Trig Graphs (Horizontal Transformations)

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

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

00:00
how do you move the graph of a trig function left or right
what is a horizontal translation in this episode we’ll practice
horizontal shifts of trig functions it’s
actually quite straightforward and quite useful let’s do some math [Music]
hi everyone welcome back uh we’re going to begin by reviewing horizontal trans
transformations we talked about these in uh previous episodes
um under under the name of phase shift so we’re going to take
perhaps a slightly different approach than you might have seen before
but it’s actually going to be very straightforward and quite easy and but
first before we get started on that i wanted to mention that in the previous
episodes of trigonometry is fun we talked about sketching sines and cosines
secants and cosecants and tangents and cotangents so we talked about all these

00:01
before um but in this episode we’re going to focus on horizontal shifts
um yeah and so the link for the series is below in the description trigonometry
is fun step-by-step tutorials for beginners i recommend checking out the
full series this episode is just um one of the episodes in the series
all right so let’s go ahead and begin let’s start off by sketching this graph
right here so this involves a horizontal shift
and since this is positive pi over two we’re going to shift left
and we’re going to have the shape of sine
so let’s sketch this graph right here so what i usually like to do is to first
sketch the graph of the base what i call the base function which in this case
sometimes we call it a parent parent function not a parrot a parent function but
anyways it’s sine and that’s what we’re shifting and so what does sign look like
yeah let’s just refresh our memory real quick so it goes up and then goes back

00:02
down the period is two pi halfway is pi halfway again is pi over two
and now we count up our pi over twos one pi over 2 2 pi over 2 and 3 pi over 2
and so has a height of 1 the amplitude is 1 and so this is the minus 1 here
so that’s just a quick sketch of sine right there
and if that was too fast then that’s understandable but keep in mind that
we’ve practiced graphing the sine function already before in previous
episodes and so you definitely want to check those out if that was too fast
then you want to go back and check check those out so now how do we actually do
this right here so what i’m going to do is i’m going to take these and this is a
general approach that you can take in all of these examples here so we’re
going to just practice this in this episode here but i’m going to take these
five tick marks here 0 pi over 2 pi and then 3 pi over 2 [Music] and then 2 pi

00:03
and since uh we’re subtracting or sorry we’re adding uh we’re going to move to
the left pi over 2. so i’m going to subtract the pi over 2 from each of these
and that’s going to give us our new tick marks
so this first one is minus pi over two this one is zero
and now we’re looking at two pi over two so that’ll be pi over two
and here we’re looking at uh three pi minus pi so that’s two pi
over two which is just pi and then this will be
think of this as four pi over two so this will be three pi over two
however you want to add those fractions together but in the end what we’re going
to get is the same shape it’s going to start coming up and then
go down and so on but we’re going to have different tick marks here
so instead of starting right here at 0 now we’re shifting to the left so now

00:04
we’re going to be starting right here at minus pi over two
and we’re going to reach the height of one and this is going to be pi over two
so this is minus pi over two so we’re gonna reach a one here and then
we’re gonna come down and then we’re gonna keep going up like this and down
like this and so this tick mark here is 0 and then that’s pi over 2
and that’s pi and this is 3 pi over 2. we got our new
labels right here we got a height of one and we have our you know relative
minimum relative maximum that minus one there and so here’s the sketch of
sine of x plus pi over two right there looks a lot like the cosine function
doesn’t it they’re co-functions um they’re complementary
in any case oops um so there’s our first um graph right there let’s do another

00:05
one let’s do cosine of x minus pi over six
so let’s first recall what cosine looks like
so we’re going to sketch the graph right here of cosine and it’s going to start
up here and then come back and then it’s going to start repeating
the height here is one this is minus one the period is two pi
halfway here is pi halfway again is pi over two
and now we’re going to count up our pi over twos one two three pi over twos
and so there’s where it hits a height uh maximum height and
that’s a one right there all right so there’s a typical cosine graph right there
now since this is minus five or six when you plug in the pi over six that’s
when you get back to zero so we’re going to be shifting it to the right by pi

