Using the Distance Formula (The EASY Way)

Video Series: Functions and Their Graphs (Step-by-Step Tutorials for Precalculus)

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

00:00
in this episode you’ll learn about vertical and horizontal shifts
reflecting about an axis and scalings this knowledge greatly enhances your
ability to understand the graphs of many commonly used functions
let’s do some math [Music] all right everyone um welcome back
everyone and so in this video here we’re going to work on
applying transformations to the functions that we talked about in our
previous episode so we talked about the constant function identity function
quadratic function cubic function and for each of these functions we
stated the domain and range and we sketched the graph
and we even looked on how to do these uh on using a computer so i recommend
checking out that series where we covered each one of these videos here so

00:01
the series is called functions and their graphs step-by-step tutorials for
beginners the link is below in the description
so in this episode we’re going to work on applying transformations to these
functions right here and so by applying transformations to these right here
we’re going to be able to sketch many more graphs just by knowing these basic
graphs right here all right so let’s get started let’s first talk about vertical
shifts so of course we’re going to be talking
about functions in this video here this episode is we’re going to have a
function and if we add a constant to it then it’s going to be called a vertical
translation now we have two different types we can
have a positive constant so f of x plus k and that’s going to be shifting the
graph upwards and then if k is um and then if we have minus k then then the
graph is going to be shifted downwards so uh let’s look at a couple of examples
so in our first example right here we’re going to look at what is f of x

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so here we have the identity function right here which is just this function
right here y equals x and now what if we have the function
right here x plus a constant so let’s make the constant positive let’s say
plus three and so what that’s going to do it’s going to take the same graph
right here but it’s going to shift it up three
so i’m gonna have one two three so now it’s going to be the exact same shape
oops but it’s just going to be shifted up three i really missed that
one two three and then there we go so it’s the same exact shape it’s just
been shifted up three units so that’s a three right there
and so let’s look at another one let’s say let’s look at x squared
and then now let’s look at x squared and now let’s do a minus two
so we can see an example of moving down two units right here so this will be
this part two right here so now the constant uh is two

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but we have minus two so we’re gonna be shifted down so again we need to know
what this graph looks like first looks nice right there going through
there like that and so this one right here has been
shifted down to so one two and it’s going to have the exact same shape excuse me
so this is the uh just a quick sketch of this function right here
um you know you got some zeros right here
but it’s basically the same shape right here it’s just been shifted down two
units so that’ll be minus two right there okay so
um and let’s do uh another one let’s do a cubic and this time let’s do a cubic
and let’s go up four and so let’s see what this would look like
so the idea is to sketch this just by applying some transformations so

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we know what this looks like we graphed this in our previous episode it looks
something like this it’s symmetric with respect to the origin
so how would it make how would we make a
quick sketch of this right here x to the third plus four so it’s been shifted up
four units so i’m going to say one two three four but now it’s going to keep
the same shape so it’s going to be coming through here like this and then
it’s going to have the same shape right there so it’s basically the same shape
it’s just been shifted up four units okay so um let’s see here how about the
absolute value let’s do the absolute value so here’s absolute value of x
and here’s the absolute value of x and then let’s go let’s say minus one right
there right all right so this shape right here we have it memorized we know
exactly what it looks like we know what the domain and range are and we know how
to plot points a couple points on this so we know how to get the shape of it
this is the part y equals x this is the part y equals minus x right here so what

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happens if i shift it down one unit so instead of going through zero zero
now it’s going to go through this one right here but it’s going to have the
same shape so it’s still going to be a v right there just like that
so this will be the point here zero minus one
and this you know this was the zero zero right here
and so we know we can plot something like this right here this will be one
and minus one right here in any case there’s a quick sketch of
this graph right here just by knowing what this one looks like right here we
can sketch this right here pretty quickly just by using shifts um let’s do
one more let’s do the reciprocal function just for
fun so let’s do the reciprocal function right here and let’s give it a
vertical translation or a vertical shift so let’s see we have
1 over x and then let’s go with -3 so this one right here we have the
isotopes we talked about this one right here a lot in the last episode so it’s

