Up next in 10
How To Graph Quadratic Inequalities
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Feb 28, 2025
We've already learned how to write quadratic functions in the vertex form then graph them as well. Now we'll learn how to graph quadratic inequalities leaning on what we've learned in the previous section. Chapters:
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all right so we're going to talk about
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graphing quadratic inequalities right
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we're going to learn how to graph
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quadratic inequalities so first we know
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how to graph quadratic functions we've
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learned how to do that now the only
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thing that we're going to do here that's
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different is we're going to learn how to
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graph them now because so far what we've
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seen is y is = like X2 + 2x + 1 and
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graph it now we're going to learn how to
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graph it when you have an inequality
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that's what we're going to do in this uh
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section here is it is not hard at all
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trust me when I say it I mean it right
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so the first thing that we need to know
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is this before we graph these quadratic
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functions we have to put them in the
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vertex format so then we can graph it
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Nathan has his own way of graphing this
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quadratic function he can stick with it
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that's fine he learned how to do it he
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found some YouTube video and then he
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felt like the guy was more comprehensive
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than I was so hey more part you right
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the way we do things is this this is
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called a Socratic method if you find a
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way that works for you stick with it I
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don't have problem with it as long as
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you can get this right right okay now
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the only thing I will have is if you
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come up with a different gospel then I'm
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going to tell you you going to hell
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right because there there's no another
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gospel right it's only one gospel
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through Jesus Christ Our Lord and I
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thought was a metaphor on YouTube Sor
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didn't used to be yeah I used to be yeah
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yeah I was in a de iniquity
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I got out of that because of the grace
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of God in because of working at 7-Eleven
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no yes 7
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somebody all he did was just tell me and
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I got the Bible but then there's a
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series of event that happened that led
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to the
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[Music]
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eventual all right so now let's say we
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want to graph this function here right
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we have y is greater than X2 + 2x + 1
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the first that we need to keep in mind
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is that this is greater that means it's
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going to be what a dotted line a dotted
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curve not a solid curve right curve so a
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doter curve so we need to start by now
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we're going to start here and we're
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going to transform this into our vertex
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uh format right so I have Y = X + 2x + 1
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so I'm going to put it at X2 + 2x plus
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Dash close parth plus one right now I
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need to figure out what goes in here
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what do I do I take B iide by
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what yeah what does say what does that
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say here yeah goow right prior to
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graphing okay prior right so here I'm
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going to divide the two by what two two
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and then that gives me what one and I'm
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going to one square right which is so
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therefore I'm going to add one here both
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oh my
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goodness that's enough you're giving me
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a headache right we're going to add one
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here and then what do we do on this side
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everybody
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remembers subtract one but we're still
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doing this yeah because we have to graph
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it's a Vertex format right we want to
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subtract one here because we are not Sol
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we not completing the square to solve an
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equation but we are completing the
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square to find this in in a Vertex
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format right why are we subtracting one
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instead adding it cuz if I don't I need
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to balance out the equation right if I
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if I add it won't it won't do that right
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why don't we just not add it so this is
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zero now I have y = what this is what x
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+ one what squar right x + 1
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squ now what's my vertex uh Stephanie
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what would be my vertex here it would be
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1 Z thank you very
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mucha 1 0 right everyone knows why
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because it's always what you take the
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opposite of this number and there's no h
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it's H and K right this is H and then
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this is k k is what K is zero here and
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this is H will be what 1 so 1 is z
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thanks for catching that now we need to
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graph this function right so we need to
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graph this function we just need to make
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a quick table
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right so we know we have the vertex I'm
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going to also pick another point right
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so what can I pick here Stephanie so I
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have x + 1 s what would be the easiest
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other point to choose here to make this
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EAS zero when X is z y is what um 0 0 +
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1 is 1 1 1 square is one one right so
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now we can graph this function now
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remember we are graphing Y is greater
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than what X2 + 2x + 1 right so what we
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want to do first is we want to put our
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vertex vertex is 1 and zero right is
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right here and then we also know that
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this is going to be my Axis of what what
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do you call this axis of symmetry right
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thank you and then when X is z y is one
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is right here so now I'm just going to
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graph portion of it is it going to be a
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solid line or a do line got it got it
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line thank you very much shap and then
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the other p is going to be right here
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right and we are pretty much done now
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actually we have another step so Y is
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greater than X2 + 2x + 1 right so I'm
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going to replace both Y and X by what
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remember what we did we shaded yeah so
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we going to replace both by what 0 0 0 0
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right so I'm going to replace both by 0
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0 and I'm going to double check the
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statement right so I'm going to replace
