
checked out the previous episode the link is below in the description but
we’re going to graph these and we’re going to graph them pretty quickly too
because we already know how to do them so we want to sketch these right here
first and then we also want to sketch the
vertical isotopes the vertical isotopes are important because they show the
behavior of the graph and that’s important right here and then step three

00:18
we’ll actually try to you know sketch the graph we’ll sketch
the graph that we’re looking for so those are more or less the three steps um
all right so yeah let’s look at a couple of examples so
our first example is going to be um i’m going to give this the
sine cosine version and we’re going to graph the secant cosecant version of
this right here so sketch the graph so we did this one in our previous episode
and so now in this one we’re going to do y equals
2 cosecant x so how do we graph this right out here 2 cosecant x well first
we’re going to graph this one so this is just a basic sine graph i’m going to
come up and then down and i’m only going to be gra i’m i’m only going to be
graphing one period and so this will be pi pi over 2
and then this will be 3 pi over 2 and then this will be a 1 oh sorry no

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this will be a 2 and this will be a minus 2 right here
and so this is y equals sine 2x which we could put right there if we wanted to
so now i’m going to sketch the graph of 2 cosecant x of course this repeats
and it goes on but now i want to draw the vertical
isotopes so this will tell me right here where sine is 0 that will be a vertical
isotope for this graph we’ll have a vertical isotope right
through there at x equals zero and we’re going to have a vertical
isotope right through here at x equals pi so we’re labeling our
vertical isotopes and then we have another vertical isotope right here
at x equals 2 pi and so now in purple i’m going to sketch
the graph i’m looking for which is y equals 2 cosecant x that’s the graph
that’s new for this episode right here and we’re just going to sketch it in
right here and you want to get close to the isotopes you don’t want to do

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something sloppy like that you know you want to have a
reasonable graph that shows that you understand what a vertical isotope is
that’s good enough something like that you can put arrows or not
all right there we go you definitely don’t want to show
crossing but you want to show it getting closer that’s good so there’s our two
cosecant graph right there this is a point on it right here and this is a
point on it and we have the isotope sketched and labeled so that that shows
behavior right there and if you want we could get rid of the the the sign the
sine graph we don’t need any more if you wanted to get rid of that
there it is right there and now we can put in the axes a little better
and perhaps re-dash the isotopes there so there’s a reasonable graph of
2 cosecant 2 2 cosecant x right there all right so there’s our first example

00:21
right there let’s go do another one right here what happens if we do four
sine x what happens if we change that to a four
right all right so we did that one in our previous episode so now what happens
if we graph this one right here four cosecant x so what does the four
change we change it to a two to a four well over here we’re going to go and
change this to a four right here so this will be four
and this will be minus four and so this would be
a graph right here that you could use right here to to sketch to have the graph
now however it it it look it looks nice if you could compare the two to make
sure that you understand what’s going on what does that four do right here’s a
graph of it but you know how did how do these actually compare
so like if you had a sine graph that just went up to a height of one
and then you had a sine graph that went up to a height of

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two it still hits 0 at the same place now if we have 4 right so that will come
up to a height of 4 and then it’ll come down to
a minimum of 4 but they still go through the same zeros right there so there’s a
4 and there’s a minus four right there and so that’s that would be the fill in
part right there and so you can see what’s happening with the secant right
that’s what’s happening with the sign we talked about that last episode but now
it’s happening with the secant so the secant is coming down here to a one
and then the two is going to get you know it’s it’s going to it’s only going
to come down it’s going to go like this it’s going to come down to a 2
and then it’ll come down to a 4 and so you can kind of see
you know what that’s kind of doing there
but in terms of having to sketch a graph there it is right there
all right so let’s do another one here um let’s do a minus now
so the problem that we’re trying to solve we did this one in our previous
episode just want to keep saying that and then minus and then we have a three

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and then we have a cosecant so now let’s grab this one right here so the first
step is to graph this one right here so we have the graph of sine sine comes up
like that but now this is a minus three so now we’re going to go down like here
like this then we’re going to come up here like this and this is a three and a
minus three and we still have a two pi and a pi
and a pi over two and one pi over two 2 pi over 2 3 pi over 2
and so there’s a quick sketch of y equals minus 3 sine x
and so now let’s graph this one right here minus 3 cosine x
so next step is to put in the vertical isotopes we have vertical isotope right
there at zero we have a vertical isotope right there at pi
and we have a vertical isotope right here at two pi
and we can do more but that’s one period that’s enough and then
we’ll come in here now with our cosecant graph it’s going to come in

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here it’s going to be an isotope so let’s just draw it draw the behavior
like that and draw the behavior like this right there
um and so that would be the graph of minus 3 cosecant x and then we can erase
the sine graph there if we want all right let’s do another one minus cosine
and so now let’s do a secant so now we’re going to be trying
to sketch the graph of y equals minus secant x
that’s that’s the new problem that’s the previous one we did in the last episode
that’s the new problem right there all right so let’s sketch it
so first i’m going to sketch the graph of cosine right here here we go
uh minus cosine and so cosine normally starts up here now it’s going to start
here i’m going to go like that and come in through like that and then
go back down and then there’s one period right there 2 pi halfway is pi

