Graphing Trig Functions (Complete Review)

Video Series: Trigonometry is Fun (Step-by-step Tutorials for Beginners)

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

00:00
are you looking for a review of graphing trig functions
we’ll review the basic graphs of sine cosine tangent and the reciprocal
functions we’ll also practice the phase shift and vertical and horizontal shifts
i think you’ll enjoy this super helpful helpful review let’s do some math
[Music] hi everyone welcome back um first off
let’s review what was done previously on trigonometry is fun
so we went through these episodes right here so far we talked about sketching
the graphs of sines and cosines and secants and cosecants and tangents and
cotangents so this episode is a a review now if you haven’t catched up up on the
previous episodes make sure and check them out the series is trigonometry is
fun step-by-step tutorials for beginners so link is below for the

00:01
complete series let’s go ahead and begin we’re going to begin by doing a
um cosine graph right here so you know as we move
along and do these examples please keep in mind that you know this episode is
a review episode and it’s you know just going to have some more fun but if i’m
going too fast for you then you definitely should back up and try to
check out some of the um previous episodes so let’s go with cosine first
so the way i’m going to sketch this graph here is by starting out by looking
at cosine and so the cosine graph is uh coming up here
and then it starts to repeat and the period of cosine is 2 pi
and so halfway is pi and then halfway from that is two pi over two
and then we have three pi over two if i can get that in here three pi over two
all right and so the height here is one the amplitude of cosine is one
and so we have minus one and one right there and so we’re labeling these

00:02
points right here and we can get as many cycles as we want
simply by adding the period so this point right here would be pi over 2 plus
2 pi okay now to do pi cosine of pi x so cosine of pi x
the period is going to be 2 pi divided by this pi which is just two
and so we’re going to sketch the graph and keep the same shape so we’re going
to come through like this and then go start repeating
but now the period here where it starts repeating is two
and so halfway is one and then halfway again is one half
and so one half two one half’s three one halves and so this is one
and this is minus one and so here’s the sketch of cosine pi x right here

00:03
yeah so let’s do a another example let’s look at minus one-half sine x
so for this one here we’re going to sketch the graph of sine sine first so
in fact um i think we can do all this at once let’s just check it out and see
yeah so let’s just sketch the graph right here now normally sine will
start going up but this has been reflected so it’s going to start going down
and then it’s going to come back up and it’s going to go back down again and the
period of sine is two pi and so this is halfway
and this is halfway again so pi over two and three pi over two
and this height right here so the amplitude is the absolute value which is
one-half so this height right here is one-half
and this is minus one-half right here and i’ll just go ahead and label these

00:04
right here all right so here’s a sketch of number two right there
we got all the points we got the zeros the three zeros right there for one full
cycle and now let’s do this one right here another sine graph
this time the period has been changed it’s been vertically stretched and it’s
been vertically in its and there’s a vertical shift here
so i’m going to first uh ignore the two and i’m going to sketch this part right
here so i’m going to look at this graph here first 3 sine of 2x
now the period for this graph is 2 pi divided by 2 which is pi
so when i sketch this graph right here this is sine and it’s positive so it’s
just going to start going up and then come back down and the period is pi
and then halfway is pi over 2 and halfway is pi over 4
and then to get this point right here which is a minus 3
and this is a 3 right here and so this is um

00:05
one pi over four two pi over four three pi over four
and so we have there one full cycle now what we need to do is we need to shift
everything up two units so this three height of three right here
would get shifted up to five and this minus three right here will get
shifted up to a minus one so let’s sketch the graph we can sketch
it say about right here and now instead of here the x-axis actually i’m going to
draw the line y equals 2 to help me get a good sketch so i’ll put about y equals
2 about right here and um the three is going to go up to a five
and so you know we’re going to sketch this up coming up here like this
and then it’s going to come down to a minus one
so it’s going to need to come down below the x-axis here to about a minus one

00:06
there and that look too um not curvy enough so let’s try to do that again
like to make them nice and curvy and so that’s a minus one here
and this here a height of five um so this uh tick mark right here comes from
this one right here it’s been shifted up to so that’s halfway and
where we start to repeat here at the peak right here at five
that right there is the uh pi and this this tick mark right here is
a three pi over four and to get this minimum right here is pi over two
and this tick mark right here is pi over four um
and so this is just a rough sketch i kind of made it go a steeper right here
than this part right here but it’s okay um so we’ve got the height there we got

