Graphing Trig Functions Using Transformations

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

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

00:00
what is the amplitude of a sine graph what about the period what about cosine
in this episode graphing trig functions using transformations we’ll practice
graphing lots of functions that involve sines and cosines
you’ll learn what all this means and best of all you’ll learn how easy it is
let’s do some math [Music] hi everyone welcome back we’re going to
begin with the sketching the graphs of trig functions
um so we’re going to start talking about um the um
even and oddness of a trig function of in the following form
so let’s get on here going um so we’re going to be looking at uh graphs of
these four types here uh where a and b are real numbers
so a is going to be a positive or negative real number

00:01
and b is going to be a positive or or negative real number um [Music] now
in the previous episode um we talked about um trig functions of these forms
here but we were assuming that b was positive so now we’re not going to
assume b is positive so b could be negative and we’re going to deal with
that case and we’re going to see how easy it is
but before we get started on this episode i wanted to mention
that this episode is part of the series trigonometry is fun step-by-step
tutorials for beginners link is below in the description
and so you can follow along in the whole series
and so like i said in the previous episode we talked about uh what happens
uh how these graphs look of these type of functions when b is positive
so now when b is negative so we’re going to use the fact that sine
and cosecant functions are odd functions
and cosine and secant are even functions and so what does that actually mean

00:02
well if we assume that b is positive and then we have a minus sign in front
of here then we have a negative number in front of the x
and what we can do with that negative so we can pull the negative out because uh
the sine of negative x is equal to sine of x
and so this is what i mean by that sine is an odd function
just like if you put a negative x into a third power here
you know that would be minus x to the third here right
so the negative comes outside of the third or here the negative comes outside
of the sine function right here so um now cosine is an even function and so
what that kind of means is it absorbs the negative right so if if b is if
we’re assuming that b is positive then this will be a negative
times a positive number for example something like 5 cosine
minus 6x right so now the b so the a is 5 and the b is 6

00:03
and we have negative 6 here and so how can we deal with that negative right
there so we could this will be just 5 cosine 6x because
cosine is an even function cosine of minus x is cosine of x
um and you can see all these um identities you can if you look on the
graph you’ll you can see them on the graph so if we enter say pi over six
that output will be the same output as if we entered minus pi over six so
cosine of minus pi over six is equal to cosine pi over six
of course that’s just not true just necessarily for
um the quadrant one and quadrant four angles but it extends on but in any case
the secant is also even so secant will absorb a negative sign
and that’s true because well the secant is the same thing as 1 over cosine this

00:04
is our reciprocal identity that we talked about since the beginning since
we defined cosine and secant and so if i was to put a negative into
secant negative x a negative angle for example that would be 1 over cosine of
minus x but that is one over cosine of x and so that is secant x
yeah so secant of minus x is the same thing as secant x
and then lastly the cosecant as you might imagine because cosecant is the
reciprocal of sine so if we have a cosecant here then this will be one over sine
and so if we try to substitute in a negative into a cosecant then we’ll have
sine here and then we’ll pull the minus sign out and so that’ll be minus sine
and so that’ll be minus 1 over sine and so that’s just negative cosecant x

00:05
okay so cosecant minus x however you want to think about it is just the same
thing as minus cosecant x so if we know how to sketch these graphs over
here we did this in a previous episode when b is positive
and so if you have negatives in here well we can use these identities right
here so this is assuming that b is positive because we know how to graph
these and so now we can graph these right here if we want to graph something
that looks like this we’ll just sketch this graph right here because they’re
equal to each other all right so um keep in mind though that the period is 2
pi over b and again we went over that in the
previous episode so here we’re just kind
of extending this using these identities here and practice graphing some more so
let’s get started let’s uh graph our first one here 2 sine 3x
so this is definitely something that we covered in a previous video because the
b here is positive and the a is just a 2. so let’s just refresh our memory how

00:06
to sketch this real quick so the period is 2 pi over the b and the b is three
so the period is two pi over three and the amplitude is going to be two or
the amplitude is absolute value of two which is just two all right so we can
sketch the graph now now what i’m going to do is i’m going to
sketch the shape of sine and then i’ll adjust the tick marks for
the period so it’s important that we get the shape in now you don’t want the
shape to be like making made up a bunch of you know lines right there you want
to add the appropriate curviness to the graph
the shape of the graph is important so i’m going to focus on that
now when we substitute in 0 we get out 0 sine starts right here at 0
or at least the primary period of it and so i’m going to go up

