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Graphing Quadratic Functions Using The Vertex Form
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Feb 20, 2025
In our last session, we learned how to use the complete the square method to turn quadratic functions in their standard form to the vertex form. Now we'll learn how to graph quadratic functions using the vertex form. Chapters: 00:00 Introduction 00:49 how to turn a quadratic function in the vertex form 02:12 how to graph a quadratic function using the vertex form
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0:00
anyway with that say let's start so we
0:03
talked about graphing quadratic
0:05
functions and we learned how to use the
0:07
completing the square to write an
0:10
equation from the quadratic form to the
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vertex form that was last week and now
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we're going to learn how to graph them
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right we've learned how to like put them
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uh from the regular a x² + b x + C to
0:27
the a x - h square plus K where H and K
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are the vertex so we learned how to do
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that now we're going to learn how to
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graph it right it's very practical so
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now let's say I wanted to graph a
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function that is in the format 4x2 - 16x
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- 40 right I want to graph this now
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before I graph it I have to be able to
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change this into the vertex format then
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I can use it right so now before we get
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into this let's learn how to graph these
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functions first and then we're going to
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use that to graph uh
1:00
a function that's given in the form ax2
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+ BX plus C all right so let's say I
1:05
wanted to graph this here x - 3 2 - 2
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and then y = 1 14 x + 1 2 now the first
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thing I want to do is this um there's
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two ways we're going to do it right we
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you can identify what the vertex is so
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what's the vertex in this function here
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what's the vertex two two no that's not
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just the vertex what's the vertex 3 -2
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is the vertex but 3 -2 H and K right now
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that we have the vertex we can graph
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this function okay now the book teaches
1:35
using Transformations where you have to
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do a vertical stretch like all of that
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stuff is great but I don't find it
1:42
necessary because I feel like there's an
1:44
easier way to do it so I have reserve
1:46
the right to do that I don't have to
1:47
teach necessarily what the book is
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saying because especially if the book
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way I feel like it's too long right you
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want to make things easier so what I
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want to do is first I want to identify
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what the vertex is what's my vertex 3 -2
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right
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my vertex is 3 -2 so I know that and I
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also know that my vertex is either what
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the highest point on the curve or what
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lowest right so I already have my vertex
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3 -2 so I'm going to try to point it out
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1 2 3 1 -2 right here right this is my
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vertex now I go back to this and I'm
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looking at what's the value of a in this
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equation here one one so since a is
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positive that means graph opens what
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upward or downward upward right a is a
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is here this is a is equal to 1 right so
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the form is a x - h 2 + K because a is
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positive right it's going to be a
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parabola that opens upward so I already
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know that it's going to be going up like
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this now all I need to figure out is
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what I need to just find one point and I
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find this image using my Axis of
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Symmetry right why is going to cut my
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graph into two halves x = what 3
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remember we talked about the axis of
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symmetry right HK so this is your axis
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of symmetry is going to cut your graph
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into two halves so 1 2 3 we're going to
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put this here like this and now we're
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going to try to graph this so this is
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our vertex right now I just need to find
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one random point on on the curve right
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when X is three I can use this here when
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X is 3 Y is what -2 I already have that
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so I'm not going to use that I'm going
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to use another Point find your AIS to
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cry it's always going to be the x
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coordinate of the vertex three yeah okay
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your axis of symmetry is always going to
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be the x coordinate of the vertex right
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your axis of symmetry which Cuts your
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graph into two halves is always going to
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be the x coordinate of the vertex okay
3:42
now I just need to find a random point
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right I already have my vertex my vertex
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is 3 -2 I just need to find another
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point right when I'm going to replace X
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by two here right when X is two I want
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to find a y value right so Random point
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when X is
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2 I'm going to replace in here 2 - 3 is
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what -11 s is 1 1 - 2
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is1 right 1 - 2 is1 so once I have that
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I have my curve so I'm just going to go
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here so X is 2 y is1 right right here so
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I just need to go like this and guess
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what I'm going to repeat it here and I'm
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done see CU this is going to cut the
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grass into two halves if this first half
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go this way the other half is going to
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go this way and you're done right the
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book teaches a long way of doing this
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where you start with Y = x² and you
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doing all this shift shift three to the
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right two down and we're not going to do
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that unless you guys want to learn that
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do you want to learn how to do that okay
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so just let's stick with the vertex form
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and it's easy it's easy to graph the
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curve knowing this right we just need to
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find the vertex we know what the axis of
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symmetry is and then we just choose a
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random point and then that's
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it get one portion of it and then get
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the other one yeah how do you get the H
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in K we going to we did that last week
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I'm going to show you how to do it my
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not for no I'm going to do one I'm going
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to do an example here right so now let's
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say we graph this let's graph this one
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and then after that I'm going to do an
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example here to show you how to find
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those right so if I have y = 1/4 x + 1 2
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what's my vertex so let me write that
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first 1 4 x + 1 s what's my vertex here
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1 0 1 0 because this is is positive and
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this is like an imaginary zero somewhere
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we don't have it right but because we
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don't have it does not mean that it
