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Master Exponential Functions | Graphing, Domain, Range & Transformations
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May 7, 2025
In this lesson, we dive deep into graphing exponential functions—a key concept in algebra. Whether you're learning for the first time or need a refresher, this video breaks it all down in a simple, relatable way. What you'll learn: How to recognize and graph exponential functions Understanding the domain, range, and the role of asymptotes Real-life examples of exponential growth (email chains, viral sharing) Step-by-step graphing using key points How transformations affect the graph: shifts, vertical stretches/shrinks, and reflections
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0:00
so what we're going to talk about you
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weren't even here yesterday so skip both
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we're going to talk about uh
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graphing
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exponential functions exponential
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functions are functions that grow really
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fast example let's say you know you ever
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receive those emails where somebody goes
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hey i'm going to send you this email to
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send you to five more people no no and i
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mean it used to be something that was
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used to happen forward you forward to
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five more people right
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so to figure out let's say you said you
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send it to
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like 10 people right 10 people do this
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so how would you figure out how many
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numbers of email was sent out it would
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be what 5 to the^ of what 10 right so
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the function that would best represent
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that would be f ofx = 5 to the^ x
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depending on the number of people that
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are forwarding this email okay so
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because you have if you have 10 people
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doing this so you that means you send
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about five to the power of 10 emails it
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used to be something that people used to
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do before or send this to five people or
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something's going to happen to you today
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i never bothered with it cuz it yeah
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just chain stuff that didn't work for me
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i'm not that one that never scares me
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i could care less so that doesn't i'm
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not superstitious like that so i don't
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really care i i just send it back to
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them what they did that don't bother me
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i don't need to send it okay so that's
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that's the kind of step that we're going
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to do today we're going to learn how to
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graph these type of functions right now
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uh an exponential function is always
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going to be in the form f ofx= b to the
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x b to the x okay now this function has
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some specific things that we need to
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know about the first thing that we need
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to know is what is the domain of this
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function and when i say domain of the
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function i'm talking about for what
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values does this function exist right so
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if you have a function in the form b to
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the x what are the values for which this
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will exist i mean sort of what what is
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it -21
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i mean not here i mean right here the
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domain will be what all real numbers
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right every single number you put in
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there you'll be able to evaluate this
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function if you put 3 to the 100 you're
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going to find it so any exponential
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function in the form b to the x has for
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domain all real
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numbers now the good thing about this
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exponential function is this what's the
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range when the range of this function is
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this when b is positive right when
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there's no negative sign before b the
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range will be where all positive values
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because we're going to have something
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called asmtote right asmtote asmtote
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yeah asintote is the point for which the
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function does not cross over right so as
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it's called an asmtote so
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basically so you're going to see that a
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lot in calculus right in calculus you're
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going to see what we call an asmtote
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okay so it's spelled a s
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y b t o t e asintote right so basically
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at this specific point the function is
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like a limiting value a limiting it's
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like a border you don't cross over that
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right
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so now how do we graph this function how
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do you graph the exponential function so
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let's take an example here if i graph i
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want to graph y = 3 ^ x right 3 ^ x we
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want to graph this so to graph this we
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need to first find the domain of this
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function the domain will be all real
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numbers and then to graph it we're going
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to choose some random points right now
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for to keep to keep this easy to make it
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easy i always want to choose like this
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one here like -2 1 0 1 and two right
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we're going to graph this and then we're
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going to use those as your guiding line
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to draw the function so for example when
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x= -2 i'm going to plug it in here
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y = 3 to the -2 we've already worked
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with uh exponential um numbers right so
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33 is basically what 1 over what 3 to
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the 2 right which is 1 over 9 right
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that's 1 n so this is 1 n so when x= -2
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y is 1 9 and then when
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x=1 it be 31 which is 1/3 right 1/3 here
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and then when x is zero that's pretty
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one so because 3 to the 0 any number
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raised to the 0 is one right and when x
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is one pretty easy 3 to 1 is three and
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when x is two is nine so with this four
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point i can graph this exponential
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function right i'm going to start by
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okay when x is zero y is
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1 when x is uh
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-2 y is 1 n 1 9 is somewhere right here
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somewhere close to that right when x is
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uh
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-1 y is 1/3 1/3 is going to be somewhere
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here and when x is uh 1 y is three uh
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one two and three somewhere here when x
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is two y is nine so
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two 3 3 4 5 6 7 8 9 somewhere here so
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the function is going to look something
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like this and like i say we have an
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asmtote right so the function is always
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going to get closer and closer to the
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line this line y = x but it's never
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going to cross it because that's your
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borderline right it's never going to
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cross this line but the function is
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going to look like this this is called
