Transformations of Functions (and Their Graphs)

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

00:00
in this episode you’ll learn how to use the graph of a function of a given
function to sketch the graph of a transformation of the function
what does that even mean let’s do some math [Music] hi everyone welcome back
so um i’d like to uh remind you that uh this uh episode
is part of the series functions and their graphs
and in the previous episode we talked about these functions right here so we
talked about them in a couple of different episodes
so i just want to remind you that this as part of the series functions and
their graphs step-by-step tutorials for beginners
and so yeah in the previous episode we talked about each one of these parent
functions and then in another episode we talked
about applying transformations to these functions such as vertical shifts

00:01
horizontal shifts reflections and compressions and stretches
and we also saw how to do that on using a computer using python
programming language so check out those previous episodes link is below in the
description so in this episode we’re going to um not
concentrate on these parent functions um always so sometimes you want to just be
able to take an arbitrary graph and apply those type of transformations so
we’ll see how to do that so but for our first example we’re going to uh look at
um some graphs and we’re going to kind of reverse the question we’re going to
say here’s the sketch of the graph let’s find the equation
so let’s look at our first example right here so we’re going to have this
function right here and it’s going to be
coming through here like this right here it’s going to be coming like this and
then it’s going to come up here and bounce and then it’s going to come back

00:02
down and this point right here is 2 0 let’s make that a 0 to 0
and then let’s say this is the point right here
where it just keeps going down and this is the point right here zero minus four
and so yeah we have this uh you know another point over here
um in any case the parent function is y equals x squared
so what will be the equation for this graph right here
so the equation for this graph right here is well first off it’s been um you
know the parent function right here is say y equals x squared so first off we
notice is that it’s been reflected it’s upside down
and then we notice that it’s been shifted to the right by two
so um you know the parent graph right here the parent function y
equals x squared it’s been reflected and it’s been shifted to the right so the

00:03
equation that i’m going to write for this graph is going to be minus because
it’s been reflected through the x-axis and then we’re going to say
x minus 2 and then we’re going to say squared now it hasn’t been
shifted up or down it’s still hitting the it’s still on the x-axis right here
at this point and same with this point so it has
hasn’t really been shifted up or down but it’s been reflected and then shifted
over now you might say y minus 2 well if you use a 2
and you plug in a 2 then you’re going to need a minus 2 to get us back to 0. so
that’s why even though we’re shifting to the right we’re actually going to use a
minus two in here so this would be the equation right there that’s just scratch
work so there’s the equation of the graph for this graph right here
let’s do another one let’s call this example a so let’s call this one example b
um and so for this one right here it’s going to be just coming in here like this

00:04
and we’re going to have this point right here 0 minus four
and we’ll have this point here and then we’ll have the corresponding point over
here and this will be minus five six and this will be the point right here
five six okay so um we don’t know what the function is
it’s not the square function but it looks like it
but we’re just going to call it an f and then we’re going to ask what did we do
what transformation did we apply so we shifted it down so the
parent function the parent function we’ll call f
we don’t know that it’s x squared because when you input 5 and you square
it you’re not going to get out of 6 right so it’s not the normal y equals x
squared but it’s a little bit shaped like it it’s symmetric with respect to

00:05
the y axis um in fact let’s just call this a g so
we don’t we don’t know that it’s f y equal
f of x equals x squared right it’s just we’ll just call it g so what will this
uh right here be this will be g of x plus four
because we’re sorry minus four because we’re shifting it down four so whatever
g is the parent one this one right here will be g of x minus four because we
shifted it down four all right so now let’s look at one more
so let’s look at something like this right here and we’re gonna be
shaped like this right here and this is the point right here minus three zero

00:06
now let’s change these to 25s if these are 25s and now we know that this is the
uh quadratic function here then what will the equation for this right here be um
so let’s change them back to sixes and minus five five and six so
this is zero minus four so what would this graph right here look like
or let’s just say y equals and then we said we shifted it down so x
squared minus four so let’s see here what if we input a zero then we get
minus four what if we input a five then we’re going to get 25 minus 4
which will be 21. so let’s call this a 21 and a

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minus 21 here i want to be symmetric with respect to the y-axis
and so this will be the equation for this graph right here so the parent
we’re assuming is f of x equals x squared and so this right here this
equation right here would represent this graph right here this is 5 which we did
25 minus 4 so the output would be 21 there and so for this one right here
this is a minus 3 and the parent is y equals square root of x
and so we’re shifting it to the left so i’m going to say this is the graph of
let’s put it over here this is the graph of y equals square root of x plus 3
the reason why i use the plus is because this is a minus 3 over here so i need a
plus 3 to counter the minus 3. it’s been shifted minus 3 so minus 3 plus 3 gives
me an output of the 0 there so the parent would be square root of x
and by the shape of it and then this right here would be the

