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how do you move the graph of a trig function up or down

what is a vertical translation in this episode we practice vertical

shifts of trig functions it’s actually straightforward and quite

useful let’s do some math hi everyone welcome back i’m dave

so in this episode uh trick graphs vertical transformations let’s go ahead

and get started um and as we get started here i want to mention what was done in

the previous episodes just very briefly so at this point we have sketched the

graphs of sines and cosines and secants and cosecants and tangents

and cotangents in our previous episode today we’re going to sketch some of

these graphs again but this time we’re going to concentrate

on vertical shifts so we’re going to be shifting graph up and down

in this episode and stay tuned for the next episode

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where we shift left and right we’ll do some more practicing of that

before we give a final and complete review uh episode

all right so let’s go ahead and get started vertical translations

so we’re going to begin by sketching the graph of this one right here so we can

see how it’s accomplished so this minus 3 right here is a vertical shift

now we could write the graph like this we could say y equals

minus 2 sine pi x minus 3 but you see that kind of is ambiguous

then it would be better to put parentheses here and even still then

it’s not like 100 clear if the minus 3 is part of the angle or not

it just looks more elegant let’s just say that

if the minus 3 is written in front so basically we’re going to take this graph

right here which we already know from previous episodes and we’re going to

then shift it down three units so let’s practice doing all that

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so first thing um i’m going to sketch the sine x graph

now i’m going to take this step by step so i’m going to make a series of graphs

it just depends on how many of these you’ve done already if you’ve done like

50 of these you don’t need to necessarily to sketch every single step

out like i’m going to do but you know as you get further along

you need less of these intermediate steps but i’m going to start off by

sketching the graph of sine because we have a sine here and so let’s just

recall what sine looks like so i’m going

to start going up and then down and then i got a period of 2 pi here and then pi

and then pi over two we get a height of one and then three pi over two

we get a minimum right there of minus one and so that’s just one quick cycle

of sine uh but now we have a pi x here not just an x so

the way i like to think about it is when you input a value for x a number a

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real number for x what’s the next thing that you do

well we do multiply it by pi right so that’s going to change the period so

that’s the next uh transformation i’m going to do is so i’m looking at you

know when i’m going to input the x i’m going to do the pi and then i’m going to

do the sign and then i’m going to do the minus 2 and then i’m going to do the

minus three so i’m going to make those sketches in order in which we are

actually calculating the output here so next thing is to say

okay so the period is 2 pi because that’s the period of sine

divided by the b which in this case is pi so the period is two

so instead of going from zero to two pi i’m now going to go zero to two

so now i’m going to sketch the graph like this right here so let’s come up

like this and then back down but now the period is two

and so halfway right here would just be a one and then halfway again would be a

one half and that’s where i’m getting the one

and this is one half and then two two one halves and then three one halves and

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then four one halves and so this is a minimum right here minus one and so this

is the sketch of sine of pi x right here or it’s just one cycle of the graph but

there we go um okay so yeah it’s nice and curvy just keeps

repeating over and over again now what happens we put the minus two on here

so let’s do that next so the minus is going to be a reflection

and the two is going to be a vertical stretch

so instead of going up now we’re going to come down

and then we’re going to go back up come back down and the period is 2

and we have a one and we have a one half and we have a three halves

but now here the height here is um here it was a minus one

and when i take that minus one and multiply it by minus two [Music]

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i get a positive two out this is the height of two

and this is minus two right here so here’s the sketch of minus two sine pi x

um it’s been reflected reflected and it’s been stretched so no longer a

height of one now height of two right here all right there’s a

sketch of this one right here like i said you don’t necessarily need

to sketch both of these to get here maybe just this one or or that one or

you know but i’m going to go ahead and erase this one right here

and because what we’re going to do now is we’re now going to do the vertical

shift and we’re going to shift it down three units

so this is the x-axis right here and it’s symmetric it’s not symmetric but it’s

it’s you know got these zeros right here and it’s going down and then up so i’m

going to repeat that right here but instead of the x-axis right here i’m

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going to draw a dash right here i’m going to call this right here