00:07
this point right here labeled pi over 4 2 right there and we have this
point right here labeled at pi over 2 minus 1 and we have this point right here
labeled 3 pi over 4 2 output 2 and then pi 5 so we have those points
labeled and we clearly showed how this right here was shifted up
two units right there so it has the same shape
it’s just been shifted up this one’s been shifted up to so there we go
there’s number three um let’s try this right here uh we have a 2 sine of pi x
so let’s go here um so we have a sine so the period is 2 pi over pi
the period is 2 and the vertical stretch is is a 2. so here we go we got sine

00:08
and this is a two now the period it’s a one this is a one half
and this is the three halves and the height here the amplitude this right
here is a two and so this one right here is a minus two right here
and there’s one cycle right there of 2 sine pi x right there
a rough sketch real quick all right now let’s do a horizontal shift
so we’re going to be shifting left by pi over 4 so horizontal shift so i’m going
to take the sine sketch right here the just the normal sine sketch where
this is 2 pi pi pi over 2 and 3 pi over 2 height of 1 and
-1 here and then so that says 2 pi and so i’m going to take all of these
labels right here these ticks and so 0 pi over 2 pi 3 pi over 2 and 2 pi

00:09
and what are we going to do well we’re going to put a minus pi over 4 in
here to get back to 0 sine of 0 0 so this has been shifted to the left pi
over 4 so i’m going to subtract a pi over 4 from all of these right here
and we’ll get our tick marks here and so this one is minus pi over 4
and think of this one as over four so we’re going to get two pi minus pi
in other words pi over four and this one right here think about it as four pi
minus pi so three pi over four and think of this right here as six pi minus pi
so five pi over four and think of this one is eight pi
minus seven uh sorry minus one so seven pi over four
and so there’s the new tick marks there so um i think we can sketch it uh let’s
sketch it over here i can move out of the way um

00:10
and so yeah let’s sketch it over here so basically uh this has been shifted to
the left and so i’m gonna sketch it about right here
and i’m gonna try to keep the same shape right here this is minus pi over four
and the next one is pi over four so it’s going to come up hit about right there
so that’s about a minus pi over four and maybe a little bit more
about a pi over four right here and so that’s that part right there
and then it’s going to start coming down and then it’s going to go
about like that and you can’t see that they ran off the screen there so let’s uh
bump that and try it again it’s a good idea to usually shape it first
uh and then that’s wrong right there because it’s not hitting the pi before
let’s try a third time so let’s try about like that let’s try a fourth time

00:11
so maybe i should draw it like this first and
it’s hitting the high point here i still didn’t get good symmetry
so let’s go to the highest and then we’ll come back down
and then i think we’ll be able to see it this time
there we go all right so here’s the highest point right there it’s pi over four
that one and there’s where it’s starting at minus pi over four so we’re keeping
the same shape here right there and then
the next tick mark is three pi over four and then to get this one down here is
five pi over four and then to get this last one is seven pi over four
so the pi over two has been shifted to the left by pi over four and so now it’s
seven pi over four and we still have the same amplitude here
i have one and minus one just make sure and not put that one right there um

00:12
anyways there’s the shape of sine of x plus pi over 4 right there so i’ve just
taken this one and shifted it over and we get something like that so
yeah um let’s go on to the next one number six so this one is fun this one has a
horizontal shift it has a vertical stretch and it has a vertical shift down
three units so um i’m going to prepare this one here
by writing equals and then let’s keep the minus three and then plus three
and then for the sign here so what we need to do is to factor out the pi over 3
factor that out and then we’re going to say x minus
now we’re already factoring out a the pi so
think of this as 3 over 3 multiplied by 3 over 3 we’re going to get a 12 down

00:13
here so i’m going to say 1 over 12 and so these are equal to each other
right the minus 3s and then the positive 3 and then the sine and here i’m
factoring out the pi over 3. so let’s check that pi over 3 times x there we go
pi over 3 times negative 1 half is certainly sorry negative 1 12 is
certainly negative and then it’s pi on top and then we have a three um actually
nope that’s not right um i’m putting a three over three and
i’m taking the three on the bottom out so it’s actually three over four
so now let’s check pi over three times minus three over four the threes cancel
and we’re going to end up with minus pi over four which is what we have there
all right so there we go so this is going to shift right by three over four
we’re gonna have to deal with that somehow and we have the period right here
all right so to sketch this graph here i’m not going to worry about the
vertical shift yet i’m going to just focus on this part right here