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going to shape like this right here and shape this right here
and now i’m going to shift it down three units so i’m going to take the same
graph right here and i’m going to shift it down three now notice when i’m
graphing this i’m using these isotopes here to help shape
and help provide structure to the graph so now i’m going to shift down
so so this is still the same vertical isotope
but this horizontal lines and shape has been shifted down now
so this is the horizontal isotope is y equals 0 and here it’s y equals minus
three so that’s the horizontal uh isotope and so now i’m going to keep the
same shape i’m gonna keep that same shape right there so it’s gonna it’s
gonna hug right here and it’s gonna hug right here so i’m gonna keep the same
shape right there and i’m going to keep the same shape right here
and just to make it clear i’m going to extend that down there right there there
we go so we’re going to hug we’re going to be really close we’re going to be

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really close so i’m going to do the same thing right here and here
all right so there’s that graph right there
it’s this graph right here but it’s been shifted down three units right there
all right very good so now let’s look at um the next one right here what are the
horizontal shifts so let’s look at this real quick
so the horizontal shifts are going to be shifting left and right
and we’re going to be shifting the opposite in sign so now the change
or the shift or the transformation is right next to the x as compared on the
vertical uh it was the x is still inside the function right here and now the
transformation came after applying the functions right there so this is the
difference between the two this one right here is right next to the x the
transformation and so this these are also going to be shifts but instead of
shifting up and down now we’re going to be shifting left and right
and so here we go if our shift if if our constant here is positive then this

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right here is going to be shifted to the left so if we have a plus h we’re going
to be shifting to the left and if we have a minus h we’re going to be
shifting to the right so this is a summary of how horizontal shifts look
but you know what let’s see some examples here so
before we start mixing them together let’s just see what horizontal shifts
look like so let’s look at a function like absolute value of x minus 2
what does that look like all right well so first let’s see what is absolute
value of x look like and what does this one right here look like x minus two so
we know what this one looks like right here straight through there nice v
right so this is just practice right here but how does this right here look
well it’s minus two so i’m going to be shifting to the right two units so let’s
say one two and now i’m going to keep the exact same shape so let me shape
just like that right there nice nice v going right through there so
here when i substitute in zero i get out zero

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here when i is when i substitute in two i get out zero
so that’s why it’s two minus two and so here would be just a quick sketch
of that graph right there what about if we have something like absolute value of
x plus four right so that would be shifting it up
for so let’s say one two three four and now i’m going to have the same shape
right here nice v shape right here and so that would be the tick mark for four
all right and so now let’s graph y equals x squared
um but let’s say we’re going to do plus three squared
so how would this graph right here look like
so first i would sketch the y equals x squared graph
let’s sketch that one right here let’s say so y equals x squared
so y equals x squared is the parent function for this
so i’m going to have a nice shape right here
and now what is this going to do it’s going to shift it left so if it’s plus h
then it’s going to be shifted left units so it’s going to be shifted to the left

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so 1 2 3 and then i’m going to keep the same shape so this was zero zero here
and i’m going to keep the same shape here and now i’m going to keep the same
shape right through here and then there’s the nice parabola right there and
there’s the x y axis right there all right very good
so it’s basically this graph right here y equals x squared but it’s just been
shifted you know to the left three units so this will be minus three here
all right very good so let’s look at another some more horizontal shifts
um let’s um get some space cleared up here here we go perfect
so now let’s look at something like how about a cube root of um let’s go with x
minus one what is the cube root of x minus one what’s a sketch of this graph
so first i’m going to identify the parent function and

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oops and i’m going to identify my graph for this that i already have memorized
from the previous episode so let’s remember this graph right here comes in
right here really steep and then it comes back out like that all right so
there’s a graph of this one right here goes to the origin here 0 0.
all right so there’s what we know already so how do we sketch this one
right here now so i have to notice the difference between this one
and this one right here i’m going to sketch this one right here separately
cube root of x and then minus 1 out here so what’s the difference between these
two graphs so here the transformation is right next
to the x so that means it’s a horizontal shift
and i’m looking like this one right here
so i’m going to be shifting to the right by one unit so i’m going to shift right
so here we go we’re going to have the same graph right here but i’m going to
shift it to the right so instead of going through 0 0 now i’m going to go
through one zero right here and then i’m going to get in here
really steep right here like that like really steep and then i’m going to come