right so this is better known as y equals zero i mean not better known as

but it’s y equals zero this is x axis this is y equals minus three

because we’re shifting it down and now i’m going to keep the same shape

here so it’s going down and so i’m going to go down and then come back up

and then go back down and so this right here is the same tick

marks right here this is a 1 this is a 2. and the height right here

is at 3 pi over 2 and the minimum right here is at the one half

and so now what are the tick marks over here along the y axis so the

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tick marks are so this is a minus two and i shifted it down by minus three

so now this is a minus 5 right here and this was a 2 right here

and we’ve shifted it down by -3 so this right here has a height of -1 right here

and there we go this is minus three minus two sine pi x

so we’ve took into account the period the period is just a two

we’ve taken into account the the amplitude of the sign and shifting it

down by -3 right there so it has the shape right here it’s just going up and

down up and down along the y equals minus three axis there i dash that in not

necessarily because it’s an isotope but just because it’s just helping me sketch

the graph right there all right so we got our x labels our y

labels everything looks good right there so let’s do some more

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um let’s graph uh this one right here let’s graph 2 plus 2 sine x so

i’ll just sketch an intermediate graph this part right here first

and so this is just going to shift it up to

so we’re going to have this nice sine graph right here period is two pi

so we have a pi we have a pi over two but now we have a height of two

and a height of minus two and this is three pi over two so this is y equals

2 sine x now if i did that too fast for you

then i just want to remind you to check out their previous episodes where we

went through graphs like this very diligently

and so what’s new right now is the two you know how does this come in how does

this affect it so we’re going to take the whole graph and

we’re going to shift it up too so again this is the x-axis and or or

if you want to label it as a line it’s y

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equals 0. and so now i’m going to sketch in here the y equals 2 graph line

i dash it in because it’s not really part of my final graph but i’m going to

go up by 2 and then down by 2. so when i go up by 2 i’ll hit a 4 and

then when i go down by 2 i’ll hit the x axis again

so let’s see if we can get that sketch in here up and we’ll come back down

and i missed it let’s go back down and then up and so this right here is

still two pi where we hit that and then right here is pi

and this is power two and the height right here is four

because we shifted this height of two but we shifted it up by two so now we

get a four and this right here is three pi over two

and that’s where we hit the zero right there

all right so this is the graph of two plus two sine x and

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i’ll move out of the way right there so this is just the line y equals two right

there all right there we go that’s a lot of fun so there’s number one

let’s look at another one so this will be three minus one half sine x

so again i’m going to sketch the part without the vertical

shift first so minus one half sine x so let’s sketch that real quick sine x

now this is a minus in front so it’s going to be reflected so it’s going to

look like this and then like that and the period is 2 pi

and we’re going to come through here to pi and pi over 2 and 3 pi over 2

and the height here is a one-half and this is a minus one half so i’ve

taken into account the minus sign by starting by reflecting it so it goes

down first and then up and the amplitude’s one half that’s the height

and so this is a minus one half here and so this is the sketch of

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minus one half sine x and so now we’re going to shift this up by three

so this height of one half will be three plus a half it’ll get shifted way up

here and this minus one half will get shifted up so let’s sketch the graph

right here it’s going to get shifted up high so i’m

going to put this pretty down low here and this was the

x-axis right here so i’m going to dash in the line

y equals minus sorry y equals 3 just to help me sketch the graph

and so we’re going to go down first and then up

and we’re hitting a tick mark right here of 2 pi

the period is 2 pi for this we’re not changing the x

and so halfway right here is going to be pi

and then halfway right here is going to be pi over 2 and then 3 pi over 2

where we hit this height right here and so what will this height be

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3 plus a half so think of 3 is 6 over 2 right so that’s just

