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How To write And Graph Absolute Value Functions
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Feb 19, 2025
Welcome back to our Algebra 2 class, in today's lecture we'll continue and conclude chapter 2.6 today and look at part of this section as we discuss how to write and graph absolute value functions. This can be a bit confusing for some students and I'm hoping that this video will be helpful to you. Chapters: 00: 00 Introduction 00:45 How to express an absolute value function in its piecewise form 02:55 How To graph an absolute value function
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0:00
so uh we're going to move on from the
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pie wise Define function to another type
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of pie wise function now these are
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called absolute value functions and they
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are not that hard you just have to
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understand how to build them okay so
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quickly I will explain like absolute
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value of x is always what positive right
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because that's what we know now here's
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the thing like when I have absolute
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value of x as a function I to graph it
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it's a complex process cuz I first have
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to be able to express this
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function in its uh different um part for
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example I don't know the value of x do I
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I do not know what x is because X is
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just X so now the question is absolute
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value of x will be equal to x if x is
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positive do you understand what that
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means we need to understand that because
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if you don't understand that it's hard
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to grasp the uh this kind do you know
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why is X x is positive
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why abute value it's be all right so
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yeah so for example what is absolute
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value of two two two right what's
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absolute value of -2 two right so
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basically absolute value of x is going
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to be x if x is positive meaning if x is
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a positive value it's going to be the
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same thing correct M now it's going to
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be zero if x is is zero that makes sense
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right because I just have to plug it in
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here correct if x is zero because zero
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is my what again remember zero is my
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base my base my limiting value that's
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the everything is built around is zero
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here right now here you Bex if x is
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negative what am I trying to say
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here um um it's the absolute Val because
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the absolute value is always positive
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exactly so if x is negative thetive X
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would be positive that's right did yall
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get that right so for example I put
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value of2 that gives me two
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right -2 is what is really -2 that's
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what it is right so I'm saying here the
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absolute value of x ISX if x is negative
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if x is negative I have to add another
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negative to the X to make it what a
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positive value does that make sense
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right so this is why I I created this uh
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piecewise function here so absolute
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value of x is X suppose X is positive
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and it's going to be zero if x is z and
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it's going to Bex if my x value is
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negative because I don't know what x is
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so I have to consider all three options
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does that make sense right now this is
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where uh some of us get like a little
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bit like confused now we have to graph
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this function correct now after value
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function is a fun function because it
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always look like a V or if V that's
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reverse so I love this function so now
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if I to build this function right how
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many tables am I going to make three
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three table thank you very much right so
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I have X Y now remember let me explain
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this
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again this is not this x here
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means in other word Y is equal to x y =
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0 and Y = tox that's what this means
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right because f X and
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Y mean the same thing correct we talked
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about that in our previous chapter okay
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so now here what would be my first value
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for x zero zero choose another one one
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so when X is zero Y is and when X is one
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y is one right so when X is zero Y is
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zero right and when X is one y is one
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now here's the problem so I'm supposed
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to be going where in this direction do I
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go past this no no right so now I will
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have one here normally open circle right
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but you going to see that we we going to
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eliminate that open circle in a minute
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now now when X is zero Y is what zero
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zero right and when X is one y will be Z
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zero X is z so 0 0 is just one we're not
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going to change that so it's going to be
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right
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here right because when X is z y is what
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zero we only have one option cuz X is z
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y is z so that's that's only one correct
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and now we are here so I'm going to
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start again with what Z zero when X is z
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y will
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be which is zero and I choose another
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one and then y will
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be posi one y 0
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01 1 and
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look this is the absolute value function
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again it's a beautiful function if you
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know how to slice it correctly right
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it's always going to look like a V or
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it's going to be reverse it depends on
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what we have okay now this example is
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pretty simple but I'm going to do
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another one and then we will understand
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how to build this function once and for
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all okay now coming here right so
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how without even having this here let me
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see if I can establish this right this
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is what I have I'm starting with this
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absolute value of x - 2 suppose this is
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what I'm I'm giving you I say graph this
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function right the first step is to
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build our