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doesn't exist it's just zero right so
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your vertex is going to be 1 0 does
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anybody have a problem with Y if the
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vertex is 1 and 0 no all right so now
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that we know that the vertex is 1 and 0
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we just know that it's going to be
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-1 and zero right right here so what's
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my Axis of Symmetry xals what sh 1 so
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this going cross your graph in two half
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right so now we have our vertex so we
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going to go back
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here and we just going to write this 1 0
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and I'm going to PE point on the graph
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right when I can replace X by what here
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anything find choose a point one all 1 +
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1 is two right 2 square is what 4 4 *
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1/4 is 1 right so when X is two did you
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say one or two I said one one y is what
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one one all right so that's it when X is
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one y I is one right here right now A is
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positive so he opens what upward right
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so we're going to go ring and what R and
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that's it see pretty easy stuff we don't
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have to use the Transformations I just
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feel like it was a waste of time I I
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looked at it last year I I put my other
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student through hell cuz they went
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through and crying all you know what
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let's just use this it's better okay
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we're not going to do that this year
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we're just going to choose this way and
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we're going to stick with it all right
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so now now that we know how to do it now
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we have a a graph here y = 4x2 - 4 16 x
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- 40 so we need to use the we're going
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to put this this graph into the Y = A x
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- h² so which I will just tell you right
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so now let's go and do that so I have y
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= 4
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X2 - 16 x
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- 40 all right so here's what we did Co
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so when you have a function like this so
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you want to First isolate these two like
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that right you want to isolate the one
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with BX okay now step number one we're
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going to factor out four right so it's
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going to be 4 * what X2 - what 4x right
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plus Dash right Das here and then - 40
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plus d again right so that's what we did
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now next step we just completing the
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square we're going to divide this number
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by
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what what rememb
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M right and then two and do what we
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divide this number by two so what's 4id
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two what's 2 squ four so we're going to
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add four here but
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remember so because this is 4 and 4 16
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we need to find a way to do what the
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equation balance it out right so this is
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going to be what - 16 16 but yeah- 16
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isn't that supposed to be a minus right
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there well yeah plus but you can put
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plus and then and then -6 here yeah to
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balance it out right it's the same thing
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you know like if you want just Min - 16
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cuz sometimes if this is negative then
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that will be positive right vice versa
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right so is anybody having trouble with
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this here yeah wait what do you do with
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the x
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s uh what x s so you do so so hold on
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let me let me let me put it this way so
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do you understand up to this point right
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what I did here right so we completing
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the square now what we want to do is
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this we use the same step that we used
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in the other um chap chapter right so
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you take this number here you divide it
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by what two so4 by 2 is -2 right and -2
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s is four so in the last chapter we
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added four and we also added four here
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right subtracted because when you add it
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on one side you have to subtract it on
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the other one so we want to balance out
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the equation but here one thing that we
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we can't forget is that this four is
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also being multiply by what four that
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means this is 16 really so instead of
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subtracting four here we going to
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subtract what 16 so this is why this is
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now- 16 all right and now we have
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finished so because we have y = 4 right
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x - what 2 2 because this is factored
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out right and then4 - 16 is minus what
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56 so so now we have our vertex uh
10:02
component here right so now what's my
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vertex here two and
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what 56 2 and 56 and my Axis of Symmetry
10:10
is also two right so if I have to graph
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this right so 2 and 56 let me assume
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that somewhere down here right it's not
10:21
quite accurate but I'm just going to do
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that here right that's also my Axis of
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Symmetry right and now I just need to
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figure out I'm going back here right X
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and Y this is
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256 I choose one uh what what could I
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choose here to make it easy for myself
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three three all right three the x is
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three 3 - 2 is 1 1 square is 1 1 * 4 is
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4 4 -
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5652 right so when X is 3 y is52 maybe
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somewhere here right I'm just making an
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assumption now because a is positive it
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opens what way upward or downward upward
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right so it's going to be boom right and
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then boom and then you're done so again
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make it easy
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explain it opens because a is
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positive if a is positive opens upward
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right and if a is negative it opens
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downward so it depends all depends on
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the value of a a determines whether the
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graph opens upward or downward if it's
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positive you go upward you open upward
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and if it's negative you go down
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okay all right that's pretty much it for
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this how to graph this now tomorrow
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we're going to talk about graphing
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inequalities right so now we going to
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have stuff like this it's Y is less than
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= X2 - 3x + 1 we're going to learn how
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to graph this yeah do line solid line
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and all that kind of stuff that that
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good stuff okay so this is it for this
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one