7 halves so 7 over 2 and this minimum right here will be three minus a half

so six over two minus one over two so it’ll be five halves

so that’s the minimum right there and that’s the maximum right there for one

cycle right here and so this is just 3 right here so let’s just erase that

so there we are there is the graph of number 2 right there three minus one

half sine so let’s do another one so five minus sine two x

so let’s work on sine two x first uh with a minus in front of it and then

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we’ll shift it up five so here we go um y equals minus sine two x

so the period is two pi over two or just pi

and so let’s just uh this is reflected it’s a minus sign it’s going to go down

and then up and the period is pi so halfway is pi over two

halfway here is pi over four and then count up our pi over four so this is uh

one two three pi over four and the height here is one

and the minimum right here is at minus ones so we got a relative max relative

min right there all right so now let’s shift this all up

by five so this one this height of one right here is going to get shifted up to

six and this height of minus one right here is going to get shifted up to four

i usually like to take note of those before i uh sketch this so i can kind of

figure out you know where it’s going to be the best

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way to put the origin right here so i’m going to say here is the line y equals 5

and then i’m going to take the same shape to it it’s going to start going

down so it’s going to go down and then up

and there’s one period right there at 2 pi and then here’s halfway pi

and here’s where we’re going to get our minimum at pi over 2

and our maximum right here at uh wait a second our period is pi my bad

period is pi and then halfway is pi over 2 is right here and then pi over 4

and 3 pi over 4 and now these right here these marks right here uh the height is

one and then we shifted it up five right

so it’s it’s starting at five it’s going to climb up to a six so there’s a six

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and this was at a five it’s going to go down one to minus one so it’s going to

be out of four there we go here’s the sketch of one cycle of five minus

sine two x and that’s it so we’ve taken this one right here and we’ve shifted it

up five and that’s what we get right there let’s do another one

ah let’s do a secant all right so again you want to check out the episode

where we graph sequence if you haven’t seen it yet

but in that episode we didn’t do vertical shifts um at least i don’t

remember doing vertical shifts in it but in any case so two things

secant is an even function so what i mean by that is this is one over cosine

and cosine is even so secant is even so in other words the minus sign

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right here gets absorbed into the secant so to simplify things we can just say

this is equal to 3 minus 2 secant of x so this is what we’re going

to sketch because these are equal to each other now the next thing is

when we graph secant i’m going to first graph cosine

so i’m going to first sketch the graph of minus 2

cosine x and that will help me sketch the graph of the secant and then once i

sketch the graph of secant then i’ll lastly do the

vertical shift there so here we go um so we have a cosine here

and so let me sketch the cosine here and cosine usually starts up here at one

but it’s a negative so it’s going to start down here at minus two

and it’s going to go up and then up and then down and then it’s going to start

repeating itself and this marker right here is two pi and that’s zero

halfway where it reaches its height right here is pi

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and then this is pi over two and then this is three pi over two

and so this is the height of two and this is minus two right here so this

is minus two cosine x right there or there’s one cycle of it okay so now

um for the cosine uh sorry for the secant of this is well we’re going to need to

make some isotopes so now let’s sketch the graph of the secant

and so we’re going to need to know where the cosine is 0 because remember secant

is 1 over cosine so where is the cosine graph right here 0 so we’re going to

have an isotope right here at x equals pi over 2

and we’re going to have a pi over 2. let me write that better

and we’re going to have an isotope right here at 3 pi over 2 3 pi over 2

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and so let’s put the graph and say orange for secant so we got this right here

and maybe i’ll extend my isotopes a little bit

and so there’s the there’s one branch of the secant

and then here’s another branch right here and another branch right here

so we have a upper branch and we have two halves of the lower branch right

there so there’s one cycle for the secant so the one in

orange i’ll write that in orange so this one is the graph of minus 2 secant of x

all right so there’s the sketch of that and now we

need to shift this secant right here up three

so um this height right here of two is going to be shifted up to a five

and this minus two here is going to be shifted up three and we’re going to get

a one there so let’s see if we can sketch the graph i’ll try to do it over here

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and so let’s put it down to about right here oops

let’s put it down about right there in fact let’s just bump it over here a

little bit and so our isotopes so shifting it up

and down is not going to change the vertical isotopes so we still need an

isotope at five two and the other isotope the other vertical

isotope is three pi over two three pi over two good and so now um

we’re going to have a branch sitting right here and this this minimum this

relative minimum right there is 2 it’s been shifted up to five

there’s a tick mark right there for five and

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this branch right here it’s minus two but it’s been shifted up three so it’s