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an exponential function because it's
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growing very fast you want exponential
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meaning something that is exponential
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growth something that grows bigger right
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grows big so this is how the function is
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going to look like so this is 3 to the x
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okay so basically if you want to figure
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out how many emails were sent by two
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people would be nine emails right man
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three people that would be 27 if you use
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that example of forwarding the emails
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right so this function actually give you
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um that type of like uh interpretation
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okay now as example how would you graph
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uh y = 4 to the x it would be pretty
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much the same steps right y =
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4x the domain of this function will be
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what all the numbers and then all you
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have to do is just go through the same
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step i will choose the same numbers -2 1
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0 1 and two and then draw the function
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okay now the problem comes when you have
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like transformations okay
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transformations and today i debated how
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to teach this and i think i figure out
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the best way to teach the
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transformations for example if i have
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this function here 2 to the x + one and
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i want to graph it right you guys
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remember what we did in a very like in
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the second quarter no no well some of
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the things that we did when we transform
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a function right what do you see here 2
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to the x + one so you're moving to the
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left well this is the function you're
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adding it do you are you adding this to
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the x value or the whole function what
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are we adding the one to the x value
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only or the whole function here the
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whole function so we're going to move on
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what axis
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up right so we're going to move up so
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basically we're going to go one unit up
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right so that's where we do the
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transformation so let's say i wanted to
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graph this function 2 to the x + one
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right i'm going to go through the same
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step i'm going to erase these here and
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i'm going to do the same thing right so
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i have 2 to the x and i want to graph
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this 2 to the x + one right this is what
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i want to graph so what i want to do is
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this there's two ways to solve this we
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can go the transformation way or we can
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go the traditional way of just plugging
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in stuff right so let me let me use the
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same values here right so let's say i
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want to use a transformation so i'm
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going to start just graphing 2 to the x
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okay i'm going to go when x is -2 right
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this will be 1 2 -2 which is 1/4
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right and then when x is -1 it' be what
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12 i don't need to do all of that right
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when x is zero is going to be one when x
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is one will be two and when x is two be
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what four right so i can put those on
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here
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i'm going to put those here and then
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we're going to do something so when x is
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-2 1/4 is somewhere here right when x is
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-1 1/2 is in the middle right when x is
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0
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one x is one we got two
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here uh and x is two we got four
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somewhere here right so the function is
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going to look like this but the the
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question is this this is just 2 to the x
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right but what am i trying to graph 2 to
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the x + one so that means i'm going to
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choose those key units and i'm going to
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move them up how many units one unit
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right so this guy is going to go up one
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unit it's going to be right here this
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one's going to go up one it's going to
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be right here this one is going to go up
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one up one and it's going to now the the
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new graph is going to look like this
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right so this is called a transformation
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so you are basically transforming from 2
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to the x you can transform it right by
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just adding one unit okay i can do that
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now what happens is it gets a little bit
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more technical when we have more than
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just one transformation
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example if i have if i want to graph
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this
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graph right y = 2 to the x
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-1 +
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1 so how would you graph this 2x - 1 + 1
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so how would you graph that function
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move that function over right right did
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you see what he says here and move that
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function to the right or you can start
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by doing what first what would you do
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here once you graph 2 to the x what
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would you do first move to the right one
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unit up one and up one right so you
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could do that with this you can move to
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the right one unit and then move up one
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because they now the transformation is
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occurring at the x right so if he's up
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here when he's negative we move to the
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right and when he's positive we move to
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the left yes what do you mean move to
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the right all right so if i have 2 to
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the x - one right this is the new
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function if i give you this function you
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have two transformations that are
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happening okay so the first
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transformation is this you start by
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graphing 2 to the x right which we did
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here we start by graphing 2 to the
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x all right we start by graphing this
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function and then now the next
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transformation is this all the x values
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what is happening to all the x values
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subtract one right but we say when we
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subtract we go to the right right so
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this guy here is going to go one unit
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here this one is going to go one here
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this one's going to go here and this is
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going to go here so your first
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transformation is going to look like
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this okay and then the next one because
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we are adding one every single one of
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those that moves is going to go up one
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that's the last one right so you're
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going to go up one up one