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over 6. so we can do that over here um so i’m going to take these
tick marks that i have zero pi over two these tick marks right here now keep in
mind that cosine of course keeps going on in both directions we’re just
concentrating on graphing one cycle now i’m going to take the 0 the pi over 2
the pi the 3 pi over 2 and the 2 pi and i’m going to take um this says minus
pi over six so we’re going to shift to the right so this is positive pi over six
plus pi over six and so you know as you see we
get some fun fractions this is pi over six think of pi over two as three pi over
six and then plus the pi over six would be four pi over six which is 2 pi over 3
if i could get out of the way that’s just 2 pi over 3 here so 2 pi over 3
and then pi so let’s think of that as 6 pi over 6 plus pi over 6 so that’s 7 pi

00:07
over 6. and three pi over two so let’s think of three pi over two as
nine pi over six and then pi over six so that gives us ten pi over six
which reduces to five pi over three so five pi over three and
now we have two pi so think of two pi is twelve pi over six um plus pi over six
so that’s thirteen pi over six all right so we found the new tick marks
now i’m going to have the same shape though so i’m going to concentrate on
the shape and not the tick marks when i’m sketching this graph
but it’s going to start at pi over six so i’m gonna put the pi over six first
and now i’m gonna concentrate on the shape so it’s gonna come like that so
that right that part right there it’s gonna come down here like that
and it’s gonna go back up and then it’s gonna start to repeat itself
and so we have this tick mark here we’ve got these points here so that’s pi

00:08
over six this last one is 13 pi over six and then halfway is this one right here
which is seven pi over six which is where we hit the um there’s a
one in front so that’s where we hit the minus one the amplitude is one and
here’s what hit a one and then this tick mark right here
and so that’s two pi over three and this one is five pi over three
it’s five pi over three and so there we go there’s the graph of
cosine of x minus pi over 6. and it looks exactly like cosine the only
difference is it’s been shifted over by pi over 6 right there so when we
substitute in pi over 6 into this we get cosine of zero and we know cosine of
zero is one that’s why it’s been shifted to the right
all right so there’s our second example there
now let’s do secant of minus x minus pi over two
so recall secant to graph to sketch the graph of secant

00:09
and just you know want to mention one more time that we practiced graphing
secant and cosecant in the previous episode so you have if you haven’t seen
that i recommend checking that out using the link below
all right so let’s uh also remember that secant is a um even function
so when i look at these negative signs here i’m going to think about it like
this i just want to kind of show you what i’m
thinking in my head so i’m going to think about this as minus and then x
plus pi over 2 just by factoring out the minus sign from these two
now secant is an even function and that’s because cosine is an even function
and so this will be secant of x plus pi over 2
and so what we’re going to be doing is shifting to the left by pi over 2.
now first what i want to do to sketch this graph which is the same thing as
sketching this one because they’re equal is i want to sketch the graph first of
cosine because remember secant is 1 over cosine so i first want to sketch this

00:10
graph right here of cosine of x plus pi over 2
cosine x plus pi over 2. so let’s first sketch this graph right here
and then using that graph we’ll be able to sketch this right here pretty easily
so first i’m going to sketch the graph of cosine
and again that’s just a real quick sketch right here up and down
and then this is two pi pi pi over two three pi over two one and minus one
so we got all the important points and we got the shape of it uh nice and
curvy all right so now let’s shift it um
this is positive so we’re going to shift to the left in other words when we
substitute in minus pi over two we get back to zero which would get out the one
so i need to shift it to the left by uh pi over two and let’s see if i can move
down here and let’s see if i can use this space

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over here to get this done here so i’m going to shift it left by pi over two
and so this one right here which is zero now becomes minus pi over two
and this uh pi over two right here now becomes the zero we’re going to
shift that pi over two to the left and so you know this this shape right
here is gonna it’s gonna start up here and it’s gonna come down
and then it’s gonna go back like that and like that it’s gonna start repeating
and the height here is a the height is a one and
so this is minus pi over two that’s a zero and so this is another pi over two so
the pi has been shifted to the left by pi over two so this is pi over two here
and this is pi and this is three pi over two here