now going to be a positive one so let’s say positive ones right here

and it’s just going to be going down like that

and then we have another one over here like this and so this is at you know 0

0 1 and what is this right here so this was at 2 pi right here

it’s the maximum right there at 1 at 2 pi and then it you know continues on the

other part of the lower branch and this continues on so there’s one cycle right

there for this graph right here 3 minus 2 secant x

is this part right here and that part right there and that part right there

and there’s one cycle right there and we

have the isotopes right here it’s really

unnecessary to sketch the cosine in here you could if you wanted to but i did it

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at least once so that you know we could follow along pretty easily

all right let’s do another one ah yes let’s do a tangent all right so

if you have any questions let me know in the comments below any question about

any of the um any of these that we’re doing right here

all right so now for the next one let’s recall that the tangent is odd so the

tangent of minus x is sine minus x over cosine minus x and

sine is odd so the negative comes out cosine is even and so this is minus

tangent x so in other words tangent x is odd because

of this so so this minus sign can come out and that’ll change that to a

positive so this right here number five is equal to one plus three times tangent

of x and so this is what we’re really going

to grab because these are equal to each other so

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let’s recall what the tangent graph looks like i’ll put that down here

so we have an isotope right here at uh minus pi over 2.

sorry pi over two let me label it up here pi over two

and we have another one right over here at minus pi over two

those are the vertical isotopes there and now we’re going to shape this right

here it’s going to be increasing and halfway between zero and pi over two

pi over four that’s we hit our one and halfway right

here between zero and minus pi over two minus pi over four

and that’s where we hit r minus one right there

so there’s a reasonable sketch of the graph of tangent x

just got one cycle in there right remember the period is pi

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so what does the three do and what does the one do so the three is going to

vertically stretch it and the one is going to shift it all up one

so let’s try to make a sketch over here so stretching it and shifting it up one

those two are not going to change the isotopes so the three is is going to

make it go up faster so at pi over 4 we’re going to be all the way at a

height of 3 instead of just a 1 because the output will be a 1 for this

part right here and then we’ll multiply it by 3. so at pi over 4 instead of

being at a 1 it’ll be at a 3. that doesn’t change the isotope by by making

it grow faster it’s just getting faster uh going up but it’s not um you know

changing the isotope at all same thing with the one the shift of one

is taking it and shifting it up but that’s not changing the isotopes so long

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story short i just have the same isotopes here pi over two and minus pi over two

and so we have the same shape but i don’t want to sketch it in right here

because we’ve been shifted up one so this is the um you know right here

is the x-axis and so now i’m gonna sketch in here a

dashed line this is not an isotope it’s just helping me sketch the graph this is

the line y equals one and now i’m going to you know shift this origin up one

and then i’m gonna keep the same shape here i’m going to keep the same shape

it’s really close to the isotope it’s coming in here nice and curvy and then

it’s going to just get really close to the isotope right there

and so this is at the tick mark of one and halfway between zero and pi over two

is the pi over four and over here we would get out one but we’re

multiplying it by a three and then we’re adding a one to

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it so the height here would be a four so we input pi over four and we get out of

one times three that would give us a three and then plus one we get the four

and then over here at pi over four what’s happening at at minus pi over four

we’re going to get a minus one so that’ll be a minus three and then

plus one so that’ll be a minus one right there

so i’ll put it in about right here minus

one and this is the tick mark right here minus pi over four

so again to check that out here minus pi over four we input that in here

and we get out a minus one and so that would be a minus three and

then we add a one to it and so that’s actually minus two isn’t it yeah minus two

all right so there we got a one we got a four we got a minus two

we got our two isotopes and we got our two right there and so yeah there’s our

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graph of one plus three tangent x which is equal to that right there um