00:14
and so let’s sketch the graph of three sine of pi over three x first
let’s sketch this one here first and then we’ll do a vertical shift to it
so this one right here has period two pi over pi over three
and so what does that come out to be so that’s the same thing as two pi over
one times three over pi and the pi’s cancel and so that’s the same thing as six
so uh this graph right here has period six and so we can graph this right here
and this is just a sine graph so we’re going to go up and then come back down
and this is a six right here it starts to repeat right here at six
halfway is a three and then halfway is a three over two
and then here is going to be one two three so nine over two and then

00:15
another three so 12 over two which is six and then our height here is three
and then this right here is a minus three right here
all right so there’s one cycle and we got these points right here
and let’s go ahead and shift them so we’re going to be shifting by
uh to the right by three fourths and so we need to take
uh these numbers right here let’s do them up here so let’s start with a 0
and then three over two [Music] and then three and then
nine over two and then six and we’re going to add a three-fourths
to it the reason why three-fourths is because if we plug in three-fourths we
get out zero here and zero times pi over three is zero and so that’ll be sine of
zero which gives us our starting right here sine of zero is zero all right so
um let’s go here with uh adding three-fourths to everything

00:16
so add three-fourths and we’ll get our new tick marks here for our shifted graph
so let’s find these fractions real quick first one is three-fourths
and then we have uh think of this one as a six over four so that’ll be nine over
four and think of this one as twelve over four so that’ll be fifteen over four
uh think of this one as 18 plus three so 21 over four
and think of this one as 24 25 to 627 27 over four
so there we go there’s our new tick marks for our shifted graph
and so let’s do that right here and so we’re shifting to the right so we’re
going to keep the same shape here so instead of starting right here i’m going
to start over here so i’m going to go up and come back down
and then start to repeat right here and so this one right here is three-fourths

00:17
and this one is 27 fourths halfway is 15 4. and we could find that halfway by
finding the midpoint and same thing here we can find the midpoint but we already
have it right here nine fourths and then finding the midpoint right here
and that’ll be 21 over four so there we go and the height here is three
and the minimum relative minimum right here is minus three
um and so we got these points uh labeled now and we have one full cycle for
this part right here now we still have to shift it down by three units so
let’s do that right here actually let’s shift it down by three units so
this three is going to be shifted down to here
and this minus three right here is gonna be shifted down to zero

00:18
so let’s do that about right here and i need to put this axis up pretty high
now this is the line um y equals 0 you know
the x-axis so now to sketch this right here i’m going to sketch the line y
equals -3 just to help guide me make the shape of the curve
and we’re going up and so i’m going to start here and go up
and this 3 has been shifted down by -3 so it’s going to hit 0 right there
at this at this tick mark right here so i’m going to hit zero right there and
then i’m going to come back down and then now i gotta space it out right
and come over here and there we go so this point right here
and here and here and here and here and we’re going to write the tick marks over
here so this one is uh the three over four this one is the nine over four

00:19
this one right here is the 15 over 4 and this one right here is the 21 over 4
and this one right here is the 27 over 4.
all right and so there’s one full cycle and then our height here minus three got
shifted down to minus six and this height of three right here got
shifted down to zero and so let’s just go ahead and label this x and y and
there we go so there’s number six right there
all right so let me move to the other side of the screen and let’s
hit up number seven let’s try number seven now if you have any questions
please let me know in the comments below and while i’m erasing this also let me
mention that if you like this video please go ahead and like and subscribe um
let’s try number seven now so number seven is a cosecant graph

00:20
so and we have a horizontal shift and we have some kind of
stretching or shrinking going on here it says below one so you might call that a
shrink and then we have a vertical shift up one
now in order to sketch the cosecant i’m first going to sketch the graph of sine
and so what i’m first going to sketch the graph of is y equals one-half sine
[Music] of x so one-half sine of x i’m first going to
sketch this graph right here that one doesn’t take uh very long at
all it’s just very quick i’ll just sketch it it’s just a sine
graph it’s coming up and then going down and the period is 2 pi
and so right here in the middle we have a pi pi over two three pi over two
the height is one half because the amplitude here and so this is minus one