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this by0
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and this by
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zero right so that gives me zero is
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great than one true or
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false true false false statement right
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now look at this 0 Z is here and this is
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false so what do I sh inside or outside
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inside wow you guys are smart that right
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so that's it so this is the solution see
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how easy this graph inquality is it's
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just like doing it the last lesson the
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last
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right right so we using this and we just
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shading it so
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now so let me let me graph another one
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here right so let me use another one let
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me do another example can I erase this
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so somebody's copying this
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here all right copy all right go ahead
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and
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[Music]
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copy all right let me uh let me graph
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another one I can use the this side
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I won't okay thank you so let's say we
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have
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y is less
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than -2X 2 now I know you guys not going
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to like this but it's okay right -2X 2 +
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3x + 5 right I am trying to graph this
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okay you can erase it now right hold on
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let this ra it after so I have y is
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less2 exp + 3x + 5 right so again we
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going to use the compl square we're
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going to put this in ver simple vertex
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form right so before I do that though
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remember so I'm going to I'm going to
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assume that Y = -2x 2 + 3x + 5 I'm going
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to make that assumption first right so
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that I can put this in the vertex so I'm
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going to pull out what -2 right as a
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factor thank you Stephanie if I P out -2
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I get what X2 right now instead of plus
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be what - 3x / 2 right plus Dash
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and then plus 5 out here plus Dash right
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remember why we did it same process okay
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now this is where everything gets like
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crazy because we have n fraction right
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we just multiply everything by two and
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give the fraction well no we don't want
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to do that because you want to make sure
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there's nothing before the x² right you
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remember so now I'm going to do right
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here I'm going to divide this by what
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first M two so that gives you3 over 4
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right and I'm going to square it what3
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over2
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4 S 9 16 9 16 right so this is 9 16 well
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remember before we do this here we have
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to multiply this number by what 9 16 we
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gives you what
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9898 all right now we are pretty much
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done and I told you the key here is once
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you get this to write this function is
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easy you just take B this is b/2 right
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so this going to be x - 3 4 squ right
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and then we're going to do this here 5 *
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8 is what 40 40 - 9
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is
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31 so this is plus 31 over 8 right so
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what's my
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vertex uh 34s and 318 right so vertex is
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34 right and then 31 over 8 how do you
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even like graph that oh we can graph it
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it's like a my favorite word they
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walking it walk walk the
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[Music]
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fork thek
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yes walk in the pork baby all right
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let's get it let's get it all right so
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now we have that right so again we have
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our vertex right 3/4 what
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0.75 and 31 over8 can somebody use put
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that in the calculator for me we going
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to get some approx 87 3.875 so 34 0.75
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somewhere here and then
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3.875 1 2 three somewhere around here
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right we just using approximation right
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now look here this is -2 so it opens
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what upward or downward it's going to go
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down right our vert axis of symmetry is
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still here and now I just need to find a
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second Point let me make my table real
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quick X and Y I got 3/4 wow and then 31
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over 8 right and now I just need to find
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another point so the the best thing for
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me to do here is I'm going to replace
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this one by uh let's see uh what's the
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good point let me let's one you say one
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yeah one so one would be 4 4- would be 1
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- 34 be what 1 over 4 right yeah 1 over
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4 S is let me just do it2 * 1 4 2 + 3 1
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8 I'm just going to do it so this this
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is 1 4 2 is 1 16 right so it's -2 * 1 16
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+ 31 8 oh great which is1 8 + 3 31 8
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that is
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uh 30 8 can somebody put that in the
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calculator sorry were you a nerd of my
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school no do I look like one
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no wait wait you
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what's that 308 is
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what
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3.7 just got right so right here and
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then 3.75 somewhere around here right
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3.75 somewhere here so I'm again I'm
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just giving you like approximation so
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we're going to go down again it's a it's
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not a solid line it's going to be a down
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line it's going to go like this right
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now the next thing we want to figure out
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is which which portion of this is the
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solution I so I'm going to repl both by
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Z right 0 is less than five cuz if I put
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plug in Z I'm going to get 0 is less
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than five true or false yes true so
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therefore since 0 Z is here your
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Solutions going to be wherewhere on the
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inside not everywhere on the
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inside on the inside right because you
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have the outside and the inside so this
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is your solution all right so what I
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want to do with this section is we we I
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was going to talk about solving
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quadratic uh inequalities but I want to
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hold that for tomorrow because we can
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sit on this one do some problems okay
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from the book and then we're going to
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hold the other one for tomorrow so let's
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we can stop here for
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now is it on
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