This content introduces the process of graphing quadratic equations, which result in parabolic shapes, by focusing on finding the vertex and axis of symmetry as key starting points.
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so ladies and gentlemen so far what we
have um basically discussed is going to
be uh quad our linear equations all
right now what we're going to be doing
is with quadratics and the reason that
we um get into this would be a quadratic
is going to be based
on that little two there which is going
to be our degree okay so for this whole
chapter chapter four we're going to be
dealing with quadratics that means
you're going to have an equation or we
might be dealing with functions that
we're going to have your variable is
going to be to the second power okay
sometimes we'll have a linear variable
as well but you're always going to have
for quadratics for it to be a quadratic
and you have to have a variable raised
to the second power okay so what we're
going to learn how to do is graph these
now when we're learning how to uh graph
linear equations you know there's kind
of two different things that I you know
told you if you had a linear equation y
= 3x - 1 right um we talked about you
can always use like a table values you
right yes right do you
remember that you can take X and then
plug in y and then plot the points
connect them but that but the reason
with that is that was a lot of work
right so what we also found out was we
could also use slope intercept
form right and just find what m is find
what B is plot the point negative 1 and
then use that as your slope 1 2 3 over one
one
and then you could
connect right it was pretty simple does
everybody understand kind of with me
okay so um or we or if it was in
standard form we use the intercept
method now for a
quadratic um the shape
shape
yes yes that that was just a that wasn't
really a part of this problem I'm going
to get into that that was just a little
forward um and we what you're doing
sorry so into graphing quadratics what
we're going to be doing is we're going
to use the first method okay which is
when you have a quadratic in standard form
form
now a quadratic the shape of the
quadratic is not like a line it is what
we call a
parabola and a lot of
times you can just relate it to a
u okay so the reason why I show you guys
this what you know some examples of a
prev look like though are because if you
just pick two points does that tell you
what the graph looks like no it doesn't
right hey gu girls this
is this is what I'm going over this is
the important thing I know this is the
important things right here so picking
two points is not going to be enough all
right and there's another thing I want
you guys to understand I want you to
make sure you write down when we're
dealing with quadratics there's a very
very important Point there's actually
two there's a two kind kind of important
characteristics the minimum or the
maximum and whenever you have the
Min or the maximum point do you guys see
how that's like the minimum point of
this Parabola right here and that's the
maximum Point any point that it takes
the minimum or the maximum is what we
vertex okay so you're going to want to
make sure you understand that the next
thing is through the vertex there is a
line of
symmetry so I'm not going to be the best
drawer here but if you guys remember in
Geometry a line of symmetry is a line
about where which you can flip your
graph over and it's going to be
symmetrical with do you guys see that if
you were to kind of take that and fold
it over that dotted line okay right
Elizabeth from geometry right right no
I'm just saying from remembering from geometry
okay does everybody see what we have
here yes okay so when we're getting into
graphing now what we're going to do when
we're graphing in a when we graph a
quadratic what I'm going to expect you
guys to do is the first thing we're
going to do is find the vertex we're
going to find that point and we're going
to plot that point remember when we
graph linear equations the first thing
we did was graph the Y intercept right
now the first thing we're going to do is
graph the vertex okay then what we're
going to do is pick two
points to graph either to the left or to
the right of the vertex and once you
know two
points can you now use those other two
points or the line of symmetry to
reflect over there so really ladies and
gent all we're going to do is now
instead of two points we're now going to
up ant it up to three points that we're
going to graph and then you're just
going to reflect those other two points
across the line of symmetry and then you
have five points that you can use to
well let's go and see what we're going
vertex so the vertex the formula for the
vertex equals opposite of B / 2
a and some of you might say okay where
did you get the a and the B well if you guys
guys
remember ax plus b y yal C that was what
we wrote for that was standard form of a
linear equation right so standard form
this do you guys see that that is your
standard form for your equation so now
all I got to
do um actually kind of my steps that I
like to do is first thing identify my a
b and my C A = 1 B = 8 C = 12 does
everybody see how I labeled my a b and c
yes okay now I'm just going to evaluate
to find my vertex so I do opposite of 8
/ 2 *
-4 Everybody follow
me so now I don't know my vertex though
I don't know what that point is all I
know is that
that
shoot I'm sorry that's not the vertex
symmetry sorry if you guys remember when
I did this
graph remember here was the vertex
right remember the axis of symmetry goes
through the vertex X right notice that's
a vertical line vertical lines are in
the form of x equals correct so I'm
sorry I just been over break I I Mis
stoed my thing so Elizabeth when I'm
describing this your ver your horizontal
line is your vertex so the first thing
we want to do is not find the vertex but
we want to find the axis of symmetry
which is the x coordinate of my vertex
a I do opposite of B divid by 2 a and I
get X4 therefore I go an X4 1 2 3 4 and
that axis symmetry always equals xals
opposite of B / 2 a
yes you're going to want to have that
memorized because you are going to have
that um my apologies I it's been a long
break but does everybody see that and understand
understand
itos okay now where is this point though
lie on this graph well to do that you're
going to have to go back to your table
so X Y we know that our coordinate is
4 right so to find the y coordinate we
got to plug -4 in for X so I do y = -4
2ar + 8 * -4 + 12 does everybody see
what I did there I took my x coordinate
and I plugged it into my equation to
find my y-coordinate and I get y = 16 -
-64 right so now I go left4 and now I go
down4 1 2 3
vertex what's
necessary all you've done is do opposite
of b divid 2 a you found that value and
then you plug it back into the equation
okay now let's pick we need to pick two
more points as I mentioned we need to
pick two points to the left or two
points to the
right what would you like to do pick two
points to the left or to the
right to the left
like well my recommendation would always
be smaller numbers right it's much
easier to pick numbers that are smaller
than going larger I mean larger in like
absolute value right so let's just do to the
the
when doing that I get y = 9 - 24 + 12
that becomes uh -12 9 that becomes
-3 so that becomes
and when I get to that um I get zero
right okay so what the main important
thing that I want you guys to see now is
I've graphed one part of my Parabola
I've graphed the right side of this but
remember these are symmetrical to get to
this point I went over one over over one up
one to get to this point oh shoot I'm
sorry it's right there I went over two up
four right I'm sorry my graph is all wrong
wrong
here so now to get to my next point I'm
going to go over one up one to get this
next point I'm going to go over two up to
there anybody have any questions on
this anybody have any questions
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