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out really steep like that right there and so let’s just make it go like that
right there all right so there there’s the graph
right here and this is the point right here uh 1 0 there we go
now let’s compare that with this one right here so this one right here would
look like same shape as this one right here this
is the same parent function for this one but this one this graph right here has
been shifted down one so that’ll be like
right here so that would be like a minus one right here and then it’s going to
come in here really steep like that and then come back out on the other side
like that so they have the same shape all of them have the same shape
but this one’s been shifted to the right one unit and this one’s been shifted
down one unit and then this one is the parent function
here so i hope these three examples here help understand the difference between
horizontal and vertical shifts so now let’s see some of them combined together
so let’s see what are vertical and horizontal reflections actually before we do

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reflections let’s do one more combining them together why not let’s do one more
so how about let’s do y equals absolute value of x plus five
and then let’s go down two units right there so let’s graph this this says has
some vertical and horizontal shifts in it this one’s the horizontal shift
because the change is right next to the x and this one’s a vertical shift
because it comes after applying the parent function so this looks like
f of x plus five minus two where the f of x is the absolute value
this is the parent function right here sorry about that this is the parent
function is is i’ll put it down here so the parent
function is the absolute value of x and remember what the parent function looks
like it’s just a v coming right in through there like that so that’s the
parent function so the parent function is the absolute value but i’m doing a
plus 5 to the input and then i’m doing a minus 2 after the parent function

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so that’s we got here so here we go let’s apply this two transformations
here now now the question is which one do i apply first well actually for
shifts it doesn’t really matter if you move it up and then down or down and
then up oh sorry i said i said i’m wrong it
doesn’t matter if you move left or right
and then up or down or if you move up or down or left or right either way you’re
going to get the same graph however the change here
you really want to go by your order of operations to do them in a more
readable more um coherent fashion so whenever i plug in an x what do i do
first i add five so that’s a horizontal so i would do
horizontal shifts before the vertical shifts all right so here we go um
so we’re going to shift left by -5 so i’m going to say here’s a -5
and so i’m shifting that over here and then i’m going to shift it down by
two so i’ll put the minus 5 up here and then i’m going to say this is the

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minus 2 and i’m going to shift it right here so there’s our new point right
there so this one right here was 0 0 right this is the parent
and so i’m going to be shifting this point minus 5 and then down minus 2.
and so this will be the new let’s we can call this the vertex i guess
of the absolute value and so now this will be the new vertex so i’m going to
come in here like this and then go back up like that so this exact same shape
is this this point right here i like to keep track of it and moved five and then
it moved down so that would be the new point right
there minus five minus two and i would have the exact same shape right there
so there’s how to do multiple of them i shifted and then i moved down now it
wouldn’t have mattered if i moved down and then shifted but i usually like to
do horizontal shifts first but if actually that’s your only two
transformations are vertical horizontal shifts you can just do them at the same

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time right you can just move them and move them all in one step right there so
this is really nice and fun to do all right so let’s look at some more
type of transformations let’s look at some reflections now
so we’re going to be looking at reflections
so we’re going to have a graph and we’re going to
put a minus sign on the graph and what that’s going to do is that’s going to
reflect across the x-axis if you put a minus sign
outside of the function it’s going to reflect across the x-axis if on the
other hand if you put a minus sign on the inside of the function then it’s
going to be reflected across the y-axis so let’s see some
examples of this right here so let’s take for example y equals x squared we
know what it looks like that right there and so what does this graph right here
look like right so all the outputs all the y you know the y outputs right
here have been changed to negatives so this is going to basically turn it

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upside down so now it’s going to look like this right here
so it’s just basically upside down so this was zero zero
and this one is still zero zero and so like if this is the point right here
uh it’s say two and then you get out of four right so now this is the point here
still two but now this the output has been changed to say a minus four
right there all right so that’s a negative if you do this on the outside
now what happens if you do a negative on the inside so now
i’m going to put a negative on the inside but you know what the square
takes care of that because it’s an even function right here so
this graph right here if i was to sketch it right here and not even do that what
would happen is it would be reflected across the
y-axis so all this branch right here is going to get moved over here and this
branch right here is going to get moved over here

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and basically it’s the same graph isn’t it it is the same graph
so if your function is even then this isn’t going to have any effect on
it at all because we know for even graphs minus x is equal to f of x now
this is what the definition of an even function is now we did a whole episode
on even and odd functions so if you’re confused about what that means
either you can just say okay i get it or you can actually go back and make sure
you understand what even and odd functions are all right so
let’s do a some more examples so let’s do um how about the reciprocal let’s
look back at the reciprocal again so let’s look at this function right
here y equals 1 over x and let’s do some reflections so let’s put it f of x
and let’s sketch that graph again it’s vertical isotope vertical isotope and
then it comes in here really nicely like that and then really nicely like that