00:12
all right so this is the sketch of cosine of in the black here is cosine of
x plus pi over 2 pi over 2. so now in order to get the secant graph
what i’m going to do is i’m going to use
where the cosine because remember secant is
let’s just write that down real quick in case you haven’t seen that episode
is 1 over cosine of x plus pi over 2. so i need to know where this graph right
here is zero because we don’t want to divide by zero we’re going to have
isotopes there so we have a zero right here so here’s our here’s one isotope and
i’ll just label it down here if i can x equals zero
and then here we have another isotope right here and i’ll label that right here
at x equals pi and so there’s our vertical isotopes for
the secant graph right here we’re going to have these vertical isotopes so
now we can sketch the graph of the secant and i’m going to do that in orange

00:13
now i’m going to have to extend these isotopes up
to get a good sketch of it here and so we have a isotope here at zero and pi
and then we have another one over here but um we don’t need to sketch it i
guess we’re just trying to do one cycle here
and actually i need to like extend these down here too to get a to get a good
shape here so i’ll label these right here that say this is x is pi and x is zero
and now we’re ready to get the graph of secant so we have this right here coming
through here and this on the axis here is a 1
because the amplitude minus 1 because the amplitude of
this graph right here is 1 cosine x plus pi over two
and so that’s a minus one right there and then now we have a one right here
and so we have a half of a upper branch right here let’s

00:14
draw that in a little closer here right in there and we have one right here
and so we have two up we have two halves of the upper branch and we have a lower
branch right here i’ll put the arrows right here
um yeah and so we have the lower branch and we have the upper branch and the
upper branch right there and we have the isotope labeled at x
equals pi and x equals zero and we have exactly where we’re hitting
our relative minimum right here at minus pi over 2 1 and right here 3 pi
over 2 and 1 and then it just starts to repeat and start to repeat
and we have the slower branch here all right so there we go there’s number
three right there and let’s do another one right here
let’s see if we can get on to another one
let’s do a cosecant graph right here now
if you have any questions uh let me know in the comments below and i’ll be happy

00:15
to answer them and so let’s do this graph number four here
this one is very similar um we have a minus here so we’re going
to deal with that and we have a 2 here that’s going to stretch it vertically
and we have a cosecant so let’s first deal with the minus sign
right here so let’s say this is going to be equal to
2 cosecant and i’m going to factor out the x so i’m going to say x
and then minus pi over 3. all right and now let’s remember that cosecant is
an odd function and that’s because sine is an odd function
and so let’s pull the minus sign out so we’re going to get minus 2 cosecant of x
sorry x minus pi over 3. and so this is the function i actually
want to graph right here this is much easier to think about because we have a
horizontal shift right here we have a vertical stretch right here

00:16
and we have a reflection right here now to sketch this graph right here what
i’m going to first do is scratch sketch the graph of y equals minus 2 sine of
x minus pi over 3. so i’m going to sketch this graph right here first
in fact i’m just going to actually do this with the with the minus sign right
here um now probably be best just go ahead and take care of the whole thing
all right but to sketch this graph right
here i’m going to first sketch the graph of sine
and i’ll deal with a minus two also so so this will be the first step will be
to graph this one right here and then we’ll shift this one right here
and then we’ll sketch the cosecant on top of that so
let’s sketch this one right here first in fact let me just put that up here so
y equals minus 2 sine x and let’s sketch this graph right here

00:17
all right so normally sine will start going up but this is a minus so it’s
been reflected so it’s going to go down and it’s going to come back up
and we have the x axis and the y and the amplitude is the 2 so this is a 2
this is a minus 2 and the period here is 2 pi and this is pi
and this is 3 pi over two and this is pi over two
all right so long story short there’s the sketch of minus two sine x right
there it starts to decrease first and then it increases and then it starts to
repeat itself after that so now let’s shift this and let’s sketch the graph
uh where we’re going to have a minus pi over 3 shift in here
minus pi over 3 shifting here so that means we’re going to shift it to the
right by pi over three so i’m going to add a pi over three to each of these
tick marks here so i’m going to say 0 pi over 2 pi 3 pi over 2 [Music]
and then 2 pi and to each of them i’m going to shift