00:07
and then i’m going to come back down and that looks reasonable i guess but
anyways this is two pi over three that’s one period we go up and down
and you know if you don’t like it i recommend you know just keep practicing
go up come back down and so this i’ll put this mark here as two pi over three
because that’s one period and we repeat start repeating after that and so what’s
halfway halfway will be two pi over six but two pi over 6 reduces to just
2 pi over 3. and then we’ll cut halfway again to get
the height here which is happening at 2 and so cut halfway again is pi over 6
and now we can add up our pi over six one pi over six two pi over six
three pi over six what’s three pi over six that’s just pi over two
and the minimum right here is minus two and so there’s one period and we can

00:08
extend this to to other period we could extend this graph as far as we’d like
we just go down and then come back up and so yeah we can extend this again
this is 1 pi over 6 2 pi over 6 3 pi over 6. this is 4 pi over 6
right it just reduces to 2 pi over 3 but we can count keep counting our pi over
6’s so that’s four pi over six so the next one will be five pi over six
and then six pi over six right and so we can keep counting our pi over sixes
and we’ll get all the uh tick marks that we see here and we can just keep
repeating that and so on and we can you know we can
subtract our pi over sixes and we can get the graph this way as much as we
want now typically i just sketch though one period of graph there all right so

00:09
there’s a graph right there for number one here
all right and so let’s graph if we have a minus three in there
so now we have to work out our identity here
so let’s say this is going to be minus 2
sine 3x remember sine is an odd function so
this is the same graph that we just had but now has a minus so it’s been
reflected so oh we’ll just sketch this one right here let’s just uh
remark on the period the period is 2 pi over the b which is positive which is 3
and the amplitude is absolute value of minus two so the amplitude is still two
and we can sketch the graph let’s sketch it say oh right here
now instead of going up because it’s being reflected now i’m
going to sketch it going like this and then like this
and then there’s going to be one period this is 2 pi over 3

00:10
and then this is halfway which is you know two pi over six or pi over three
pi over six one pi over six two pi over six
three pi over six which is pi over two and the height here is two
and the minimum right here is minus two and so there’s the graph of number two
there um and so you know the minus there gave
us no problem at all because we know that sine is an odd function
all right so now what happens here if we have this one right here
so now let’s write it this as this minus sign can come out which case
we already have a minus sign so it’ll be a minus times a minus so this will be 2
sine pi x and so now the period on this graph is 2 pi over the b which is pi

00:11
pi’s cancel the 2 period is and the amplitude is absolute value of
well it’s just 2. it’s always positive just just remember
the amplitude is positive there all right and so let’s go ahead and
sketch the graph uh i might be we’re making it a little bit sideways all right
um and so yeah we have a we’re going to go up
and come back down and we have a nice sine shape to it right there
and this is period is two we’ll chop it in half we get one
we chop it in half we get half and now we count up our halves one half
two halves three halves so three over two and then two
and our height here is a two and our minimum right here is minus two
all right so there’s a graph for number three there

00:12
and let’s look at something like this now so this is
a minus sign again sine is odd function so the minus is going to come out
so this right here is going to be minus 1 4 sine of pi over 3x okay
and so the period is 2 pi over pi over three
or so differently two pi times three over pi pi’s cancel we get six
and the amplitude is uh the absolute value of minus 4 which is just 1 4.
all right so let’s sketch the graph uh let’s put it down here
um so our amplitude is 1 4 the period is six and it has a minus on it

00:13
and so i’m going to start going down so it’s been reflected
and then come back up and so on and so here’s one period so the tick mark is six
i’ll chop this in half is three i’ll chop it in half again three over
two and now i’m gonna count up my three over twos one three over two two three
over two three 3 3 over 2’s which is just 9 halves
and then another 3 over 2 which would be 12 halves which would be 6. and so
here the height here is 1 4. and the minimum right here is minus one
fourth there right there so there’s a graph of number four right there
all right so now let’s do um let’s do the cosecant graphs now now
we’ve done cosecants before in a previous episode also but we focused

00:14
when the b was positive so let’s do some here where the b is negative
so let’s look at this graph right here now i’m going to be
thinking of this as 3 over sine of 2x so the point is is that in order to
graph the sketch the graph of cosecant i’m going to look at the sketch of of
the graph of sine so of sine 2x so i’m going to be sketching this graph
but before i do so i’m going to first sketch the graph of sine 2x
so let’s do that right here so i’m looking at the period the period is um
2 pi divided by the b which is 2 and so the period is pi
and this is a sine graph and there’s a positive sign in front of it so it’s
going to be going up and this will be coming down

00:15
something like that the period is pi so that’s pi pi over two pi over four
and this is one pi over four three two pi over fours three pi over fours
and then of course four pi over four and the height here is a 3 and -3
and that’s the graph of let’s see here so yeah so we have the amplitude here of
minus three three and minus three so for example
what happens when we use pi over four here so sine of two times pi over four
and then so sine of two times pi over four
that’s the same thing as sine of pi over two
which is 1 which is 3. so pi over 4 we hit a height of 3.

