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We've officially started chapter 4 and in the very first section, we're going to learn how to graph quadratic equations.
Chapters:
00:00 Introduction
00:20 Why learn this?
01:35 Definition of a quadratic equation
02:38 What is the vertex?
05:15 The axis of symmetry
06:40 Example: Graphing a quadratic function ( the steps)
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0:01
we starting chapter 4.1 and chapter 4.1
0:05
has to do with uh quadratic equations
0:08
and how to graph them now again like I
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told you
0:13
guys before we even start the chapter we
0:15
always have to figure out why do we do
0:17
this right because that that has been
0:19
one of the questions that you guys have
0:20
always asked why we doarn this is this
0:24
even necessary so I give you a quick
0:26
example right suppose James some random
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guy named James is organizing a charity
0:32
uh tournament right and he plans to
0:34
charge a $20 entry fee for each of the
0:37
80 players that are going to show up
0:39
right now D he changed his mind say
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listen I'm going to raise this by
0:44
$5 and then what happens when he did
0:47
that five FAL players are shown up right
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so now he uses info to determine how
0:53
many entry fees will Max his profit and
0:57
if you do that you can get if you try to
1:00
put this as a graph this is what you get
1:03
so this is a this is called a quadratic
1:05
equation so the quadratic equation
1:07
pretty much gives you if you collect the
1:10
data it gives you pretty much uh an
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overview of how his profit is changing
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in terms of whether he increases his
1:19
fees or not right now in a quadratic
1:23
equation you have several things that
1:25
you need to know now that takes me to my
1:28
next thing my next uh Point here so in a
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quadratic equation the greatest exponent
1:33
is always a two this is a two right and
1:37
any quadratic equation is always of the
1:40
form a x² + BX + C this is what any
1:45
quadratic equation that you come up with
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is going to look like f ofx which is the
1:49
function equal a x² + b x + c a b and c
1:54
are coefficients of A and B are
1:57
coefficient of x² and X and then C is
2:00
just a constant right so any quadratic
2:03
equation that you get is going to look
2:04
like this now if C is z you're just
2:08
going to have something like X2 + 2x
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that's just an example right but most
2:13
quadratic equation have to follow this
2:16
specific format okay
2:18
now I remember in in our last chapter we
2:22
talked about quadratic equations and
2:23
this is the one that looks like a u you
2:25
guys kept saying U so this is now we're
2:27
going to go into in depth about this
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this type of equations okay now in a
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quadratic equations you have several
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things that you need to know first is
2:36
the vertex right vertex now the vertex
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is either the highest point on the graph
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or the lowest point on the graph for
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example here my vertex is going to be
2:48
right here the summit right this is the
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highest point so here will be the
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maximum profit that this guy makes right
2:54
so this is where he's going to make the
2:55
maximum profit yes what does it say when
2:58
it says $5 and something fewer f e w e r
3:03
you don't have to copy that it's just an
3:05
example but if you want to retain it
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that's that's fine yes what sport is
3:09
that what sport yeah is it I don't know
3:12
what tournament it's just a tournament
3:13
that he's organizing okay I don't know
3:15
maybe whatever sport maybe
3:18
hockey if you want chess I won't playay
3:20
20 buckess I don't want to play 20 play
3:23
chess most boring game that's ever like
3:26
created I don't like chess chess I like
3:30
check
3:32
Checkers Checkers for people like me you
3:34
know we like that I mean people like me
3:36
in terms of like my type
3:38
of in a good way in a good way like I
3:42
like Checkers it's quick you get it down
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like as
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CH I was trying to act all intelligent
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you like man get up you know I like CH
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check is my game all right so now this
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is the vertex the vertex is either the
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highest point or the lowest point on the
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graph on any given quadratic function
4:03
the vertex is going to be the highest
4:06
point or the lowest point now sometimes
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you have quadratic equation that go like
4:10
this this is where you want to maximize
4:12
something and when they are upside down
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you want to minimize something right you
4:17
either want to maximize it or minimize
4:20
it you want to minimize loss but you
4:21
want to maximize profit right so I
4:25
remember one day I I was studying a
4:27
certain Serv shoe and when I when I
4:30
collected the data from the shoe I put
4:31
it in the um in my table and when I when
4:35
I uh graft it it looked like a quadratic
4:38
equation basically the shoe came out
4:41
right it wasn't selling for a lot and at
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some point it picked I'm thinking yay
4:45
and then it start to go down so that you
4:48
can use you can use quadratic equations
4:50
because that is the best uh the equation
4:54
of best fit it fits the data well right
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so that's all we use this for we use it
4:58
to see if you get a maximum profit
5:00
somewhere minimum loss that type of
5:03
stuff or whatever else you want to study
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you can use a quadratic model if it fits