00:18
it so i’m going to add a pi over 3. so that makes sense because
this one right here which is pi over 3 that’s what we input into here to get
back to 0. all right and so i’m going to add pi 3 to all of them
and let’s add these fractions up and get our new tick marks
so think of this as over six so that’ll be three pi plus a two pi
which is five pi over six um think of this one as three pi over three
and that says pi over three so that’ll be four pi over three
and then think of this one over six again and so that’ll be multiply by three so
that’ll be nine pi and then multiply this one by two so
that’ll be 11 pi over six and then this one right here think of

00:19
that over 3 so that’s 6 pi plus pi so 7 pi over 3.
so basically we’re just adding 60 degrees to each one of these right here um
okay so there’s our new tick marks there so i’m going to sketch the graph over
here if possible let’s see if we can get it
and so it’s shifted to the right so i’m actually going to bump that over a
little bit and let’s go right there and now learning from the last one right
here i don’t want to make this so big i want to make it a little bit
so i can shape the cosecant nicely on it but here we go we have pi over three
right here is the first tick mark let’s just
say that’s pi over three right there and let’s keep the shape right here because
all we’re doing is shifting it so we’re gonna go down first so we’re gonna go
down and so i don’t wanna go down too far because i wanna leave lots of room for
the cosecant so i’m going to go down and then i’m going to go all the way up

00:20
and we’ll try to match that height right there
try to make it look symmetric there and then it starts to repeat
all right and so this was pi over three and this end one right here was seven pi
over three and let’s see if that makes sense
there’s one pi over three two pi over three three four five pi over three um
actually multiplied by three so two pi is six pi over three
pi over three is seven pi over three oh i forgot these two right here
probably okay anyways um halfway right here is four pi over three
and let’s see if that does that makes sense if i um add these two together i get
eight pi over three and cut that in half and then that’s four pi over three okay
um just wanna make sure i didn’t make any mistakes there

00:21
and then uh the next one is five pi over six and 11 pi over 6 right here
and that gives us those right there those two marks right there
so 5 pi over 6 and 11 pi over 6 and our height right here is a 2
because the amplitude of this one that right here that we’re graphing just to
be clear we’re graphing minus 2 of sine of x and then we’re graphing the shift
right here also so that’s a 2 and this is a minus 2 right here
all right and so now we’re ready to sketch the isotopes we’re ready to
sketch the cosecant graph now so cosecant is one over sine so we need to
uh put the isotopes in here and we have another one right here
wherever the sine graph is hitting zero and so let’s go ahead and label these

00:22
right here this one is at x equals pi over three we can’t see them
down there let’s put them up here x equals pi over three
and then x equals four pi over three these are our vertical isotopes and then
x equals seven pi over three all right and now we can put the graph of the
cosecant in the orange and it’s going to get really close to the isotopes
right there so there’s an upper branch and here’s the lower branch right here
and so there’s the sketch of this one right here in orange
minus two cosecant of x minus pi over three there we go right there
it’s in orange right here we have a lower branch and an upper branch and
this sine graph right in here is just to help guide me and the
isotopes help to guide me but the actual points on the graph are the ones in
orange of course it repeats uh lower branch

00:23
upper branch lower branch upper branch and so on okay so now let’s do some um
change in the period so let’s do some that look like this now
so here’s going to be our next example here let me erase this real quick
so if you enjoying this video please go ahead and like
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here we have a change in the period because we have a 2 here
as well as some kind of shift however this is shifting the 2x here and
so what i want to do i mean this is not shifting the 2x right
here because to be a horizontal shift you need to be changing the x and this
is changing the 2x so what we’re going to do is we’re going to actually factor
out the two here and to and i’m going to use brackets like this
also although some people don’t um it’s 2

00:24
and then i’m going to factor out and get x minus the 3 4.
and i know it’s three-fourths because if we multiply that by two
the twos cancel here and i get three pi over two out
so this is a horizontal shift horizontal shift the shift has to be
the first thing that you do when you substitute in a number is i’m going to
shift it all right and so what we’re going to do
is we’re going to first sketch the graph of cosine
and then we’ll sketch the graph of 4 cosine 2x
and then we’ll sketch the graph of the whole thing
so let’s recall what cosine looks like i’ll try to sketch it over here so
cosine looks like this roughly uh a minute two
two straight there gotta make keep it curvy two pi pi pi over two