00:21
half right here all right so there’s a just a real quick sketch
of this one right here and we have these um
tick marks right here one two three four five of them
and now i’m going to now the reason why i look looking at sine is because
remember cosecant is one over sine right
so i need to look at this right here and this will help guide me when i’m making
the cosecant graph but now what i need to do is to make a shift now this is
positive pi over 4 so we’re actually going to shift it uh
to the left so i’m going to take these tick marks here so here we go 0 and then
pi over 2 and then pi and then 3 pi over 2 and then 2 pi
and we’re shifting it to the left so i’m going to say minus pi over 4
minus pi over 4 minus pi over four minus pi over four and i’m going to get
my new tick marks so this one’s minus pi over four
now we have pi over two minus pi over four so i’m going to multiply by two 2

00:22
over 2 so 2 pi minus pi is pi over 4 and then 4 pi minus 1 pi so 3 pi over 4
and then 6 pi minus pi so 5 pi over 4 and then 8 pi minus pi so 7 pi over 4.
all right and so there’s going to be our new tick marks so um
you know i’m not going to deal with the 1 yet before i shift it up i’m going to
sketch this graph right here so let’s do that over here
so let’s go about right here and let’s sketch the sign first
so what i’m really sketching on at on over here will be the one-half sine of
the x plus pi over four all right um so let me see if i can scoot that over
here one-half sine of x plus pi over four
all right so i want to sketch this right here with the shift in other words i’m

00:23
taking this graph right here and i’m shifting it to the left and so now it’s
going to be minus pi over four and then uh this upper part right here
will happen at a pi over four so we’ll come over here and say minus pi over four
and pi over four and this is where we’ll hit my height here
and then i’ll come back down and then i’ll come back around like that and
let’s make that up there and so this is where a height of the one half is at
and the um halfway place here is now three pi over four
and then to get this one right here we have five pi over four
and then the last one is seven pi over four and so that’s this graph right here

00:24
or one cycle of it now let’s go to the isotopes i like to put my isotopes in red
so where is the sign remember cosecant is one over sine so i’m worried about
where the sine is zero so it’s zero right here at x equals minus pi over four
and it’s zero right here at uh x equals three pi over four
so these vertical lines are the isotopes the vertical isotopes for the cosecant
graph and i’m going to put that in or orange here if i can
and so what we have here now is a nice shape right here this is getting closer
and closer to the isotope getting closer
and closer to the isotope and then right here we have this shape right here
so we have an upper branch and a lower branch and then an upper branch and a
lower branch and so on and over here we would have a lower branch and then an

00:25
upper branch and a lower branch and so on and so this is what would happen if we
didn’t have a one here right the one in orange is the
y equals and then it’s just a one-half cosecant of x plus pi over four
so there’s that graph there so now let’s try to sketch the shift the
vertical shift up so for example this one half will get
shifted up one to three halves and this one right here which actually i
didn’t label it’s about right here and that is a minus one half yeah minus
one half and that will get shifted up one to a
one half so let’s see if we can sketch the graph over here
and i’ll put it about it’s being shifted up so put the x-axis about right there
and give that a try now shift it up one so i’m going to sketch the
line uh y equals one right here just to help guide me it’s not part of the graph

00:26
but so i’m going to have this right here
okay let’s draw the isotopes next so the isotopes so when we’re shifting up the
isotopes don’t change because we’re just
shifting this up and we’re just shifting this branch right here we’re just
shifting it up so let’s go ahead and sketch the isotopes here
so we have this one here at minus pi over four so let’s sketch it about right
here and we can’t see that so let’s label them up here so minus pi over four
and then we have another one right here at um three pi over four
and then we have one over here at seven pi over four
all right so there’s the isotopes and now we’re going to come in with the
upper branch right here and the height here is the three halves

00:27
so we’ll come in here and sketch this right here and it’s about right here
and we’ll come in here so this minus one half we’re shifting it up one so now
this is a positive one so it’s about right here let’s just um
actually um this is minus one half plus one okay should be just a positive
one-half all right that makes sense all right and so let’s just go ahead and
label that so that right there is a one-half and this one right here is a
three-halves and we have the tick mark for this one right here
which is at a pi over four and we have this right here isotope
label three pi over four and we have this uh tick mark for this
relative height right there which is um five pi over four
and then we have the isotope here seven x equals seven pi over four great so we