00:16
so that’s a 1 sine of pi over 2 is a 1 so we get a height of 3 out there
all right so in any case now we’re ready to sketch cosecant all
right so what we’ve sketched so far is y equals 3 sine 2x
and now to get the reciprocal so now i’m going to sketch that in red
and so i’m looking where the sine is zero so i’m going to put the vertical
isotopes in here x equals zero and another one right here at x equals
pi over two and another one right here at x equals pi
and i’m going to sketch the graph here in purple for cosecant
and so the graph is going to look like this so these are isotopes so we got to
get close but not too close and then down here
right there we got that point and we got this point right here the height is

00:17
three and this is minus three right here at three pi over four
and so there’s the graph in purple and so
you know we may erase the graph in black there that was just there to guide us
the graph of cosecant is the one in purple there
all right so let’s do another one let’s do this with um a minus 5x in there
all right so let’s sketch that one now so again i’m going to first sketch the
graph of y equals minus 3 sine of minus 5x
and so let’s sketch that one right here now this is the graph of minus this is
the function minus comes out remember sine is odd so this is just
going to be 3 sine of 5x in other words this one right here is just

00:18
the 5 comes out the 3 com sorry the minus sign comes out so that gives us a
three cosecant five x so this is what i’m really sketching right here
so i first want to sketch this one right here three sine five x
and what’s the period on this one it’s two pi over five two pi over five is the
period so i’m going to sketch a regular sine graph as a guide
and so i don’t want to make it too tall because i want the cosecant graph to
stand out more all right and so there’s one period
and the period marks in at uh this is period period and so that’s 2 pi over 5.
so halfway would be just pi over 5 and then halfway there would be power ten
and so what we’re really doing is counting up pi over tens one pi over ten
two pi over ten and so this would be three pi over ten

00:19
and now i’m looking for the isotopes to get the cosecant graph so where is sine
0 right because this is this is cosecant 1 over sine we cannot divide by zero so
we’re looking at isotopes right here x equals zero x equals pi over five
x equals two pi over five and you can see the pattern you know
zero pi over fives one pi over five two pi over fives three pi over fives four
pi over fives and so on those would be the isotopes
so now we can see the different branches
and maybe i’ll try to put this in orange maybe it’ll stand out a little bit more
right here there’s that branch and now we’re graphing the this one
right here in orange and then we got this branch right here
and there’s the graph of cosecant y equals 3 cosecant 5x

00:20
and of course we have the x and y axis all right so there’s a reasonable sketch
there of the cosecant all right so you see that this episode
is very similar to the last one we’re just making the small modification
and the usefulness of because knowing the cosecant function is odd
all right and so let’s do this right here and so i’m going to first sketch this
this is odd so this is going to be pull out this minus sign it’s going to cancel
with this one or it’s going to multiply to a positive cosecant of 2 pi over 3 x
so this is what we’re going to sketch the graph of
and so to do that we’re going to sketch the graph of y equals
sine of two pi over three x and so what’s the period is two pi over

00:21
two pi over three which is the same as two pi times three over two pi
in other words just three i can’t see that just three
all right so the two pi’s cancel you just get a three all right and so what’s
the sketch look like all right so um let’s go here with this is
so this is a negative times a negative this is positive here so i’m looking at
a positive so i’m going to go up and come down
and this is a 3. i chop three and a half i chop that in half
now i add up my three over fours one three over four two three over four
three three over fours which is nine over four
and then twelve three over 4s which is of course three
and then the height here is a one and minus one and now i look where the

00:22
isotopes are coming in here where sine is zero so x equals zero [Music]
x equals three over two x equals three and we can see the pattern
zero three over twos one three over twos two three over twos
four two three over twos and so on and now we can sketch the graph now that
we have the isotopes we can shape it nicely without the isotopes it’s not quite
clear what the graph is doing so i like the isotopes on the graph there
and then if you want we can just erase the
sine graph it’s not really part of the graph
and occasionally i like to do that just so i can see the graph all on its own
i’ll put this back in here x and then y and then in red you have the isotopes
there and so that’s the cosecant graph and so
this would just repeat you got the upper branch the lower branch an upper branch