00:08
shift right there so this would be the equation for this right here
all right so i just wanted to kind of give some examples there um
where we’re still talking about these parent functions but actually that’s not
the main topic for this video right here um so at this point right here is yeah
okay good so let’s go with so i said parent parent and here i
didn’t say parent so the parent right here was f of x equals x squared
i probably said that somewhere along the way all right so there’s those um abc
examples there let’s go on and talk about um what happens if you’re given a
graph of something that’s not necessarily a one of the basic eight
parent graphs or parent functions that we’ve named so let’s look at something

00:09
like that so now we’re going to be just given a regular graph just given a graph
not not necessarily a parent function we’re going to apply some
transformations to that so let me make a good graph over here for us so we’re
going to take this right here and it’s going to go from this point to
this point right here and then it’s going to go to this point right here
and then it’s going to go to this point right here
and so i’m going to name these points right here and
so i’ll try to label them over here so this would be the point right here
zero minus one and this will be the point right here one zero
so this is a graph of a function f and it’s just a you know just
just a couple line segments glued together or whatever you want to think
about it but this is the point right here 3 1. so i’ll put a 3 1 here

00:10
and this is the point right here 4 2. so there’s the graph of our function f
we don’t know what f is it’s not the cubic it’s not a quadratic square root
it’s none of the basic eight functions that we call parent functions it’s just
a graph so now what i want to do is i want to do transformations to this graph
so the first transformation is i want to
sketch the graph of y equals f of x plus two what will this graph look like
so we know we recognize the plus two from the previous episodes as being a
vertical shift so i need to shift all of these points up up to
now shifting it up to means we’re increasing the y values so the minus the
minus one here we’re going to add a two we’re going to add it to we’re going to
add it to and we’re going to add it to but it’s going to keep its same shape
here so what would that look like here so i’m going to say now i have the point
um so we can try to sketch the graph first

00:11
and then put the axes or i could put the axes first let’s try to
um let’s look at this point right here so now what is it it’s zero and then say
minus one plus two so it’s going to be a one so it’s gonna
be zero 1 so it’s about right here so that’s the point right here 0 1
so this point right here becomes 1 2 so let’s say it’s about right here
so that’s the point one two and then now i have this point right
here three one so now it’s going to become three three
so let’s put it up about right here so three three
and then now we have four 2 which now will become 4 4.
so let’s put a 4 4 but right here and so this is the graph the
one two three these segments right here one two three

00:12
have all been uh transformed and now we have the graph of f of x plus two
y equals f of x plus two so there’s the graph of this one right here
um now let’s see what happens if we do the graph of y equals f of x minus 2.
so this is shifting to the right we recognize this as shifting to the right
and so let’s do that so now but remember you always have to
come back to the original right here this is f and it’s f that we’re shifting
so this point right here has been shifted to the right by two so now this
becomes two one so let’s say this is about a two one
right here or sorry two minus one and so that’s the first point right here two
minus one and now we’re going to shift this point
right here so that’ll be three zero now so let’s label this as three zero
and then we have the connection we’re going to connect them and now we have

00:13
the 0.31 and we’re going to shift right so that’s going to be now um five
five one so we’ll say five one’s about right
there or let’s go about a little bit like about right there um
actually probably more like this but 5 1 will be right here
and then the next one we’re shifting it to the right so it’s
the x that’s going to change it’s going to be 6 4. all right so
this graph right here will one two three those side
line segments get shifted right there so we’re going to shifting to the right
added two to this right here actually that should be six two right yeah six
two sorry there we go let’s do another one let’s do y equals 2 times f of x

00:14
so this is a vertical vertical stretch and so now let’s see here what we’re
going to get so now when i input a 0 the output is going to be -2
so this point is going to go to 0 to -2 which is going to be down here
so let’s see how i got that right there this point is i input 0. remember we’re
always going to go back to the original function because we’re transforming the
original function here so i need to multiply
two times the outputs in other words two times the whole f of x
so when i input a zero right here in other words the x’s the x is zero
then what’s the output going to be well f has output of minus 1 so it’ll be 2
times -1 so i’m going to get a minus 2. so that’s going to be my starting point
there now for my next one how is this point
going to be transformed 1 0. so let’s input a 1