00:28
got everything labeled here is it is an orange right there and then right here
and so there’s number seven right there we got the isotopes relative maximum
relative men we got an upper branch lower branch we got one full cycle
everything’s labeled so let’s try number eight now
all right so number eight is a tangent graph so let me um
get those out of the way there all right tangent graph now uh in order
to sketch this tangent graph we have a 2 here so in order to
see the horizontal transformation what i’m going to do is i’m going to factor 2
out so let’s say this is equal to minus 3 and then tangent of
and we’re going to factor out a 3 so we can get sorry we’re going to

00:29
factor out a 2 so we can get x minus so now we have a horizontal transformation
and i think think of this as uh 2 over 2 so we get a 4 down here so this is pi
over 4 and we can always check that 2 times x
is 2x 2 times uh minus 1 4. so the 2’s cancel we get a half there with a pie
all right so our transformation is we’re going to be shifting to the right by pi
over 4 and we have a two here so that’s going to affect our period here so the
first thing we might want to do is uh remember what tangent looks like so
let’s just sketch that graph real quick so tangent has an isotope which i
usually like to put in red let’s get that back out
so we have a minus pi over 2 right here and i’m getting these uh isotopes
because remember tangent is sine over cosine so we’re looking where cosine is
zero so that’s where i’m getting the uh pi over twos

00:30
and then tangent’s gonna come in through here and it’s always increasing
and at the pi over four we’re going to get a height of 1 and at minus pi over 4
we’re going to get a height of -1 and so there’s just a quick refresher of
course it just keeps repeating and it just keeps repeating over and over
again but there’s one full cycle the period is period is pi
and so just it’s just increasing right through there it’s getting closer and
closer and it’s getting closer and closer here and we have these two
uh points right here at height of one and pi over four one and then this right
here is a um minus pi over four right here
with a height of minus one right there all right so let’s see here um
now we going to take into account the two here

00:31
and let’s see if we can do that so what happens if we try a
minus 3 tangent of 2x let’s see if we can do all that in the next step together
so the period is pi and then divided by the 2 so now our period is pi over 2
so we’re going to have a sketch like this it’s going to have a very similar
shape however the minus sign says it’s reflected so here where it’s increasing
everywhere now it’s going to be decreasing but first let me graph the
or sketch the um the isotopes so instead of having a period of pi
like tangent does this one here has a period of pi over two so our isotope
right here is going to be uh x equals pi over four and
x equals minus pi over four and now we’re going to be because of the
minus sign here we’re gonna we’re still gonna go through the origin but it’s

00:32
gonna be decreasing like this and halfway right here at pi over eight
instead of having a height of one because of the minus three now it’s
going to have a height of minus three right here
and halfway between zero and pi minus pi over four
is minus pi over eight and that’s where it has a height of three right here
all right and so here’s a rough sketch of minus three tangent two x right there
so now what we need to do is shift everything to the right by pi over four
and then we’ll have our our final graph right there
so let’s see if we can do that right here
so when we shift this to the right by pi over four
this is already a minus pi over four so that means we’re going to be shifting it
back here so let me sketch the uh vertical isotopes
so this uh minus pi over four now becomes x equals zero and this pi over four
this vertical isotope right here is going to get shifted to the right by pi

00:33
over 4 so what happens we have pi over 4 plus 2 pi
plus pi over 4 well that’s 2 pi over 4s in other words it’s pi over 2. so i’m
going to have pi over 2 right here as this isotope right here let’s just say
pi over 2 right here so we have 0 right here and
pi over 2. let me move that label up out
of the way here let’s call this one here minus pi over 4.
all right and so here we have our label for the vertical asymptote in the
vertical isotope and we’re going to have the same shape
here coming down here like this because we’re shifting it to the to the
right doesn’t change the shape but halfway in between um will be pi over four
in other words we’re shifting this zero um the x on the x-axis is the zero we’re
shifting it we get pi over four so it’s still going to become through here like
that and halfway right here will be and then halfway right here will be

00:34
this will be pi over eight so this will be three pi over eight
and this height right here will be the 3 right here
and this height right here will be -3 right here
and so we got the isotopes and we got the behavior of the graph
for one full cycle and that’s it for number eight
all right very good so there’s the graph
of minus three tangent of two x minus pi over two
or if you like it looks better like that
because you can see the horizontal shift right there so we shifted this graph
right here to the right and this power eight for example this
point right here became three pi over eight okay so let’s go on to another one
and this one is a secant we haven’t done a secant yet