00:25
and three pi over two height of one and minus 1 there so that’s just the
regular cosine graph there just a quick sketch
now what happens if we try to sketch 4 cosine 2x
well we have an amplitude change of 4 and we have a period change so the
period is going to be 2 pi over 2 which is pi
so 2 pi over the number in front of x the coefficient of x
so the period is going to be pi now so let’s sketch this graph real quick
so these are just intermediate steps to help us graph the final ones we’d have
to do something like plot points right we just want to plot the important
points that that give us the shape that help us with the shape all right anyways
um so i’m going to still start up here because we have a positive 4 in front of
here so we’re going to start up here and
i’m going to shape it down and then back and then it starts to repeat
and where it starts to repeat now is pi not 2 pi and so halfway

00:26
it will be pi over 2 and then halfway again will be pi over 4
and now i’m going to count up my pi over 4s 1 2 3 so this is 3 pi over 4 right
here and of course pi is 4 pi over four now the height here because the
amplitude is four so the height is four and then this right here is minus four
here so here’s the sketch of four cosine two x right here
now what we’re gonna do is we’re gonna shift it to the right by three pi over
four so to sketch this graph right here i’m
going to shift this one right here to the right by three pi over four
and the reason why three pi over four is
because if i substitute in 3 pi over 4 i get back to 0 2 times 0 is 0 so we get
back to this height of 4 right here all right so let’s do that let’s take
each of these tick marks here to help me sketch the new shape the new
graph i’m going to take each of these tick marks and i’m going to add 3 pi

00:27
over 4 to them so let’s do that over here so 0 and then we have pi over 4
and then pi over 2 and then 3 pi over 4 and then pi
and then i’m going to add the 3 pi over 4 to each of them
and we’ll come up with our new tick marks and so the first one is easy three pi
over four and then we have four pi over four or just pi
and so this one right here is think of this one as two pi plus three pi so five
pi over four and this will be six pi over four and six pi over four
six pi over four reduces to three pi over two
and think of this as seven pi over four right four four pi plus three pi
all right so there’s our new tick marks and now we’re ready to sketch the graph
of the one that’s been shifted right here so we’re going to take these uh
tick marks right here and we’re going to shift them

00:28
and shifting them notice doesn’t change the period the period is still going to
be pi it’s just that the starting and ending is going to be shifted over
all right and and everything in between two um so yeah
let’s sketch this right here now so now i’m starting at 3 pi over 4
so let’s just put 3 pi over 4 about right there
and instead of 0 at 4 we’re going to have 3 pi over 4 at 4 so let’s
shape it by right here and then i’m going to come down
and then go back up i like to shape it before i put down those tick marks
and here’s where it starts to repeat this is three pi over four
it starts to repeat right here pi which is where it started to repeat has
been shifted over to seven pi over four and then the halfway right here
where we’re going to be at a four minus four but the halfway right here is the

00:29
five pi over four and then halfway between these two right here is pi
and then this one right here is the 3 pi over 2.
so there we go there’s the sketch of 4 cosine two x minus three pi over two
there’s uh one cycle of the graph of course it keeps repeating
to find the next level of tick marks just add the period the period is pi so
to get the next tick mark you would just say three pi over 4 plus pi
and then this one plus pi and then this one right here plus pi right so it keeps
repeating every pi and so that’s what you just add to the
tick marks or you would subtract pi off to get this shape over here
all right so it looks like a lot of fun let’s graph another one let’s do a
another cosine um let’s get going on that so here we go
um so i’m going to keep this part right here down the cosine shape right here

00:30
this is 3 pi over 2 because we have another cosine
now this one right here i’m going to factor out a 3.
so we’re going to view this as equal cosine of three
and this time i won’t put the big brackets and then it’ll be x minus
and then i think it’ll be what pi over 12 right is that right
three times what’s three pi over 12 that reduces to what pi over six
no pi over four oh so it’s not three pi over 12. um what is it here pi over
we’re factoring out a 3 so 18. so this right here will tell us that we’re
going to multiply 3 times pi over 18 and the 3 over 18 reduces to pi over 6.