00:15
so we’re right here at a 1 and then we’re going to output a 0 and what’s 2
times 0. so 2 times 0 is still 0 so i’m going to go right here to right here and
this is the point right here 1 0 still so i input 1 and get out zero
times two which is zero so i still get the point one zero right there
so this is kind of vertically stretching it so here we were we only went down to
minus one here we’re going down to -2 so we stretched it like there right now
what about 3 minus 1. so i’m going to input a 3 right here that’s my input
right here still 3 now what’s my output going to be well f
says the output is 1 but this new one’s going to take 2 times 1 so this is going
to be 3 2 so it’s going to be up higher so let’s say 3 2 is right here
there we go and then the last one right here is going to be at a 4.
and then what’s the output going to be so when we input a 4 the output’s a 2

00:16
and 2 times 2 is a 4 so we’re going to get 4 4. so i’m going to say output’s
uh 4 4. so let’s make it even steeper so here’s the point 4 4 right there
all right so that f right there has been transformed into 2 times f
so i’ll just put it right here two times f
got a little smudgy all right so let’s do uh one more let’s do um f of minus x
f of minus x what would this graph right here look like i’ll move up here
and let’s look at this graph right here so i think we can graph it over here
so what happens when we substitute in a minus x so
let’s start um and again we’re going to be doing the function transforming the

00:17
function not any of these intermediate ones we’re going to be graphing this one
right here so we’re going to be transforming this one right here
so i’m going to input the 0 into here and minus 0 is still 0 and what is f to
the due to the input of 0 it gives out minus 1. so this is still
going to be actually the same point right here so this will still be 0 minus
1 right here so let’s try that again make sure you get it i input 0 right
and that’ll be minus 0 which is still 0. and what is f of 0
well we go over here to find out what is
f of zero which is minus one right there so we get the same point now let’s try
one zero here so one zero so when i input here a 1
here that would be f of minus 1 but there is no f of minus 1. when i
look over here at minus 1 there’s no output

00:18
so i can’t really go point by point what i have to do is think about what does
the minus 1 do to the graph right here what i can do is i can input a minus one
though right can’t we input a minus one into here because what is a minus minus
one it’s a positive one and i know what f does to a positive one it gives me
zero so you see this right here is going to be the output
if i input -1 i’m going to get output 0.
so it’s going to go like that so this is the point right here minus 1 0.
so let’s check that out again i cannot input 1 because that will be f of minus
one but there is no f of minus one so what i have to input instead of the one
is i have to input a minus one and when i input a minus one then that’s f of
minus minus one which is f of one and we know the output for f of one it’s

00:19
zero so if i input minus one i get out zero so now what will be the
corresponding point right here three one
again we cannot input a three because we don’t know f of minus three there is no
f of minus three but what i can input is
a minus three in here so i can say minus three comes in and then what is f of
minus minus three that’s f of 3 so i’ll get output of 1 here so this
will be the point right here minus 3 and then 1.
and then last piece to check what happens to this part right here so f of 2
i cannot use in 4 in here sorry 4 2. i cannot use a 4 here but we can input a
minus 4 because that’ll be minus minus 4 that’ll be f of 4 which is 2. so i’m

00:20
going to get the point here minus 4 2. and so that’s how we can um
sketch the graph of this this one right here it’s just reflected about the
y-axis isn’t it so it’s the same graph it’s just been reflected about the
y-axis so i just gave a rough sketch right here rough sketch right there
and this is just reflecting about the y-axis but if you didn’t know that i
tried to sketch it without you knowing that but definitely if you checked out
the episode on even and odd functions then you would have recognized that
right away all right so let’s do a another function right here
let’s graph another function right here and let’s do one more example
all right here we go so our function now looks like this

00:21
and so i’m going to have a point here and a point here
and it’s going to come down and hit the y axis right there
and then it’s going to go up and hit it right here and so this will be the point
6 2 and this will be the point 0 minus 2 and this will be the point here
minus 2 2 and this right here will be the point -4 2.
and so this is a graph of some function f there we go so there’s some function
minus 4 2 minus 2 2 0 minus 2 and 6 2. and we’re going to apply some
transformations again this is not one of the parent functions but you don’t have
to apply transformations to just the eight functions the parent functions
that we have memorized you can apply transformations to any function
so let’s call that function f actually let’s call it g just so in the comments
below you can distinguish between the two examples so let’s call this function g

00:22
right here so for the first um example let’s say we have or we want to sketch
the graph of y equals let’s say let’s just go ahead and do the minus x here
like we did in the last example right there what happens if we do g of minus x
so we know that’s symmetry with respect to the y axis so let’s just sketch what
that would look like real quick so this point here 6 2
now reflected through the y-axis becomes -6 2. so i’ll put that point here
and then we’re going to go all the way down here now reflecting through the
y-axis this point is going to stay the same so this point is going to be right
down here so i’m going to have this branch right here
maybe i scoot it down too far let’s make it about right there
so this will be here minus 6 2 and this will be the point here 0 minus

