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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

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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

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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

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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

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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

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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

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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

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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

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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

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