00:31
uh let’s see if we can read that better or pi over 18. there we go and some
people like i said like to put brackets there
all right and so now we know how the period is going to change and we know
that this is a horizontal shift right here this is power 18 and it’s positive so
we’re going to add everything so let’s first graph cosine of three x
so the period is uh two pi over three uh we get the three from right here
and so let’s reshape this right here and we’re going to have this we’re
actually going to have a similar shape but this right here is going to start
repeating at 2 pi over 3 and so what’s halfway is pi over 3. and then halfway
again is pi over six and this will be one pi over six two and
three pi over sixes which is pi over two right three pi over six is that reduces

00:32
two pi over two and so we have one two three four pi over sixes
which reduces to two pi over three all right so we got our tick marks there
and then we still have a height of one and a a relative maximum and a relative
minimum of minus one and one all right so this is the sketch of
cosine three x right there all right and so now let’s go to and
shift it everything here so let’s list our tick marks here and this is minus so
i’m going to shift it to the right so let’s list our
tick marks over here so zero and then we got pi over six
and when we have pi over three and then we have pi over two
and then we have two pi over three and we’re going to add pi over 18 to

00:33
each of these pi over 18 and so let’s have some fun adding these
fractions together pi over 18 and then this one right here
i’m gonna say this is 3 plus 1 is 4 so what’s 4 over 18 will be 2 over 9.
so 2 pi over 9 and to get 3 to the 18 we need a 6. so
this would be 7 pi over eighteen and to get to the eighteen we need a
nine so this will be ten pi over eighteen and to get to the eighteen we need to
multiply by six we get a twelve so here we get thirteen pi over 18.
all right so you can see the pattern there okay and so now we have our new tick
marks we’re going to keep the same shape
it’s going to go it’s going to go up and
then down and same shape right there all right so here’s try to sketch it right

00:34
here and it’s been shifted over by pi over 18 so that’s about right there
and we got the shape right here and that’s coming and then it’s going to
start repeating and this height here and here is going to be a 1
and this will be a minus 1 right here and this tick mark right here will be pi
over 18 and this one over here will be 13 pi over 18
13 pi over 18 and then halfway will be the 7 pi over 18.
and then right here will be 2 pi over 9 and this one right here will be 10 pi
over 18. 10 pi over 18. is that right multiply by 9 so we get
that’s an 18 so 2 times 9 is 18 so we need a 9 plus 1 is 10

00:35
10 pi over 18 though reduces to what 5 pi over 9 so let’s just put 5 pi over 9
here all right here we go so there’s all of our tick marks there’s the sketch of
number six right here cosine three x minus pi over six
here’s the period once you know one complete cycle and the period then you
can get all the other cycles to get the next tick mark right here
to go from here to here we just add the period to go from here to the next one
we just add the period and so on all right so now let’s look at another one
let’s look at a cosecant now so cosecant and then it has two minus signs here
so let’s factor out the minus sign so i’m going
to say cosecant and then square brackets and then a minus and

00:36
let’s just go ahead and say minus two and then x and then plus pi over four
and let’s see if that makes sense so minus two times x is minus two x and
then minus two times here gives us a negative and the two over four cancels
and we get up we get a minus pi over 2 there all right
so now the negative because remember cosecant is odd so that minus sign is
going to come out and so we’re just going to have a 2 and
then an x plus and then a pi over 4 so the x plus pi over four means that we’re
going to shift it to the left by pi over four
and we’re going to have a change in the period
and but so first i’m going to sketch the graph of minus and then sine of 2x
so i’m going to first sketch this graph right here
and then i’m going to shift it to the left by pi over 4
and i’m going to sketch that graph and then that graph will help me sketch the
cosecant graph all right so let’s graph this one first so the period is 2 pi

00:37
over the 2 right here so the period is pi and it’s been reflected
so let’s see here we’re going to go down first and then back up and then back
down the period is pi so it’s pi over two and this is pi over four
and so this will be three pi over four and the amplitude of this right here is
one so the height here is one and this is minus one right here
and so i think that uh that’s a good enough sketch for minus sine two x right
here now let’s shift this uh to the left by pi over four
and so let’s take these tick marks here we have 0 pi over 4 pi over 2
3 pi over 4 and pi and let’s shift them over all right um so