00:23
a lower branch and so on all right let’s do one more
and if you like this video i hope that you subscribe
let us know in the comments below if you’re enjoying this video
all right so this is a cosecant which is odd just like sine so the minus sign is
going to come out and so here’s the function that we’re going to be graphing
it’s going to be minus 1 6 cosecant of pi over 4x
and so to graph this one right here we’re going to look at the sketch of
y equals minus 1 6 sine of pi over 4x and so let’s see here the period is
going to be period is 2 pi divided by pi over 3 sorry pi over four
and so yeah that’s two pi times four over pi pi’s cancel and we’re really just

00:24
getting an eight right there all right so our period is eight
and so let’s go ahead and sketch the graph over here
let’s do it about right here and so we got an 8 and this is a negative so
it’s been reflected so i’m going to sketch the graph here
and that’s it eight right there halfway is a four halfways of two two four six
and the height here is at the uh 1 6 the amplitude is 1 6. now we don’t
say the amplitude of the cosecant is 1 6 but for the auxiliary function right
here the sine the amplitude is 1 6. and so now we can look at the isotopes here
where is sine zero right here at x equals zero and then x equals four

00:25
and then x equals eight and so you can see the pattern here for all the
isotopes 0 4 8 12 and so on and then and then the negative
0 minus 4 minus 8 and so on and now we can sketch the graph for the cosecant
here so we’ll put the upper branch right here
and we’ll put the lower branch right here and actually just now notice that
we can’t see the isotopes labeled down there so we’ll put them up here
x equals zero x equals four x equals eight and
you know that just keeps repeating there’s just one period for this
cosecant graph it starts with a lower branch and then
an upper and then a lower and then an upper and so yeah that’s it

00:26
so what about the secant graph now or the cosine sorry what about cosine
cosine is even so these will be even easier
so what how how do we sketch 2 cosine 3x uh this is something that we did in our
previous episode so the the period is going to be 2 pi over 3 and now
cosine right so cosine starts up here so i like to
all right and then i’ll just erase that that’s that’s part of the graph but it’s
not in the first it’s not in the first period here
so i’ll put these tick marks here this is 2 pi over 3 so halfway 2 pi over 3
so halfway is right here and this will be two pi over six or pi over three

00:27
halfway again is pi over six and so we have one pi over six two pi over six
three pi over six two pi over pi over two and so this just keeps repeating
now the amplitude right here is two the height is two right here and then minus
two right here and so there’s the sketch of this graph
right here uh on on one uh period all right and so yeah what happens if
there’s a minus sign in there so cosine is an even function
so this is 2 cosine 3x which we just graphed so we don’t need to sketch it again
all right so what happens if we have a minus pi in here so this right here is
the minus sign gets absorbed so this is minus 2 cosine pi x
all right so this is easier to think about how to sketch the graph of

00:28
the period is 2 pi over pi 2 pi over pi or in other words just two
and so we have the cosine graph again but it’s been reflected so now i’m going
to start down here and this is a minus two and the height here is a two
and where did it start to repeat so right about here and this is a two
halfways one and so this is a half and so this is uh three halves and so
that’s it um and now let’s do this one right here
let’s see did i forget anything right here i got the period i got the heights um
yeah okay so that’s it it just keeps repeating
okay so um let’s look at this one right here now

00:29
um the minus sign gets absorbed so this is going to be equal to 1 4 cosine of
pi over 3 x and so the period is 2 pi over pi over three
or two pi times three over pi fives cancel we get six and
so this is a positive number in front of here so we’re gonna start up here
so we’ll just start like that go down come back up
and this height right here is uh sorry this x is a six and the height here is
one-fourth the amplitude here is one-fourth so we get one-fourth right here
and this will be minus 1 4 down here this is a 6 so let’s chop it in half we
get a 3 and then a 3 over 2 so 1 2 3 so 9 over 2. and
i’ll just put an x and a y here and we’re good to go there

00:30
and then this just keeps repeating on and on and on
now maybe that doesn’t look like it’s the same height there
all right there we go all right so what about seek it now all right
so yeah if you um like this video please subscribe i would
really appreciate that help support the channel
and so now let’s look at uh 3 secant 2x so this one right here is going to be
equal to um i only put this down to help to help someone understand
but you know i wouldn’t necessarily do that so i need to
think about the cosine graph in order to sketch the graph of secant i like to
draw the graph of cosine 2x in fact i’m going to draw the graph of 3
cosine 2x to help me sketch the graph of this one right here
so i remember secant goes with cosecant because the reciprocal functions right

