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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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