so pi over four get out of one three four and then minus three

plus one is minus two okay yeah looks good let’s do another one

and this one’s number six ah uh oops number six so this one’s cotangent so

let’s briefly recall what cotangent looks like let’s do that right here

so i’m going to sketch one period of the cotangent graph and

in fact let’s go ahead and do [Music] the minus cotangent graph

in fact let’s just put them both right here so this will be x equals 0 x equals

pi for the uh isotopes right here so again if you don’t understand cotangent

graph check out the previous episode where we went over that very nicely so now

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the cotangent graph would look something like this right here

which i’ll put in let’s say orange so this would be the cotangent graph right

here halfway right here would be at pi over two

it’s just decreasing very nicely like that but now this is a minus on it so it’s

going to be reflected so all these y values these y values right here they’re

going to get multiplied by minus so now they’re going

to be negative so this graph is going to be reflected and it’s going to look

something like i’ll put it in blue right here

but zero right if you multiply zero by negative you still get zero still going

through the right there all right and so there’s there’s a

sketch of minus cotangent x and the one in orange is just the

regular cotangent x all right in any case what we’re going to do now is shift it

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up one and so let’s make that sketch right over here so i’m still going to have

again shifting it up doesn’t change the vertical isotopes so we’re still going

to have the same vertical isotopes at 0 and pi and pi

now to help me sketch this graph here so this was the x-axis right here where

it went through the zero right there and so now i’m going to sketch the line

y equals one and that’s where it’s going to hit the

zero right there and because it’s a minus right here it’s gonna it’s gonna

be this one in blue here that we’re shifting up and so i’ll put it right here

we’re still going to get close to this isotope

and we’re going to come through here like this and then we’re going to get

close to that isotope right there let me extend that isotope up a little bit

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just to make sure that you know graph looks good there halfway is pi over two

and this says uh x equals pi so there’s the two isotopes at pi over 2

is where it hits the y equals one line let’s label that right here y equals one

and so that’s we’re hitting right there and now i usually like to go like um

halfway so something like three pi over two uh not three pi over

two three pi over four and pi over four what’s happening at three pi over four

so um well i didn’t even bother to put those over there uh that’s okay let’s

just leave it like that and there’s really nothing else to see

here it gets really close right here it gets really close right here

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and there’s the point pi over two uh one right there and then we got the

isotopes all right so let’s look at uh another cosecant here

and this one’s got a change in the period

so we haven’t looked at it cosecant yet in this episode so let’s look at it

cosecant so i’m going to look at the sketch of y equals a minus sine 3x first

i’m going to sketch that one first and then we’ll apply the cosecant and then

we’ll apply the 2 to there so here we go minus sine 3x so what’s the period so

the period is 2 pi over 3 and it’s negative so it’s going to start

going down first but here we go so we’re going to start going down

because it’s got been it’s been reflected so we’re going to go down and

then we’re going to come back up and then there’s one cycle right there

at two pi over three and so halfway is two pi over six chop it in half

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uh two pi over six so is the same thing as pi over three

which of course makes sense that’s one pi over three that’s two pi over three

but anyways halfway here is pi over six and now we got one pi over six two pi

over six three pi over six three pi over six as known as pi over

two all right and so what’s the height here right so this is a

minus one in front of here so the amplitude is one

so it’s a one and that’s a minus one there

all right so here’s a reasonable graph of minus 3 sine 3x

and now let’s try to put the cosecant on there so the reason why i graph sine is

because cosecant is 1 over sine so i need to be worried about where the sine

is 0. so we’re having a zero right here and we’re having zero right here

and a zero right here and so these are the isotopes x equals zero

00:32

x equals pi over three and x equals 2 pi over 3.