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right there all right so there’s the function right there so now what happens
if if we do a reflection so what will the graph of this right here
look like minus 1 over x and so [Music] now
the we’re going to reflect that across the x-axis right here so this part right
here will now all right so we still have the same isotopes
but now this another graph will look like this right here
so this part will get reflected about the x-axis and this part right here will
get reflected about the x-axis so this this part right here will get reflected
up here so it’ll be up here now all right
and then what would the graph of f of x one over minus x look like
well that’s the same thing isn’t it yeah so they’re both the same so

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if i reflect this about the x-axis i get down here but if i reflect this about
the y-axis then i get this one right here right so the end result is going to
be the same the same graph of these two right here that’s kind of interesting
kind of fun there it’s symmetric about the origin right
um to follow up on that why not check out the cubic ones right is the same
thing is going to happen isn’t it so let’s recall what this graph right
here is it’s also symmetric about the origin so x to the third
and what happens if you do a minus outside of the whole function
right there right so now we’re reflecting about the x-axis
so this part right here is gonna fall down so it’s gonna look something like
this right here that part right there has been reflected
up here so it’s just gonna go right through there
and now what with the graph of f of x equal to minus x now put the minus right
next to the x so what would the graph of
this look like well it’s the third power

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isn’t it so it’s minus x to the third so it’s going to look just exactly like
this one right here and so so the exact same thing happened as the reciprocal um
so let’s see here we have look at the yeah all right so there’s good for
reflections so what’s coming up next is the stretches and compressions so let’s
look at some of these now stretches and compressions
so now we’re going to be uh having a factor in front of the um f of x
so for example what if you want to graph 2x squared or what if you want to graph
one fifth x squared so these would be some examples of some stretches and some
compressions so what we do if there’s a multiple
a constant in front so we’re going to get something like this

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if the constant is greater than one then the graph is stretched vertically
and if it’s between zero and one then it’s going to be compressed vertically
so stretched vertically means like if you’re standing on
top of the axis right here and you’re pulling it vertically so
if i were to graph these two on the same axis right here
2x squared and x squared how would i show that this one is vertically stretched
so this one is the basic one y equals x squared
and it goes through the point here 2 4 what does this one right here do though
right so this one is just y equals x squared right there
now what does this one right here do let’s put this one right here um in red
what is uh y equals 2x squared to do so when i input a 2 into here i’m going to
get a 4 but then i’m going to multiply it by 2. and multiplying it by 2 is

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going to make it grow so it’s going to grow faster
because at 2 it’s already going to be up
here higher than that 4 it’s going to be 8 in fact
so this one right here is going to grow faster so that’s going to like stretch
it vertically it’s going to get it stretched vertically so this is the
graph of y equals 2x squared now what about one fifth x squared so
what will that do so again now when i’m doing these
stretches and compressions i like to sketch both the parent and the
transformation on the same set of axes so that we can see
what’s happening on the graph so you can actually see the stretch and see the
compression so i’m going to graph both of these now so the parent is x squared
and i’m going to see that this one’s a compression right it’s between 0 and 1
here so let’s see here what this will look like so first i’m going to graph
the parent just the regular old standard you know parabola right there going

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through 0 0 right there now what will one-fifth look like so i usually like to
keep a point in mind when i’m stretching and compressing so i can understand it
in case you don’t you know really understand these words or you know you
just forget them whatever but what happens when you have a 2 well you’re
going to get a 4. you know that’s y equals x squared but
what if you put a one-fifth there so now is it going to go is it going to be
stretched inside no now it’s going to be stretched outside because if i
input a 2 now into this one now i’m not going to get a four i’m going to get a
four-fifths so it’s going to be the height is going to be smaller so
that would be four-fifths right there and so it’s going to be wider think
about it as being wider it’s gotten fatter or it’s stretched horizontally
you’re stretching it out this way so that’s why we say compressed
vertically or if you want some people say stretched horizontally

