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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

and now what about this point right here now it’s going to be 1 2

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

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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

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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

means added two so added two to this added two to this added two to this and

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

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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

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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

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and 2 times 2 is a 4 so we’re going to get 4 4. so i’m going to say output’s

about right well let’s make it higher right about there

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

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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

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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

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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

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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

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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

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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

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

and then now what about this point right here it gets uh reflected through the

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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

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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

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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

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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

now what about this point right here what does that get transformed to so

zero minus two so we’re shifting it to the right so

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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

all right so now what about this point right here so this point right here is

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

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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

there now what about this part right here so this point right here it’s been

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

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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

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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

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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

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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

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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

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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

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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

now what about this one right here so i’m going to input a minus 2

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

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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

and now the last one right here what about this six two right here so this

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

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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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