00:38
we’re gonna um subtract a pi over 4 from everything because it’s positive here
so let’s do minus pi over 4 minus pi over 4
minus pi over 4 from these tick marks here
and we’ll get our new tick marks for our new sketch here this is minus pi over 4
and this is 0 and this will be pi over 4 where i think of that as 2 pi
minus pi so pi and think of this as 3 pi minus pi that’s 2 pi
2 pi over 4 or set differently pi over 2
and then think of this as 4 pi over 4 so taking away 1 we get 3 pi over 4 here
all right so there’s our new tick marks right there
now see if i can move over here and get this new sketch going over here and
let me draw it more straighter or straight um so

00:39
we’re going to shift to the left so let’s maybe move it the other way
all right that’s okay um so now we have this shape right here
but starting back over here and then we’re gonna hit through zero so
this part right here is going to be shaped right here
at 0. all right so it’s going to be shaped like this
and then it’s going to start to increase so let’s go increase it and
then come like this and then go back down and then it’ll just start repeating
so this tick mark is minus pi over four and this is of course zero
and then pi over four and then pi over two it hits the height right here

00:40
of what um so the amplitude of this right here is one so minus one right here um
and then the last one is three pi over four right here all right and so
that’s the sketch of y equals minus sine of two times um
two of and then the shift and then shift is pi over four
all right so that’s the sketch right there in black for this one right here um
it’s in the way though what i’m about to do next so let’s just
put it uh let’s say right here so let’s say here this is the graph of
minus sine brackets of 2 and then x plus pi over 4.
i want to make clear that it’s in black right there because right now
we’re going to need to sketch the isotopes in order to
in order to let’s move up here maybe so in order to sketch the graph of

00:41
cosecant cosecant is 1 over sine so i need to know where the sine is 0. so the
sine of 0 right here and so this will be x equals minus pi over four
and it’s zero right here at pi over four
so this is the vertical isotope x equals pi over four and then a zero right here
at x equals three pi over four all right and so
now we can sketch the graph of the cosecant and
actually let’s bring back the red and say let’s extend these a little further
so i can get a good shape in and so we’re going to have an upper branch
right here and kind of get smaller there i’ll move back down here
all right so there’s our upper branch and it needs to be nice and curvy

00:42
and this one here is our lower branch right here
so that that’s the sketch of the cosecant graph right here so
this minus cosecant here and notice how the sine is going down
and that’s taken into account the minus sign right there
and we’ve taken into account the period which is pi um so we can get the
uh upper and lower branches as many as as many of them as we want so for example
this uh one right here um the middle of this lower branch right
here would just add a pie to it so it’s going to happen over here at pi
and that will be the next relative maximum over there
so you can get more of these tick marks more cycles just by using the period
right there all right let’s look at one more

00:43
let’s look at a cosecant sorry a secant graph now yeah
so now we’re going to use a sign to help us sketch so here we go
we have a minus 4 here so that’s what i’m looking at first
so i’m going to factor out a minus 4 so this will be equal to -3 secant
and i’m going to factor out a minus 4 to get an x right here so i can get a
horizontal shift remember the horizontal
shift has to be plug in x and then first do the shift
so since i’m factoring out a negative this has to change sign to a negative
and i’m factoring out a 4 from this so think of this as a 4 over 4
so this would be a 12 down here then and
i’m factoring out that 4 but i’m leaving in that 4 down here that’s going to be
pi over 12 right here there we go we can double check it minus

00:44
four times x boom minus four times minus one over twelve
and that gives us the one third and then of with a pi
all right now remember secant is an even function
so that means it’s going to absorb that minus sign right there so this is equal
to minus 3 of secant of just 4 and then x minus pi over 12. there we go
so we just need to sketch this graph right here and you know
in order to do this we’re first going to sketch the graph of cosine first
and well let’s just go ahead and do that let’s sketch the graph of cosine
um actually let’s i’m going to sketch the graph of minus 3
cosine of 4x first can we can we just jump to this one right here
minus 3 cosine 4x and then we will shift it and then we
will do the secant so let’s see if we can sketch this one first it’s cosine