1. i’ll put it over here 0 minus 2 right there

00:23
y-axis so now it becomes over here so now it becomes 2 [Music]
uh sorry that can’t be right that’s minus two minus two my bad there
minus two minus two so now this gets reflected and it becomes two minus two
so here we go there’s a two minus two and now this part right here gets
reflected so it becomes positive four two so let’s say it’s about right here
so four two um actually when i um reflected this one six two should have
been minus six and but the two stays the same
there should be a two there and they should have about the same height so let
me adjust that one right there and so this is 4 2.
okay so that point’s been reflected that

00:24
point’s been reflected that point’s been reflected and this point stays the same
all right very good so there’s the graph of g of minus x
given the graph of g here’s g of minus x it’s a reflection through the y axis so
now let’s look at something like what if we want to shift it up four units
so what would that look like now i’m shifting g up so i want to come back to
g and make sure i pay attention to g here not this one right here right so
here we go we’re going to shift up four and so this one’s going to get shifted
up four so it’s going to be minus four six let’s just plot it right here so -4 6
and this one’s going to get shifted up also so it’s going to be 6
and then up by 4 so it’ll be 6 6 and this one will get shifted up four so
it’ll be up here now right so minus two and then plus six so
this will be at about a four now so we’ll say this is zero four

00:25
and we’re going to connect it and then we’re going to go across here and get
that point there and what would this point right here be it’ll be minus 2 and
then we shift it up 6 so it’ll be minus 2 4.
so minus 2 4. so there we go there’s the graph of g of x plus 4 right here
all right so very good so now let’s look at something like g of x
let’s scoot over here maybe we can get in three more so let’s try g of x is
let’s go with minus g of minus x minus four something like that so we’ll do a
reflection um actually that should say y not g
again so so how about this one right here so y equals g of x minus four
so let’s see what this looks like in fact let me scoot this up

00:26
all right so we’re going to be looking at this graph here g
and we’re going to be shifting it right by 4 and we’re going to be reflecting it
and so let’s see what we’re going to get here so when i shift this one right
um the x is going to change and it’s going to go to 10 2 so that’ll be 10 2
however it’s going to have a negative so that means it’s going to be reflected
through the x-axis so now i’m going to get the point here negative 10 2
so this point right here 6 2 has transformed into the point negative 10 2
and the way i did that is by um you know plugging in the um or you know adding 4
to the x and so that’s going to give me a 10
and then reflecting it and i get minus 10 there
zero minus two so we’re shifting it to the right so

00:27
that’s going to come out to be a four now and then we’re reflecting it
so that um sorry this is going to be put the minus in the wrong place my bad
let me let me just walk through that again and make sure i’m getting it right
here all right so we’re shifting that 6 2 what are we doing we’re shifting it
right so we’re going to get this 6 is going to become a 10.
now we’re reflecting it through the x-axis so the y changes sign so this
will be 10 minus 2 and that makes sense as to where it’s at
going to become a 4 and that minus 2 right here is going to get reflected and
so that’s going to become a positive 2. so i got this one right here now not not
right there it’s been shifted right and so it’s going to be 4
so we’re shifting this we’re adding it adding 4 to this and then we’re

00:28
reflecting through the x-axis so that’s gonna become a positive two now
so this part right here has been reflected now it’s gonna look like this
right there we go four two all right so there we go that part right
shifted to the right four units so take that point and shift it to the right now
it’s at minus two right now so add four to that and we get a positive two
so that’s about right here and then we’re going to take the minus 2
and reflect it so it becomes a positive 2.
so we get this part right here and this point right here is the point here 2 2
and again i get this point by adding 4 to this
and then changing the sign on this to reflect it through the x-axis so we get
2 2. all right and so now the last one to transform is -4 2
now we’re going to shift this to the right and it’s already four units away

00:29
so we’re going to shift it right and it’s going to be right on the x-ax
y-axis there now the 2 is going to get reflected so it’s become going to become
a minus 2. so instead of being here it’s going to be down here
and so this will be the point right here 0 minus 2
and then we’ll connect it right there so down over up
and then it gets reflected right there so this part right here
right here now goes up right because it’s been reflected and then we stay
constant and then we’re going to come down right there
so this is the graph with the reflection and a horizontal shift right here
all right very good so let’s just do one more let’s do something like
y equals say g of um let’s do something like um what if we do a um