00:25
halfway again is pi over 2 and so that would be three pi over two
all right so that’s the graph of minus cosine x right there in black
and now let’s put in the isotopes so where is cosine zero
so cosine 0 is right here so i’m going to have an isotope right there
cosine 0 right here so this will be the isotope x equals pi over 2
x equals 3 pi over 2 and then we could draw another one but
we don’t need to we could draw another one we got one period right there
so now i’m going to draw the um the secant now to do that i’m going to extend my
vertical isotope up here and here to show behavior
and so we’re going to get it like this in purple and so i’m going to come in
here like this and the amplitude here is 1
so this highest point right here is 1. put a one right there

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and so let’s see here we have a another branch right coming right through here
and another branch coming right through here excuse me all right so there we go
the purple one is the secret the minus secant graph right there
um i wonder if orange would show any better than the purple here
and then it’s hard to tell between orange and red any case
let’s go on to another one um what about one-fifth cosine
alright so that was our previous one now this one is going to be y equals
1 5 secant x so how do you sketch the graph of one fifth secant x
first i draw my cosine now it’s just a regular cosine so just

00:27
coming through like this this is a 2 pi pi pi over
one pi over two two pi over two three pi over two
now i’m going to draw my isotopes right here got one coming right through here
at zero at sorry pi over two and another one coming right through here
and so now in the purple we can sketch the secant graph
right there and right there and right there
i’ll extend this up here a little bit and what are these right here these tick
marks right here that’s a one-fifth and this one right here is a minus one-fifth
and so there’s the sketch of one-fifth x it’s in purple and let’s just go ahead
and erase this one right here just to erase one of them at least and then

00:28
this would be what we’re left with is the actual graph that we were looking for
this graph right here so there it is there it is right here
it’s going to bounce right there at the one-fifth
and bounce right here at minus one-fifth um
at pi minus one-fifth right there all right so there’s the graph of one fifth
secant and let’s do [Music] uh sine two x now now we’re gonna be
changing the period a little bit so now we’re going to sketch the graph
of instead of the sine one that was our previous one
so now let’s graph the sketch the graph of cosecant cosecant of 2x
so how does this change right so let’s see here um remember the period
is going to be 2 pi over the b the b is positive so it’s 2 pi over b
so the period is pi and so i’m sketching this one first just

00:29
like we did last time so it’s just a sine graph so i’m gonna come up
and then down and the period is pi so halfway is right here pi over two
halfway again is pi over four and then now we count up our pi over
fours one pi over four two pi over four three pi over four four pi over four
and so there’s one period of the graph of y equals sine two x
and now time for the secant graph time for the vertical isotopes first so where
is this one right here zero so it’s a zero right here zero right here
and right here so keep in mind how these are related to each other right um
this right here cosecant 2x is the same thing as 1 over sine 2x so
we have to sketch the graph of y equals sine 2x first nothing not just the graph
of sine x sketch the graph of y equals two sine x

00:30
where is the graph of y uh of sine two x where is that zero because that’s where
this one right here will be undefined okay so i just wanna mention that here
at least once but in any case time for the
sketch of this one right here let me see if i put it in orange if it will look
nice here let’s put it in orange it will go right here
and then we’ll come right here and let’s figure out the height now
the amplitude is for the sine one right here the amplitude is one so
this one right here is one right here and this one here is minus one right here
all right so there’s the graph right there in orange
is the graph we’re looking for right there i think it looks so much nicer if
you put the isotopes down but the sine graph right here which i always put
first but you know that’s optional you don’t necessarily need to do that
especially if you can visualize this if you do enough of these and you can just

00:31
visualize it but you know you got to get to tick marks
all right so the next one is cosine 4x so we’re going to be graphing y equals
secant 4x so how do we what is the graph of secant 4x look like
so let’s get a cosine graph going so cosine we’re going to be looking at
the period first is 2 pi over our positive b which is 4.
and so that’s pi over two and so our period is pi over two now
we’re going to be looking at cosine cosine is going to start right here and
so we’re gonna we’re gonna start here and then come down and come back up
uh don’t make it look too straight it’s got to be nice and curvy
all right and so here’s where it starts repeating at pi over two
and then i chop that in half pi over four and i chop that in half again pi over

00:32
eight so we get one pi over eight two pi over
eight and this will be three pi over eight and so now we’re going to look for
vertical isotopes and so where is this graph zero right here at pi over eight
so that’s the vertical i um let’s make it a little longer so this is the
vertical line x equals pi over eight and we have another one right here this
is where cosine four x is zero at three pi over 8 right so we’re going
to go right through here and say this is 3 pi over 8
and now we’re going to have here the graph
we’re going to come in through here like this and like this and
maybe we should extend this one up a little bit more
and then we’re going to come in here like this one right here there we go