and so those are the isotopes for the cosecant graph

so now where is the cosecant graph let’s put that in let’s say blue

so the cosecant graph is going to come right through here

there’s the lower branch and here’s the upper branch

and so this is the sketch of in blue right here is y equals minus cosecant 3x

so we’ve changed the period the cosecant has period 2 pi but not cosecant 3x

all right and so there’s the lower branch upper branch and then lower

branch upper branch and it just repeats all right so so far so good

so now we need to shift it up two units so this one right here would get shifted

up to a three and this minus one will get shifted up to a one so let’s try to

00:33

put that in here if we can i’ll uh try to squeeze it in right here

and so we need the now keep in mind that vertical shifting doesn’t change the

isotopes here so we’re still going to have x equals 0 as the isotope

x equals pi over 3 as the isotope and x equals 2 pi over 3 as an isotope

and so now what i’m going to do is sketch it in black

so this minus 1 now gets shifted up to so now it becomes a positive one

so we’ll put it right there so here’s the lower branch right here

there’s the lower branch of the uh function right here the whole function

and the upper branch right here this is a one that gets shifted up two so now

it’s a three so i’ll put it about right here and i’ll

put a three and this is the upper branch right here

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so it looks something like that so it’s just this graph right here in blue

but it’s been shifted up so this branch right here has been shifted up and this

branch right here has been shifted up and there’s the graph right there y

equals 2 minus cosecant 3x right there in black right there

all right and let’s do one more let’s just do a cosine we haven’t

done a cosine yet so let’s identify the vertical shift it’s one half

and we’re going to change the period and we have a minus in front of the the

3 pi over 4 and we’re going to remember that cosine is an even function so it’s

going to absorb that negative so this is equal to one-half minus and then this

will be cosine of 3 pi over 4 x so cosine absorbs that minus sign because

00:35

it’s an even function there and so we need to graph this so let’s see if we can

do this part right here first so i’m going to sketch the graph of minus

cosine of 3 pi over 4x just this part right here without the vertical shift

now the period is 2 pi divided by 3 pi over 4

which is the same thing as 2 pi over 1 times 4 over 3 pi

but in the end the pi’s cancel in fact we just get eight out of that

so the period is actually eight and it’s been reflected because we have

a minus sign there so let’s see if we can sketch this graph so instead of

starting up here i’m going to start down here like this

and go up and then back down and then back down again

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and this part right here where it starts to repeat itself is an eight

halfway is a four halfway is a two two four six eight

and the height here is at a one the amplitude is one

and minus one right here and so here’s the sketch of

minus cosine three pi over four x right there so actually

i just realized i completely ignored the three so that’s eight over three

eight over three i don’t know how i just completely

missed that so these tick marks are different this is eight over three

halfway is eight over six which is four over three which makes sense

um and then halfway again is four over six which is two thirds

so if it’s just the same numbers before it’s just that over threes right so two

00:37

four six six over three but six over three is better known as two all right so

almost missed that three there all right anyways the height is one here

and minus one here and now we’re going to take all this and shift it up by a

half so this one is going to get shifted up

to three halves and this minus one is going to get shifted up to minus one half

so let’s put this x axis here and now i’m going to try to sketch it right here

and i want to put it about right here and instead of the

so i’m going to try to sketch this line right here y equals a half

y equals one half and now i’m going to try to keep the

same shape right here so there’s a half so let’s say this is about another half

this is minus one half and that’s what we’re getting right here

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because we’re taking the one half and we’re shifting it we’re taking the minus

one we’re shifting it up a half so we get minus one half and then i’m gonna

now keep the same shape so i’m gonna go up like this come back down

and then start to repeat and this is the tick mark eight thirds

so we got the same exact tick marks right here so halfway um is right up here

which is four thirds halfway again is two thirds

add up the thirds this is six thirds which is two right there so

and this height right here is one plus the half so three halves right there

so there we go there’s the relative maximum relative minimum um now actually

i put this right here but it’s been shifted up one

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so actually that’s not labeled very nicely it’s been

so at two-thirds here at zero and zero plus a half so right here

that tick mark is the two-thirds and then four-thirds is still giving us

the max and this one’s not very good either

so this is right here where it crosses so right here is the zero but it’s been

shifted up a half so that’s right here and so this tick mark right below this

point is the two and so the two is not going to be right there

all right yeah so there we go there is the last one right there

so if you enjoyed this video i hope that you uh like and subscribe and the next

one is what about horizontal shifts so today we

talked about vertical shifts a lot and we’ve already talked about horizontal

shifts before but let’s go through them and practice

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some more so that’s we’re going to do in the next episode is vertical shifts and

that episode starts right now