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the model okay now here in the quadratic
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equation we also have what we call the
5:14
axis of symmetry right and this axis as
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you can see what does it do to the graph
5:20
you can observe it and see it cuts it in
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half right so the axis of symmetry
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always cuts this in half and that's
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going to be handy when we try to graph
5:29
this function we're going to learn how
5:30
to use this to graph a function now the
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next thing we're going to talk about so
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so far all we've talked about is what
5:36
the quadratic equation is like and what
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does it look like now we're going to
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talk about how to find the axis of
5:43
symmetry and how to find the vertex
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right so to do that we need to find an
5:47
example so let me go to the textbook
5:50
here so we have a function f
5:53
ofx equals
5:57
to remember this is being it so you
6:00
don't have to wor but you can go back
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and watch it right let me just erase
6:03
that whole thing yeah you can it's going
6:05
to be available online my not right so
6:09
x² +
6:12
4x right - 3 so this is a quadratic
6:16
equation this is a quadratic function so
6:18
now with this quadratic function we're
6:20
going to try to find the axis of
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symmetry we're going to try to find the
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Y intercept and we also going to try to
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find the vertex right so there are three
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things that we want to find we want to
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find the
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axis of symmetry right we also want to
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find the
6:38
vertex and we also want to find a y
6:41
intercept with these three pieces we
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will be able to graph this quadratic
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equation now why is this quadratic
6:49
because it fits the model what ax² + BX
6:53
+ C right it fits this model this is our
6:56
model A X2 + b x + C that's the
6:59
quadratic Mar so now here what is the
7:01
value of a in this
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equation a is one B is and then C is C
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is3 right so a we need all this a is one
7:15
B is 4 and then C is -3 now the first
7:19
thing I want to find is what my Axis of
7:21
Symmetry right I want to find it so
7:24
number one we need to find the
7:28
axis of
7:32
you want to find this so to do it we
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have to follow the
7:37
formula and the formula is what x equals
7:40
what B over 2 a so this is how you find
7:45
the axis of symmetry right so now here
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what is B four so negative V will be4
7:53
over 2 * what 1 right be2 so that'll be
7:57
-4 over 2 which is 2 so our axis of
8:00
symmetry is x
8:02
= -2 is going to be is the axis I just
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put as I don't want to keep writing axis
8:10
of symmetry I'm just getting tired of
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that right so this is our axis now the
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next thing we want to find is we want to
8:17
find the vertex right the vertex so next
8:21
two we want to find the
8:24
vertex so to find a Vertex this is the
8:27
formula for the vertex right B over 2 a
8:30
and F of B / 2 a so do I already have B
8:36
over 2 a yes or no yeah yes so I already
8:39
have the x coordinate of the vert is -2
8:42
now to find the y coordinate all I have
8:43
to do is what take this guy and do what
8:46
to it plug it into the equation plug it
8:49
into the equation right so I need to
8:51
find F of -2 I remember we did this in
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one of the first chapters right I need
8:56
to find this point so I need to find F
8:59
of 2 to find F of -2 all this is saying
9:02
is what I'm going to take -2 and I'm
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going to plug it into the equation so
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let me find F of -2 right so F of -2 is
9:11
-2
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2 + 4 *
9:17
-2 - 3 right which is 4 - 8 - 3 4 - 8 is
9:27
-44 - 3 is7 so F of-2 is7 so therefore
9:33
my
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vertex the vertex
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is -2 and then -7 right now I need to
9:47
find this the third piece is my my Y
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intercept Nathan how do I find the Y
9:53
intercept you're going to plug your -2
9:57
now the y intercept
9:59
Y intercept yeah the Y intercept is when
10:02
X is
10:03
what zero right X is zero so I need to
10:06
find the Y intercept so I'm going to
10:08
find F of what F of 0 right so that is 0
10:13
2 + 4 * 0 - 3 which is -3 and again the
10:18
Y intercept is not just is a point so
10:21
you mean it has a x coordinate and a y
10:23
coordinate right so the x coordinate is
10:25
zero and the y coordinate is what we
10:27
just found so our y intercept is what Z3
10:31
and3 now with these three pieces of
10:34
information we can graph this function
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okay so now let me show you how the
10:38
graph this based on that so
10:45
now there's also one thing that we need
10:47
to learn right a is equal to what we say
10:50
is one right because a is one the graph
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opens
10:54
what upward right it's going to look
10:57
like this because a if a is positive
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right the graph looks like this when a
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is negative the graph looks like this it
11:06
opens downward so this is also another
11:08
piece of information that we need to
11:09
keep when a is positive you have a u a
11:13
regular U and when a is negative you
11:15
have a flipped over you right wait we