00:30
how about g of 2x something like that what if we make the change inside the uh
right there in right next to the x so that’s different than uh
i didn’t do any scalings on this one right here so maybe we can do two more
maybe we can see the difference yeah that would be fun so let’s see the
difference between these two right here if if i put a 2 here because this one
right here is usually hard for most students starting out so let’s see
if we can do this right here and then we’ll try to compare it with if the 2 is
out here so this will be a vertical stretch and so let’s see if we can put a
2 in here and let’s see what happens here so we’re going to take each of these
points right here and transform it so let’s see what we can do here um
we’ll try to start off with um this one right here

00:31
um so right now we have -4 is the input and 2 is the output
so if -4 is the input for g we know what the output is so
how can i get this whole thing right in here to be minus four
that’s when x says what minus two so this i’m going to start here with minus
2 for the x instead of the minus 4. so my input’s going to be minus 2
and then i get a minus 4 out of all that and then i go with the minus 4 into g
and then g tells me i have a2 so this point right here will get
transformed into minus 2 2 so that point right there is transformed
into minus 2 2. all right and so now let’s see what
happens to this point right here now when the input is minus 2 i know what i
know what the output is so how can i get an input of minus 2
so i need 2x to be equal to minus 2 in other words x is you know just divide

00:32
by two x is minus one so at minus one i can input into this function
and i get out minus two right here and then i go into g
and what happens at minus two as i get out of minus two so that is going to be
the minus one and a minus two so this point right here transforms into
uh minus one let’s put it about right there you know it’s a little bit closer
a little bit closer there so let’s call this -1 minus 2 right here and then
let’s connect those and then now what is this point right here
going to transform to so again i know what g does when i input 0. so
what can i get what can x be so i input 0 into g
so in other words what is the x going to be so that 2x is 0. in other words x is

00:33
zero so if i input zero into here two times zero zero so that’s zero so take
zero into g and what does g do to zero it gives me
minus two so i’m actually at the same exact point right here
so this is the point right here zero minus two
all right next point right here the last one here is six two
so i know what g does to input of six it gives me two
so i need this 2x to be a 6. so 2x equals 6 so x equals 3.
so this point right here is going to be 3 something
so let’s find out what it is if i input a 3 i get a six so what’s g of six
g of six is two and so this height here is two
and i should probably make them look more
the same height here so here we go three two
there we go this is coming up right here all right um

00:34
in fact i probably should have made it more level right here
in any case so there’s that one right there there’s g of 2x there’s the
transformation now let’s look at the difference between
that and and 2 times g of x so i’ll put it up here so let’s say here
y equals 2 times g of x what would this look like
all right so what is this going to look like let’s try to sketch that graph here
so now because there’s no sorry now because there’s no change in
the x it’s not shifted left or right it’s not scaled it’s not anything
i can use the same input values here so it’s going to be a lot easier than this
one right here in other words if i input the minus 4 i can do that right into
here we could input -4 into here we had we inputted minus two so anyway
minus two this is just going to say minus two sorry minus four

00:35
and then what’s g of minus four is two what’s two times two it’s four
so we’re going to come up with this point here minus 4 4.
so let’s make sure we get that if i input minus 4 into here
what’s g of minus four g of minus four is two
so if i input minus four i get two and what’s two times two is four
so this point right here transforms into this one
let’s say minus 2 is about right here so minus 2 in here
and what do we get out we get out of -2 and so what’s 2 times minus 2
is a minus 4 so this minus two has gone down to a minus four here
so this will be the point here minus two minus four so you can see
this transformation of two stretching it by two vertically

00:36
is shifted this up and stretched that point up and it stretched this point
right here down and so that’s just so you can see it minus two minus four
and then we’re going to have this point right here the same if we input zero
then g of zero here is minus two and what’s two times minus
two so minus four again so this will be the point zero minus four so it’s going
straight across here and it’s coming down here and this is a steeper incline
here than it is over here because it’s being stretched vertically we got to
take this part right here and stretch it out and we stretched it out right there
and this part right here is level we keep it level
six two let’s see what the point is i’m going to take these input of six
here we go input six so what’s g of 6 g of 6 is 2 and what’s 2 times 2 is 4 so

00:37
this is going to be the point here 6 4 and there is the sketch of the graph
right there just like that all right so uh there we have it there’s um
a bunch of examples on how to take a graph that is not necessarily one of the
parent functions but you can still apply the transformations to that well i hope
you enjoyed this episode and i look forward to seeing you next time bye bye
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