Trig Graphs (Vertical 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 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

00:01
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

00:02
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

00:03
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

00:04
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]

00:05
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

00:06
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

00:07
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

00:08
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

00:09
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

00:10
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

00:11
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

00:12
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

00:13
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

00:14
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

00:15
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

00:16
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

00:17
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

00:18
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

00:19
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

00:20
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

00:21
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

00:22
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

00:23
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

00:24
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

00:25
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

00:26
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

00:27
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

00:28
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

00:29
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

00:30
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

00:31
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

00:34
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

00:36
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

00:38
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

00:39
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

00:40
some more so that’s we’re going to do in the next episode is vertical shifts and
that episode 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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