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piecewise component correct so absolute
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value is x -2 so we're going to go like
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this just like the other
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one ABS value of x will be what x - 2 if
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what somebody help me here if x is
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greater than Z no x - 2 is greater than
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0 right x -2 because it's whole thing
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it's not just X if this is positive the
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abute Valu is positive right if x - 2 is
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more than Z does that make
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sense right and you'll be what you'll be
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zero if x is equal to
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what
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no posi two you mean because if I set
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this equal to zero I add two it would be
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two right x = 2 that make sense yes or
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no no all right let's build it again
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let's confused all right let me explain
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right the point not Z so here right this
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is what we have
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right absolute value of this number will
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be what x - 2 if this number is
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whater z greater than zero right there
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you go in other words if this number is
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greater than what zero two two two right
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it's the same thing I'm saying the same
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thing here does that make sense
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right now so let's say this is our first
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option right
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now what if this thing is zero the
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inside is zero right x - 2 is 0 that
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mean X is equal to what zero zero X not
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not z x equal to pos2 for this to be
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equal to zero that mean X has to be
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equal to what two two right so therefore
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this will be zero if x is is equal to
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two does that make sense yes or no not
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really just keep on going we'll figure
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it
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no I want you to show I want to show you
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what's actually going on here right
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confus is the is the point not on the y
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axis then it's on the x axis uh which
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point um the um um the the point of the
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absolute value function is on the x-axis
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yeah yeah so here so what you need to do
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is this this number
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here is what is
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causing this absolute value to change
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right so now I want absolute value is
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always going to be positive correct
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right so now if this number inside is
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positive then this is going to be the
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same right so absolute value of f
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ofx is going to be x - 2 if x -2 is
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positive yes or no
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yeah does that make sense yeah okay
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because if this is positive then you be
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this one right in other
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words x is more than two because I'm
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just adding two on both Sid does that
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make sense to this point yes right all
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right now what if this number inside was
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Zero what if this was
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zero right that
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mean X is equal to
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what two because you going to do x - 2 =
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2 0 x =
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2 right yeah if this is zero that mean X
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is equal to 2 correct now wait would it
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be
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-2 x - 2 is equal to0 you add two on
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both side that gives you x =
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2 if this is equal to
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zero yeah solve for
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x w what would it
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two does that make sense now to anybody
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yes or no all right now what if this
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number was negative right what if this
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is like
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this that means X is what x - 2 is less
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than Z right which another word X is
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what less than two that's all this mean
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so this is how you build this function
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is x - 2 if the number is positive is 0o
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if x is 2 and therefore is
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the opposite of this number if x is less
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than two does that make sense now all
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right now we can build our
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graphs right XY XY X Y now here I'm
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going to start X is what my first value
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will
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be yes I'm waiting two two right and X
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is two Y is what
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zero zero and another value when X is
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three Y is 3 - 2 1 right here we have it
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when X is zero when uh X is 2 Y is what
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Z zero right and here we're going to do
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the same thing when X is we're going to
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start at two Y is what Z zero and when X
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is uh take another one less than two One
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X Y is 1 - 2 is 1 two negatives that's
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positive right and then we go back here
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we do the same
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thing the rest is just understanding
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this process so when X is
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two Y is zero right and when X is 3 Y is
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1 right here right so and I want X to be
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more than two so I'm going to go this
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direction right and then when X is 2 Y
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is z I already have that over here and
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then when X is one y is One X X is one y
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is one I'm
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here this is the function it's always
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going to look like a d right if your
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function does not look like a V guess
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what something here right okay it's
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going to look like this so the the most
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important part is knowing how to build
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the function based on the number that's
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given so we're going to do several
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examples and then tomorrow we're going
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to do some more examples I'm going to
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stop here and I'm going to let this you
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can marinate on that right I don't want
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to move on until we all understand this
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so let's let's do that let's work some
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more examples and then we're going to go
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from there
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