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uh not here up one and then up one so
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now the new graph is going to look like
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this okay i know i'm using the same
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marker that's why you can't see the
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distinction but this is what we're doing
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so these you have two transformations
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here the first transformation is the x
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values are going to change because this
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is minus one you go to the right if it's
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plus one you go to the left okay and
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then the next transformation is the y
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values on the horizontal is called a
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vertical shift you're going to shift it
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up one unit okay all right now there's
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also something else that will occur and
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i do want to talk about it but i'm not
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sure if i should mention it now or save
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it for tomorrow probably today i think
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it's a good idea to state it no i'm not
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listening
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oh stop stop stop
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stop stop
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all right you know what i want to do is
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i want to save this for
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um
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tomorrow
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all right now let me mention this one
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thing here
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all right so now let's say i wanted to
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graph um
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12 right 2 to the
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x minus 2 like
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this
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right so here's what i suggest we do
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here right now you can see that there's
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something that's happening with
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our y values here right
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there's no y well i'm saying whatever
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there's a y there y is equal to this
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right so i have 12 and the entire
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function is being multiplied by what
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no the entire function is being multiply
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by what 12 right so here's what's going
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to happen here
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so it's one now before we do that right
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let's just go ahead and graph this
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function again and we're going to we're
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going to do the transformations together
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and figure out what's happening why did
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multiplication stop being in x
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because when we add in
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right so let's start by graphing we're
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going to start by graphing the function
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that we know right pay attention here
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because this is critical okay so we're
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going to start here like i said again i
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put in two i have a 1/2 here and the 1/2
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is multiplying the entire function right
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this is the function that we have and
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the 1/2 is multiplying the entire
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function so we're going to start again
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by graphing 2 to the x okay so i'm going
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to start by graphing 2 to the x so when
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x is -2 1 2 y is 1/4 so right here
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okay when x is -1 1/2 is right
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here i put it a little bit up and then
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when x is zero one is right here right
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and then when x is one uh we have two
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right here and when x is two we have
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four three four somewhere right here
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okay so this is the function that we
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have right here this right here is 2 to
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the x right that's just 2 to the
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x all right are we good on that now the
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next transformation is i'm doing what
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after that
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right i'm going to move to the right how
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many units two unit right to the right
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to the right so that means this guy's
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going to go one and two right here one
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and two right here one and two right
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here and this is going to go one and two
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right here right so we have
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this right this is a new uh curve now
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pay attention here what is happening i
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got 1/2 right that's
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multiplying all the y
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values the x's are not changing right
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only thing that's changing is the y
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values here do you does that make sense
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right so now what's going to happen is
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this when x is two right now what's
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going to happen is all the y values are
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going to shrink by what 1/2 okay so
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instead of um i ended up here now this
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guy is going to end up where two two in
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the middle somewhere right here right
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two uh so this is a let me see one two
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three four so it's going to end up right
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here right where it was before no no no
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you divide by multiply by 12 so it's
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going to end up right here um
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right and then this
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time the x value stay the same yes sir
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um you're um you're not dividing the x
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by two the y by two yeah y by two or the
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y by two yeah the y by two so if you
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here four
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right so this way it's going to go where
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right here right so it's gonna
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be right here now right because we have
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two and two this is gonna go here and
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then this guy because this is here and
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then this was uh right
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here now it's going to go in the middle
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here so we we're going to have it right
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here somewhere right and then this guy
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here was here now it's going to go down
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here and down here so now we're going to
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have this
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curve this is what you're going to get
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right now the only reason why the reason
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why this is a little confusing is
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because i'm not using different colors
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can you give us color pencils i can use
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i can give you color pencils but what
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you all you need to remember is when you
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do the
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transformations two to 2x to the two the
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first transformation is you move two
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unit to the to the uh right and then
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next all the y values are going to be
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divided by two the x stay the same i'm
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going to repeat that again tomorrow so
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that we can understand
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all right so we're going to redo this
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again tomorrow because i want to re
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reemphasizes that okay so we're going to
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do that again tomorrow so we're done
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we're not done what are we what are we
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doing now looks half dead