00:45
but it’s been reflected and what’s the period on this one so the
period is 2 pi over the 4 which is pi over 2.
so let’s see if we can sketch a graph so normally i would start up here for
cosine but then it’s been reflected so i’m going to start down here
and i’m going to get a nice good shape right here and then
down here and then it’s going to start repeating itself
so we got one full cycle here and i’ll just see if i can get smaller
here or down here um okay and so you know this right here
is one full period which is pi over two so let’s put a pi over two here and then
this is halfway right here where we have
a height of three the amplitude is three and so this is minus three right here
and so the halfway is pi over four and then halfway again is pi over eight
and then halfway again so we count up our pi over eight one two three pi over

00:46
eights four pi over eight all right and so that
this is the sketch of minus three cosine four x right there roughly um
and so now what we need to do is take all these tick marks right here and
shift them and this is a minus pi over 12 so we’re
going to add a pi over 12 to all of them so see if we can do that right here so
let’s take our zero for right here and that’s where
we’re right here and then pi over eight and then pi over four
and then three pi over eight and then pi over two and i’m going to add a
pi over 12 to each of them so that’s our horizontal shift we’re going
to be shifting right pi over 12 so pi over 12 and then add up pi over 12
and up pi over 12 and a pi over 12 and a pi over 12.

00:47
if you don’t like adding fractions you can just name these a b c d and make the
same shape and then just label them a b c but no i’m just kidding let’s just do
the uh fractions here so let’s think of this one as over 24
and i’m going to need 3 pi here and a 2 pi here so that’s 5 pi over 24
and think of this one is over 12 and so i’m going to need a 3 pi plus a pi
so that’s 4 pi over 12 which is pi over 3 so let’s put pi over 3 here
and then this one over here over 24. so we need a three so i’m going to say 9 pi
plus a 2 pi which gives me 11 pi over 24 and
over here we need a 12. so i’m going to say 6 pi
plus a pi and that’s going to be 7 pi over 12.

00:48
so voila we got our new tick marks now we take the same shape and we’re just
going to shift it over and when i sketch this graph right here i want to save
room to then also make the secant on it so let’s see if we can do all that
all right so here we go i’m going to move over here
and see if we can put it in right here so it’s been shifted over pi over 12 pi
over 12 is not that big anyways i’ll just put it about right here and so um
now this one right here is instead of zero it’s pi over 12 so it’s just been
shifted over a little bit and then i’m going to keep the same
shape right here but it’s going to be about right here let’s put it about
right here so we keep the same shape right here and then it’s going to go up
and it’s going to reach a height here and then it’s going to start repeating
right down through here and so we need these tick marks here

00:49
and this we got right here so we got pi over 12 and this one is 5 pi over 24
and this one is pi over 3 and this one is 11 pi over 12 24
and this one right here is the 7 pi over 12. all right and the height here is
a 3 coming from this right here the amplitude is a 3 for the cosine so minus 3.
and so this is the sketch of y equals minus 3 cosine of 4
and then x minus pi over 12. so that’s the sketch right there now i’m going to
put the isotopes where this cosine graph is hitting zero

00:50
so we have isotopes right there and there and um
yeah so we’re going to have the isotopes
i think i can label them right here this one right here is x equals 5 pi over 24
and this isotope is 11 pi over 24 and now we can sketch in orange
the original function here and it’s going to be coming in through here
and go up here and perhaps if we need to we can extend the isotope up
and then we have two branches down here we have the lower branch and we have the
right part of it and then we have this part of it right here because we have
these two isotopes right here and then it just keeps repeating this is
repeating right here because we already got part of it right here
and but i just like to a little bend it a little bit just to make sure i get a
good shape in all right so there’s the graph of secant right there

00:51
so excellent so now the question is um can we put it all together can we um
you know put the vertical shifts and the horizontal shifts and the
change in the period and look at tangents and cotangents and do all that
kind of stuff there um and that’s we’re going to do in the
episode that starts 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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