00:33
all right so let’s see what’s missing now is this label right here which is
um the amplitude is one for this so the for the secant 4x it’s going to
bounce here at 1 label about right there and this will be -1 right there all
right there we go so the period of secant 4x is pi over 2 same period
and we have a branch going right here a branch here and a branch here so we’ve
covered enough for one for one period of this for one period of this graph right
here all right so next one is sine of 1 3 x
so what’s the new function that we’re graphing this time will be y equals

00:34
cosecant of one-third x and so now what is the period so the period is
2 pi over 1 3 which is 6 pi so i’m going to graph the sketch to
graph a sine real quick but i’m going to change the period to 6 pi
and that will guide us towards graphing this graph right here so here we go so 6
pi this is sine it’s it’s a positive sine function right
here so it’s just going to come up and then back down
and then i’ve already gone through 6 pi right here so halfway right here is 3 pi
and then halfway of that is 3 pi over 2 so i need to count up my 3 pi over 2’s 1
3 pi over 2 2 2 3 pi over 2’s and this would be 4 3 pi over 2’s which is just
you know wait one two three i forgot three pi over
twos so three three pi over twos which is just nine pi over twos all
right so nine pi over twos all right so where are the vertical

00:35
isotopes right here we have a vertical asymptote right here
and a vertical asymptote right here and right here
and so now we can sketch the graph for the cosecant we’re going to get this
graph right here this branch right here and
that’s already one period right there so all we’re missing is the
what labeling on the y-axis right here one and minus one there
all right and so these are the isotopes right here right x equals 0
x equals 3 pi and then x equals 6 pi there we go all right so next one is
cosine of one fifth x this won’t take us long cosine one fifth so

00:36
we’ll sketch the graph of secant of one fifth using this as a guide
um that looks strange so one-fifth x all right and so now let’s see
so what would be the period so let’s put the period right here period is
2 pi divided by 1 5 1 5 or in other words 10 pi
all right so that’s the period so here we go um
we’re going to sketch the graph of cosine first so cosine starts up here
here we go cosine and then right there and right about there
and so we’re going to get one period here which is 10 pi
so it grew really slowly right just go on really slow it took 10 pi to get
through one cycle one period all right so halfway is five pi
and then halfway again is five pi over two and then halfway again
or this one right here is 1 pi over 2 5 pi over 2 two of them and then three of

00:37
those this is 15 pi over two and that’s 20 pi over two
all right and so now we have our vertical isotopes right here and right here
and let’s label them 5 pi over 2 and 15 pi over 2
and so now we’re going to have the sketch of the graph which i’ll put in
purple right here so let’s go right here bounce it and go up
we get this lower branch right here and this branch right here
and so if we want to we can take off the cosine graph
here and then every now and then i just like to take one off and so
there’s roughly the sketch right there where this is a amplitude is one again
so this will be minus one and this one right here will be one right there

00:38
all right so all right so now um let’s um i think that’s good um
we haven’t done any where where are mixed right here
so these are all amplitudes and these are all periods and let’s do a couple of
them that are mixed um let’s skip ahead a little bit to how about
this one right here let’s go to this one right here
if you’d like to see these other three right here put a comment down below and
let me know and i’ll work those out otherwise let’s just go on and kind of
mix them let’s do this right here so now the graph we’re going to sketch is

00:39
the sketch of minus 2 cosecant of 2x how do we sketch this
minus 2 cosecant of 2x right there all right so the period is going to be 2 pi
over our positive b and that’s a 2 that’s positive so the period is just
going to be pi all right so let’s sketch the graph with the sign and
this is a minus 2 so instead of starting up like that i’m going to go down here
like this so i’m gonna go like this and then like this
and so here one period is pi and there’s pi and so this would be pi over two
and so this would be pi over four and so one pi over four two pi over
fours three pi over fours and now we’re going to be looking for
the isotopes to sketch the graph for cosecant

00:40
so we hit 0 right here so we’re going to have x equals 0’s and isotope
x equals pi over 2 and right here at pi so those are vertical isotopes and now
we can sketch the graph um let’s do it in purple now here we go
let’s sketch the graph right there it’s touching right there
and we got it right here and there we go so what is the uh
labels on the y-axis right here so y-axis x-axis
and it’s a 2 right the amplitude for this function right for this right here
is a two and so that’s where we’re doing our bouncing right here two
and minus two so it’s going to go up and then it’s gonna do hit that minus two
height right there and then it’s going to start falling down again start
decreasing and here it’s decreasing and then it hits that height of two right

00:41
there and then it starts to increase again right there so we got our isotopes
we got our sketch um so yeah looks good all right so um
if you’re wondering what happens when the b is greater than zero
well we worked that out today now if you’re wondering what happens if b is
less than zero then i recommend that you watch the next
video which is starting right now if you enjoyed this video please like and
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