00:35
all right so now i’m going to be looking at the graph of a cosine
so um because secant is one over cosine um and we need to check out what is the
horizontal shift right here um and this one right here the it
doesn’t have a vertical shift we’re not going to shift it up or down
so let’s just rewrite this right here first as one half and then secant of
and then i’m going to factor out a pi over 2 so pi over 2 and then x
and then what do we need we’re going to factor out the pi um oops and
and let’s see here we’re going to factor out a 2. so that leaves us 2 down here
left so minus pi over 2 all right so that seems to make sense pi
over 2 times x is pi over 2x and then pi over 2 times negative pi over 2 oops
that should there be no pi there all right that makes even more sense

00:36
so that gives us when we multiply this that gives us the pi over 4 back
all right so pi over 2 is going to affect the period and this right here is
going to be our horizontal so we’re going to have a horizontal shift to the
right by pi by one half all right so here let’s go up here and first sketch
the uh one half the cosine and then we’ll do the pi over two
times x let’s see if we can do all of that at once cosine of pi over 2 x
so for this graph right here the period is going to be 2 pi divided by pi over 2
which is the same thing as 2 pi over one times two over pi
fives cancel in fact we get four so it seems like the period of this one
right here is four two pi over pi over two so two pi is two pi over one
and then flip and then all right so now let’s go with

00:37
this graph right here so this is a cosine so i’m just going to start up here
and then go down here and then start to repeat up here and so this um
period here is four so that’s a four halfway is two and then halfway is one
and then add up the ones one two three and the height here is one half because
the amplitude is one half and so this is a minus one half down here
all right so there’s one cycle of this function right here
and so now what we need to do is to do a shift we’re going to shift
right by one half so i need to take each of these tick marks that i have right
here and add on a half to that and so i’m going to do that about right
here i’m going to try to do this in such a way that i can also sketch the secant
the final graph on on it if i can so let me move up here and try to get

00:38
out of the way here so let’s see if we can do this um put it about right here
and oh actually i didn’t get my tick marks yet but um so this is a
zero right here and we’re gonna shift it so we’re gonna get a one-half
so it’s gonna start about right here and then come back down and that let me
put that not so high it’s coming back about down here and then about here
and something like that and this is a four and this is halfway uh four two um
and a one all right but we got them shifted right we gotta shift
so this zero has been shifted over so zero plus a half is one half here where
it hits its height and then this is a one plus a half so three halves
and this two here is uh two plus a half right two plus a half

00:39
is just five halves and then this one right here will be what um
three plus a half so seven halves and then four plus a half so nine halves
so one three five seven nine and so that’s a nine halves
and this right here seven halves and five halves all right so and that’s for the
and let me just label that here in black minus sorry one half
because that’s a one half right here and minus one half right here
so it’s one half and i’m going to scoot it over here
it’s one half what i just graphed right here is one half cosine of pi over two

00:40
and then x minus one half so that’s we just sketched right there
all right and so basically we have the one half for the amplitude
and we changed the period and we shifted so now to sketch the graph of secant
from there because secant is one over cosine we
need to um worry about where cosine is zero
and that’s going to give us our isotopes so we’re right here zero at the three
halves so i can label this as x equals three halves
and we’re right here at zero at seven halves so i’ll say x equals seven halves
and so that will be enough to get uh one cycle and let’s graph it in orange
so we’re graphing this function right here
and it’s going to be coming through here like this
and like this we’ll get a half of an upper branch and another half of an
upper branch and so we have this point right here

00:41
labeled one half one half right here and nine halves one half right here
and five halves minus one half right here and we have this nice isotopes labeled
and we have the behavior right here graphed so there we are
so there is a uh secant with a change in the period and a horizontal
shift right there so yeah lots of fun hey let’s do uh one
more let’s do a cotangent we haven’t done a cotangent yet
so let’s see if we can do this so here we have a cotangent cotangent right here
and we have a three so we’re gonna have a change in the period and we’re gonna
have a horizontal shift so first thing i’m going to do is
actually rewrite this if i can and say this is equal to 2 cotangent of