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and there’s the intuitive idea behind that so this one right here is y equals
one-fifth x squared and we can plot a point on it two and then the height here
would be four-fifths so then we can see clearly the the
distinction between the two all right so there’s some
uh there’s a stretch and there’s a compression
and the parent function was x squared let’s do that again but this time let’s
choose a different parent function so let’s see here what can we go with
so let’s do square root we haven’t done square root yet let’s do square root
square root was one of the parent functions so let’s see if we want to graph
a 2 times the square root and then let’s also graph a
let’s say here let’s do a negative let’s
just throw in a reciprocal also i mean a negative sign so let’s throw in here
minus 1 3 and then square root of x so let’s see what these look like right
here without plotting any points or i mean without making a table of

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values or you know using a computer just
we can just sketch this by hand so first thing i’m going to do is i’m going to
graph the parent function because this is going to be a stretch
this is going to be a compression and a reflection
so let’s graph this one right here first we got the square root of x it’s coming
right through here and again we studied that function in a
whole previous episode all right so now i’m going to factor it
by 2. what is the 2 going to do so for example when i input a four
i’m going to get out of two and now what is the stretched function going to do
when i input a four now i’m going to get a height of four
out also this is going to be higher but it’s still going to go through 0 0 isn’t
it when you input 0 you’re going to get square root of 0 which is 0 2 times 0 is
still going to get 0 here so this graph is going to come in here like that
so it’s going to be vertically stretched like that
so this will be two square roots of x and this one right here is just square

00:27
root of x all right there we go and then how about minus one third x so
now we can do this in steps since this is our first time combining
reflections and stretches so i’m going to graph the parent function
right here first square root of x and then now i’m going to do the 1 3.
now to help me do the one-third i’m going to again plot a point so let’s say
here if it’s 4 and then the height is what 2 right square root of 4 is 2.
and then now what if i do the one-third so the one-third if i plug in
um the one-third here i want to get out don’t worry about the minus yet but i’m
gonna get out two-thirds right not two i’m gonna get a two-thirds it’s lower
it’s about right here let’s say and so this graph right here will be
y equals one-third square root of x and this is the point right here

00:28
which is which will be two-thirds all right and now what does the negative
do remember what the negative does so the negative is going to reflect
through the x-axis so it’s going to look
like this right here so it’s going to be the exact same copy of it but it’s just
going to be reflected so this will be minus one-third square root of x
and this point right here which was two-thirds it’s now negative two-thirds
so we have a nice sketch of that graph and we have a point on it it still goes
through 0 0 right there so there we go this is 1 3 square root of x
and minus 1 3 square root of x and the graph of this one here is in blue and so
that’s it yeah that looks nice don’t need to plot like tables of values don’t
need to use a graphing calculator no technology needed in fact if you do
enough of these you should be able to start visualizing them on your own
question is how many do you need to do before you get to that

00:29
all right so let’s do um let’s do just a combination of them
um yeah so let’s let’s do some more examples here so let’s go here um
and let’s do this function here let’s do a quadratic let’s do minus and
then we’ll have x plus three squared and then we’ll do a minus two all right so
the goal is to sketch this graph by applying
transformations and not plotting points so i want to identify the
transformations let’s write them down and then we’ll plot the function over here
so i’m going to say horizontal transformation or horizontal shift let’s
call it horizontal shift left by 3 right this is positive so i’m going to

00:30
shift it left by 3 and i’m going to identify my parent function
so parent function now some people use toolbox function or
different words for that but i’m going to say parent function is f of x equals
x squared and now after we do the square what’s
next i’m thinking about the order of operations after we do that now i’m
going to uh do the minus to that so i’m going to do reflection through x-axis
and then now i’m going to shift down so this will be a vertical shift
down by two units some people like uh to use the word units here so let’s
go ahead and say units there horizontally shift by three units
vertical shift down by two units so these are the transformations and this
is the parent function here so we have three transformations and we have a
parent function so i’m going to take this in steps here

00:31
for my first time doing this um and then this will be like
y equals x squared right here then i’m gonna do the horizontal shift
left by three units so shift left by three units so this
point right here becomes what if we’re shifting it left it’ll be minus
three right so we can do this on a different set of
axes or we can do all this on one if we have color we can you know maybe use the
same one here and then this will come up right here so that’s just been shifted
left there so it’s minus three and now we’re going to reflect through
the x-axis perhaps let’s do that one in orange um and so then this it’ll be look
something like that so this will be the sketch of the minus x plus 3
squared we have to we haven’t shifted it down yet and the one in red is the