00:31
here all right and so the period here is well we’ve already did this graph right
here remember the period is 2 pi over 2 so just pi uh i think we did this graph
anyways um period is pi and it’s cosine and so it’s going to start up here
and that didn’t give me much room for my
secant so i’m going to put that in again and this time try to draw a straight
line so now i’m going to just try to draw it in here like this
it’s just going to keep repeating over and again the height here is a 3
and this is the minus 3 and this right here the height right here is going to be
at pi and then pi over two and then pi over four and then three pi over four

00:32
if you can see those tick marks actually you know
that doesn’t look that great let’s do it again uh i’ll try to
make it in the middle here so this will be pi halfway will be pi over two
pi over four three pi over four and then now let’s find the isotopes
where is the cosine graph zero so right here
and this would be x equals pi over four and where else is the cosine graph zero
so at three pi over four and so now we can draw the
secant graph right here i’ll draw this in orange so we have a lower branch
right here and we have a partial upper branch right here

00:33
and another partial upper branch right here and so the
graph in orange is the seek 3 secant 2x where the height here is a 3 and this
right here is a minus 3. here we go all right so this is um
pretty straightforward now i’m hoping getting the hang of this here
so this is secant it is the same thing as minus 3 over cosine of minus 5x
remember cosine is an even function so this will be cosine 5x
so what i really want uh so what i want to sketch the graph of first is
minus 3 cosine 5x so i want to sketch this right here to
help me sketch this one right here and this this graph right here is uh so the
m the minus here gets absorbed so minus three secant of five x

00:34
um and so in order to sketch this one i’m gonna sketch this one right here first
and so i’m gonna sketch this right here here we go so let’s put it right here
minus 3 cosine 5x and this is a minus sign so i’m going to
start down here actually come up and then go back down and then
the period here is 2 pi over 5 2 pi over five
so this will be two pi over five so halfway will be um
two pi over ten or in other words pi over five
and then the halfway again will be pi over ten so 1 2 3 pi over 10

00:35
and here the height here is at a 3 and this will be a minus 3 right here
so now we’re going to look where the um where is it zero so right here
at x equals my pi over 10 and then right here at x equals 3 pi over ten
and so now we’re ready to sketch the graph i’ll do this in orange
again we have a lower branch right here split here we
have a upper branch right here and we have a branch right here split
and so there’s the sketch right there of one period of the
secant graph right there all right so what is this right here looking like so
this minus sign right here gets absorbed so this would be minus secant

00:36
of 2 pi over 3 and then for the last one right here
this minus gets absorbed so this will be 1 6 secant and this will be pi over 4 x
so this one right here has period two pi over two pi over three
or so differently two pi times three over two pi or just three
and this one right here has period is two pi over pi over four
two pi times four over pi and the pi’s cancel we get eight right
so here um we’re going to be sketching the graph of y equals minus cosine

00:37
two pi over three and here we’re going to be sketching the graph of one sixth
cosine of pi over four x so we need to sketch these graphs first
and then we’ll be able to put the isotopes in and sketch the graphs of the
secant so to sketch these two graphs right here
we really need to sketch these two graphs right here
now if if you haven’t checked out that previous episode i just want to remind
you one more time there’s a whole episode where we go
through these graphs right here but let’s just do them one more time right
here so let’s let’s sketch this graph right here first so this one’s minus
cosine 2 pi over 3x so here we go i’ll sketch it over here
and this is a minus cosine so i’m going to start down here um
let me add more room here in fact i can move the whole thing up here let’s do

00:38
that and i’m gonna start down here come up back down and this right here is a
three three over two three over four so one two three or nine nine over four
and then 12 over four right there so the height here is a one the amplitude is
a one and a minus one right here and now for the isotopes
where where’s the graph zero for cosine graph zero right here x equals
uh three over four and x equals uh so that says three over four i think
i can squeeze it in three over four and nine over four and now we can sketch the

00:39
secant in orange right here and then down here
and maybe we’ll extend these isotopes up here all right and so now
this one right here 1 6 cosine pi over four the period is eight
so let’s sketch it right here and this one is positive right here so
it’s going to start up here and looks something like that
height here is uh as at eight and it’s one sixth so minus one sixth
and i’m gonna chop it in half and then half and then add them up two four six
eight all right and then where are they 0 here right here at x equals 2
and right here at x equals 6 and i’ll draw this one in orange

00:40
so it’s right here we have a lower branch right here
for this secant graph right here lower branch and then this is split right here
and split right here so this is one period here where we’re going up right here
coming down right here and then we have this lower branch right
here and then this just repeats if we wanted to do more we could
stretch it out there and then we would have another isotope and we’d have it
again and so on yeah we can draw as many of those as we want all right so
i hope that you enjoyed the video and um yeah so
next um we’re going to be looking at the um phase shift uh in in more detail
and so that video should start 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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