00:42
bracket i’m going to pull out my 3 here and then say x minus what
so think of this as 3 over 3 so that would give me a 12 down there and we’re
going to take out this 3. so it’s going to be pi over 12 here there we go
so this is going to be 3 times the x minus pi over 12.
and we can check that factoring by just simply going backwards 3 times x is here
and then 3 and then 12 right cancels gives me 4 minus 4 right there all right
so this is what we’re going to sketch here so we see how the period is going
to change and the shifting the horizontal shifting we’re
going to shift to the right by pi over 12. so first let’s sketch the graph of
cotangent let’s just refresh our memory of what cotangent looks like
and so to do that we’re going to have some isotopes so remember cotangent is
cosine over sine so we’re going to be looking where

00:43
uh sorry where sine is zero and it happens at zero and at pi
and cotangent is decreasing the whole way and so right here in the middle is pi
over two and then halfway again would be pi over four and that’s we get our one
and then we have three pi over four and that’s we get our minus one
so there’s just a quick sketch of cotangent of x now what happens if we add a 2
and we have a 3 in here so i’m going to wait on the horizontal shift first so
next i’m going to graph 2 cotangent of 3x so this time i’m going to change the
period and i’m going to do a little bit of stretching to it so here we go um
i’m going to have the shape right here and we’re still going to have the same um
well actually we’re not going to have the same vertical

00:44
isotopes because the period is going to be pi over and then the three
remember cotangent has period pi and then divided by three so the period
is pi over three so here the period the period is pi so now it’s going to be pi
over three so here we go with our vertical isotopes we have x equals zero
and we have x equals pi over three and so halfway will be pi over six
and we’re going to have the same shape because it’s it’s been stretched by
positive two so it’s not going to be reflected so it’s going to go like this
right here like that and then halfway right here will be pi over 12
and then this will be three pi over 12 which will be pi over four
and this height here won’t be a one anymore this part will be a one and then
times two so that a height will be a height of two

00:45
and at pi over four right here the height will be right here adding minus two
all right so there’s the sketch of two times cotangent three x
all right and now we’re gonna shift uh to the right by pi over 12. so i need to
take into account all of this information right here
and let’s do that right here um well maybe we can just sketch the graph
so let’s try to um i don’t see why we can’t just finish it
let’s just see if we can do this down here
maybe i’ll scoot this over a little bit though i’ll make it nice and big
all right so um this 0 and pi over 3 has
been shifted to the right by pi over 12. so we need to add a pi over 12 to that
so for their isotopes we’re going to get here x equals pi over 12.
now we need to figure out what is the right one so we’re going to get pi over 3

00:46
plus pi over 12. and think of that as four so we’re going to get 5 pi over 12
so x equals 5 pi over 12 and we’re still going to have the same shape to it
because all we’re doing right now is shifting it to the right
we’re not going to it’s not that’s not going to change the decreasing
and so what is halfway well we can find the midpoint of these
two or we could just take the pi over 6 that was right here and then add a pi
over 12 to it so pi over 6 plus pi over 12 and so think of that as
2 pi over 12 so we’re looking at 3 pi over 12 or just pi over four
so halfway is pi over 12 now and then we can find halfway right here
and get this height of a two out and we can find halfway right here and get the
height of minus two out so to get this um again you can find the midpoint or we

00:47
can just take the pi over 12 and add a pi over 12 to it
that will give us the pi over six and here this um point right here pi over 4
and we’re going to add a pi over 12 to it and think of that as 3 pi over 12
so we get 4 pi over 12 or said differently pi over 3
so that’s a pi over 3 right here at this tick mark right here
assuming i did all the fractions right so that’s 4 pi over 12 that’s pi over 3.
um yeah that was pi over four all right looks good
all right so let’s see here are we missing anything we got the isotopes we
got the final sketch right here we got the twos and the two minus two
and we got the isotopes and we got the right shape it’s decreasing and

00:48
yep so looks everything looks good alright so if you enjoyed this video
then uh please like and subscribe if you haven’t already that would be greatly
appreciated so upcoming we’re gonna um we kind of finished with the graphing
portion and then we’re gonna move on and uh the next episode is starting
right now

About The Author
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David A. Smith (Dave)

Mathematics Educator

David A. Smith is the CEO and founder of Dave4Math. His background is in mathematics (B.S. & M.S. in Mathematics), computer science, and undergraduate teaching (15+ years). With extensive experience in higher education and a passion for learning, his professional and academic careers revolve around advancing knowledge for himself and others. His work helps others learn about subjects that can help them in their personal and professional lives.

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