00:32
let’s say here y equals x plus 3 squared we have the red one isn’t
reflected yet and then the orange one is and then the original one we started
with was y equals x squared and then now we’ll go with blue and we’ll shift it
down we’ll shift it down two units and to do
that i’m gonna probably extend this out and then let’s just say this is a minus
two right here so now i’m gonna shift this down to
right there and i’m gonna keep the same shape right here so
the distance here and the distance here should be about the same so let me like
readjust that all right so that’s that’s the point
right here minus three minus two minus three minus two and
yeah so there we go we didn’t have really a scaling or a compression on it
right there so there’s our first example right there where we did a horizontal

00:33
and vertical and a reflection so this would be the final graph right here for
the function right here so we’ll just say this is minus x plus 3 squared and
then a minus 2 right there all right so there’s our first example
there uh sometimes you like to draw it on different especially if you don’t
have color right there if you’re just using a pencil or you know a pen whatever
it might look cleaner on separate graphs so i’ll do the next one like that just
in black and white but i think it’s fun to to do them in color too
all right so let’s do another one um the next one here let’s do a
let’s do a cubic so let’s call this one here function g and
let’s do minus one third and then we’re gonna do x plus two cubed
and then minus one so the goal again is to sketch the graph of this function
right here without looking at any you know graphing calculator or any

00:34
technology or making a list of table of values and trying to sketch it let’s
just do transformations alright so let’s apply the transformations so sometimes
people ask you hey just what are the transformations don’t even sketch it
just list all the transformations right so could you list all the
transformations so actually the first thing i’m going to do is write down the
parent function so the parent function is f of x equals x to the third
and so now what are the um transformations we’re applying so we
have a so i like to keep track of it by plugging in the x and then what are the
order of operations because you already know the order of operations right so
you should know the order in which you do your transformation so when i plug in
an x first thing i’m going to do is add a 2 to it right
so that’s a horizontal shift left by two units and
so then i’m going to do the parent function

00:35
and then after i do that i’m going to multiply by minus one third now
multiplying by minus one-third right because there’s lots of ways you could
write minus one-third you could write it as minus one over
three or you could write it as one over minus three or you could write it as
well there’s other ways but you know we need to know
this number right here is positive and then we have a minus sign so that’s a
reflection uh yeah okay so here’s what i was also
thinking is that we can write it as one-third times minus one
so the point is is that you could do the reflection first and then this and then
the uh vertical compression or you can do the vertical compression first
and then the reflection so it doesn’t really matter
which order you do these because these are the same thing here
right so vertical compression and then reflection or reflection and then

00:36
vertical compression so i’m going to actually compress first let’s just go with
vertical compression so vertical compression by 1 3 units
and then let’s just say by one third and then um vertical reflection through
the x-axis excuse me vertical shift down by one unit
okay so you wouldn’t say vertical shift down by minus one unit right the down
already means it’s a minus one okay so here we go let’s try to start
sketching the graph making as much doodles as we want or you
know as much scratch work as we want here
so just to refresh our memory what does this right here look like just looks

00:37
like that right there just a rough sketch that can just be side work if you
want it to be uh but even for side work i’ll make that
a little bit better so make it nice and curvy right there all right
so now let’s shift it left by two units so that’s gonna look something like this
right here one two and then it’s gonna have the same curviness to it right there
and you know that crosses eventually so don’t think it doesn’t but any case um
there’s a nice that’s been shifted left by two units so there’s a minus two
right there all right so i got i like to keep track of this point here
now i’m going to do a vertical compression by one-third
because i can change the order here it doesn’t really matter i’m going to
reflect it first and then i’ll do the vertical compression there

00:38
so let’s see here this will be y equals what x plus 2 cubed
and now let’s reflect so this will be the graph of minus x plus 2 to the third
and so this part that’s going up is still -2 here
and so now it’s going to look nice and curvy through here something like that
and so this is -2 still here right so this part that was going up is now
coming down and this part that was coming up here is now coming down here
all right very good and then now let’s do a vertical compression by one-third
so what is a vertical compression by one-third do so
that’s going to stretch it out horizontally
so we can put that in red here so this is going to come out wider like that
and this is going to come out wider like that and so this right here in red