11:18
have
11:19
absolute no absolute value here don't
11:21
worry there's no absolute value so do
11:23
not even mention that let you cause
11:26
trouble right okay so now we need to
11:28
First first the first thing that we need
11:30
to graph is the uh the vertex what's the
11:36
vertex -27 right 1 2 -2 oh no -2 -7 so 1
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2 3 4 5 6 7 right so that's the
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vertex right that's our vertex now the
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next thing I need to graph is my Y
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intercept right I need to find a y in
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Step what is it 03 so it's right here
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right 03 1 2 3
12:07
right now here's the thing since a is
12:10
positive the graph is going to open what
12:13
upward right but I also need to know
12:15
what my Axis of Symmetry right so now
12:18
let me do this first so because it's a
12:20
curve it's going to go like that right
12:22
but it's a curve this is just half of
12:24
the curve I need the second half right
12:27
to get the second half is my Axis of
12:29
Symmetry is what my Axis is -2 right so
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this line here we usually use the dotted
12:35
line for it is going to cut my graph in
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two portions so that means all I have to
12:39
do now is do what just get the second
12:41
portion like this because this is going
12:43
to cut it in half does that make sense
12:45
right so all we need to graph it uh is
12:48
we just need the vertex the axis of
12:51
symmetry and sometimes the Y intercept
12:54
because sometimes if the equation does
12:55
not have Y intercept you're going to
12:57
need a different point in this case this
12:59
is perfect because we have a y intercept
13:02
we can graph it all right so it's pretty
13:05
simple so now we're going to look at
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some other uh way to graph this so let's
13:12
let's it's pretty
13:15
simple let's graph this function here
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can I erase this yeah all right so let's
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graph another
13:23
[Music]
13:25
function so let's say we have f ofx
13:30
right
13:32
equals -5
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X2
13:37
right - 10
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x + 6 right so what do I need to find I
13:43
need to find what your vertex my vertex
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right I guess yeah yeah I need to find
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my vertex so what's my
13:54
vertex is X x equals what negative what
13:59
2 B first first before we do let's put a
14:03
is equal to
14:04
what
14:06
B and then C six right so since a is
14:10
negative is the graph going to go up or
14:12
down like that down down right so it's
14:14
going to be a graph that looks like this
14:15
so let's just keep that in mind as we
14:17
working right we already know that so
14:18
now let me find vertex right I need to
14:20
find the x coordinate so let's put
14:22
vertex
14:23
first so X is what B over 2 a right oh
14:28
yeah so that be what
14:31
Stephanie so-1 right over 2 *
14:35
what 5 so this is 10 over 10 10 which
14:41
is1 right so the x coordinate is 1 now
14:44
to find the yord what do I do I'm going
14:46
take this guy PL plug it into the
14:49
function right so that gives you
14:52
what5 * 1 2 right - 10 * -1 + 6 this is
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negative here it's a it's Square so
15:01
this5 right + 10 + 6 16 - 5
15:08
is 11 right no n oh yeah 11 11 16 all
15:14
right so our vertex is -111 right now
15:18
what is my Y
15:20
intercept we don't have we have to think
15:23
twice what is it if you replace those
15:25
X's by Zer the Y what six so 0
15:29
right do we still need to find the xaxis
15:31
Symmetry let me let you in on something
15:33
we already have it is what1 is always
15:36
going to be the x coordinate of the
15:37
vertex right so now we can graph this so
15:41
now let's graph
15:53
it so vertex is -111 so 1 2 3 4 5 6 7 8
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right right
16:02
somewhere right Y intercept 0 6 0 1 2 3
16:07
4 5 six right so remember the curve is
16:09
going to go like that right and then I
16:12
just need to draw my what my AIS of do
16:16
you have to draw that repeat huh do you
16:19
have to draw the yeah you do why just
16:21
for the sake of making sure you're
16:23
showing that you know what you're doing
16:24
CU he cuts the graph into two halves and
16:27
this is it right
16:30
yeah I don't get you don't get
16:34
what all right what did I gra
16:38
first what is this point
16:41
called no that's what is this what term
16:45
do we use for it vertex and this is the
16:48
Y intercept right so we put the vertex
16:51
the vertex is either the highest point
16:53
or the lowest so once you get it right
16:56
you find another point on the graph and
16:59
connect those dot into a curve like
17:01
right and then you know that whenever
17:04
you get it your axis of symmetry is
17:06
always in the middle is going to cut
17:08
this into two halves so if you get one
17:10
portion you can repeat the other portion
17:12
that make sense all right so now the
17:15
terms that we have that we use in this
17:17
chapter for um the vertex right I'm
17:21
going to tell you what they are so this
17:24
type of vertex here because it's our
17:27
highest point what we going to call it
17:28
we going to call it the maximum right
17:32
maximum so maximum mean this the highest
17:35
point on the curve that's the term that
17:37
we use so whenever you here find the
17:39
maximum they asking you to find the
17:41
vertex right and then if you have
17:44
something like
17:47
this minimum that's right minimum right
17:50
there you go that's a minimum because
17:52
that's the lowest point on the graph
17:56
right so you have two things you have
17:57
the minimum or the max maximum and he
17:59
always the vertex the vertex can need be
18:02
a maximum when he's the highest point on
18:04
the graph or a minimum when he the
18:05
lowest point on the graph all right yeah
18:08
do homework don't ask me that right now
18:11
you just something my like I'm I'm in
18:12
the middle of something you ask me about
18:14
homework wait okay now let's work on U
18:17
word
18:19
problem what
18:23
what
18:25
mhm let's work on the word problem here
18:28
and then we can call it to there so I
18:30
can turn this off now
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