00:39
will be the graph of minus 1 3 x plus two to the third and
let’s scoot this one right here over and i’ll just say i’ll label it right here
so this will be the minus because it’s been reflected right here
and then the x plus 2 squared to the third there we go there’s the black one
and then there’s the one in the red right there
now i put it in red but you can put it in black right there if you want if you
don’t use color and so you can just contrast and compare
the uh graphs right there so for example if i have a skit so now
let’s go down so that’s reflection vertical compression now it’s shifted
down by one unit so now we need to shift the one in red down by one unit
and so we can do that right here if we want let’s go here minus two
and now i’m going to shift it down by minus one units let’s say there’s a
minus one there and so there it is right there

00:40
and so now i’ll try to put this one right here in blue
and so i’m trying to model the one in red right here it’s just been uh
shifted down by minus one and so let’s keep i like to keep the same distance
the same shape here there we go and then this will come up here like that
all right so there it is so the one in blue here is
uh the whole final deal here minus one third x plus two
and then it’s been shifted down by minus one and i got the minus one labeled
right here all right there we go so the one in blue right here is their final
graph right there so you know if you were to try to do this in your head
you would say you know without doing all the scratch
work right here you just try to graph this in your head it’s like okay i know
what this looks like it looks like this now i got the minus so now it’s going to
look like that and then i just shifted it down one unit right there
so because of the minus sign it’s going to get you know flipped over it’s going

00:41
to look like that and it’s going to be shifted down so
it’s going to look like something like that instead of going through here like
this now to go through here like this right there so that’s the final graph
right there and you can kind of try to visualize it just by looking at the
transformations eventually after you do more right we’ve only done a few so far
all right so but let’s do uh one more just to end the video here
and let’s check it out one more here so let’s do um how about something like um
let’s do a let’s do a cube root so let’s say g of x is 5 times cube root of
x plus 1 plus 2 something like that right there so um

00:42
you know let’s identify the transformations right so the parent function is
cube root of x so i’ll say parent function is g of x is cube root of x cube root
and we’re going to have a horizontal shift by right sorry left by my by one unit
and we’re going to have a vertical [Music] vertical [Music]
let’s see what was it called vertical it’s going to be stretched vertical
vertical stretch so we’ll say vertical stretch vertical shift vertical stretch
by factor of five and let’s see here we’re gonna have a vertical shift

00:43
and then we’re gonna go up by two units all right and so let’s just uh check out
how this looks so y equals cube root of x this is the
basic function right here it’s coming in here really steep i like that right
there like that and then it starts to taper off but it’s still increasing
all right and so now what does the plus one do it’s going to shift it left by
you one unit and so i’m going to shift it left by one unit
and then i’m also going to vertically shift it up by two units um
well i’ll just do them one at a time so here we go now we have something
that’s coming in here like this really steep and it’s coming in here really
steep like that something like that right there so this will be y equals
cube root of x my x plus one and then now i’m going to stretch it by five
uh sorry y equals cube root of x plus one and then now i’m going to

00:44
stretch it by five so vertical stretch so now this is going to grow faster
vertically like that and it’s going to grow vertically down faster like that
so this will be y equals cube root of x um and sorry about that i wrote that too
fast and then cube root of x plus 1 but then we have a 5 in front of here
and so then the last step would be to move it up 2 units so we could sketch it
like that so we’ll move it up so we’ll say here this was minus one
tick mark right there and then it’s been shifted up two units so one two
and then now it’s coming in here through nice nice and steep and it’s coming out
through right there and so this will be the final graph right here
five cube root of x plus one plus two and there’s that

00:45
point right there minus one and plus two right there so there’s there’s what the
final graph looks like roughly and it it looks rough because
it really comes in there pretty steeply and it comes out pretty steep but it
doesn’t go vertical all right and so there’s a rough sketch
of the graph right there so anyways in the next episode we’re
going to look to see how to do these uh using a computer so
we’ll be able to enter all these functions in identify the
transformations and we’ll get a sketch of the graph and we can actually do
sliders and we’ll be able to apply these transformations in real time
and see them update automatically in real time so i
look forward to seeing you in that episode let me know what you think in
the comments below i’ll see you then bye if you enjoyed this video please like
and subscribe to my channel and click the bell icon to get new video updates

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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