Okay, good morning, class,
Today are going to do graphic...
(*kids and Robert laugh*)
Graphing quadratic functions,
are you ready?
(student: YEAH!)
Yay Math? (students: Yaaayyy)
(student: Yay Math!)
It's good, now we know what to do;
Oh yeah, math
I seriously was like "what do I do now?"
Oh yeah, we're filming
y = 2x² - 4x - 5,
this is the last major topic that we are going to do
It's hard
But don't be worried, don't be afraid,
because Captain Math is here
To start... Math Enterprise
Dududuuuumm, I've never watched Star Trek before!
I'm a hero too
Oww, the pin hurt me, it's really sharp
I'm a sensitive hero
A quadratic function is a fancy way to say Parabola
Parabola is a shape that looks like a "U"
A little bow
The reason we can tell that it is a parabola is that the x value is squared
Remember, in graphing a line, when we graphed a line,
it was y = mx + b
This is a line right? 
(student: Yes)
Notice, feel free to participate
This is a line because y and x are not squared
But this would be a parabola because x is squared
If you square the x it's a parabola,
if you don't, it's a line
(student: Do you square the b?)
The b is just a number
So it would just be a number squared,
that's another number
So let's look at this, this is a parabola
First, let's identify a few terms here
The numbers in front of x², x, and here
These are the Coefficients,
and this one is called a, this one is b,
and this one is c,
So we have to identify the coefficients
of every quadratic function
Of every parabola
Judging by this parabola, what is the value of a?
Any guesses? (student: 2)
The value of a is 2
So, a is the number in front here, this is 'a'
(student: So whatever the actual number is,
is the coefficient?) Yeah, that's right
It's not x or x² or anything,
it's just numbers
a, b, and c are numbers,
we are going to write that here
a = 2
What's the value of b?
(students: 4)
4 or -4?
It's going to be -4
The value of b is -4
So it could be plus 4, or another problem would be
+4 or -3, the sign is included
So b is -4
What's the value of c?
(student: -5)
c is -5
Okay, so that is crucial,
identifying a, b, and c 
a, b, and c are always numbers
not x or x² or anything like that
Question?
(student: Can you spell "coefficients"?)
Yes, can anyone spell it?
(students: c-o-e-f-f-i-c-i-e-n-t)
Here we have a, b, and c.
Now, the way to graph this parabola
This quadratic function,
is that if we agree that it's like a bow, we need to know the starting point
That point is called the vertex,
the bottom value
It's going to look like this
So we need to know this value.
We need to know where it starts
This is called the vertex,
The vertex will be in the form...
Vertex is a point
So we have to figure out what the point is
*caption note: no pun intended*
And the value of this point has an x and a y,
the x is, and this is important...
-b/2a
-b/2a is the x value of the vertex
And we're going to do this together, question?
(student: Are vertexes only on parabolas
or also on a line?)
Well, the line can't have a vertex
because if we're talking about a line it's just point, point
There is no starting point and no ending point
Parabolas have a vertext, it's like a starting point
It's the crux, it's where it begins
So the vertex always, every parabola has a vertex
in which the x value is -b/2a
So let's plug in. Yes?
(student: What if our b is negative?)
Then this will be positive up here
Good question, you're already on it.
(student: I don't understand, does that mean that x will always be a fraction?)
Now that we establish a, b, and c
(student: Oh, you're just solving)
Now we have a, b, and c we don't need
the parabola anymore, really
We don't need this equation,
a, b, and c are the real... things
So what is -b?
(students: 4)
Yeah, it's -(-4)
(student: So it's positive 4?)
It's positive 4,
So 4 over... what is 2 times a?
(students: 4)
What is 4/4?
(students: 1)
(student: How many of these do you typically have?)
One vertex
(student: So why are we doing three?)
No, we simplify it
The ground is shaking
It's a starship enterprise!
I need more power
Okay, earthquake is done
(student: Are we in the sky?)
Oh, yeah, we're in the sky
Pretty stars
WOAH shooting star!
We have panoramic vision
What's that thing called?
Like a window...?
The window goes all the way around?
A beautiful room
Question? Who had a question?
(student: I think I answered it in my head)
Nice!
Why am I here? Ahhh...
Alright, so the vertex is (1, something), correct?
Let's make... Oh did you ask the question "why we do it three times?" (student: Yeah)
We didn't do it three times, this is the beginning,
we put in numbers so we simplified
So this is the vertex, here
Let's make a T-chart
Containing values of our parabola, so far it's
(1, something)
Any guesses how we get x's y friend?
(student: OH!)
(student: You plug in x to each of those where x is)
Yeah, you plug in x here and you will get y
(caption note: no pun intented)
Let's do that, here we go
I'm going to take this off
(student: So the vertex is your x?)
The vertex is x and y
the x value is -b/2a
The y value is when we plug x in
So let's do that
So y = 2(1²)
(student: you square the 1 first and then multiply, right?)
So this is the value of our equation
and now we are going to place some x
Here we go, 2 times 1² is?
(students: 2)
-4 is?
(student: -2)
-5 is?
(student: -7)
This simplifies to -7, questions?
(student: so 2*1 is 2 and 2² is...)
No, no, no. 1² first, order of operations
Exponents before multiplying
Exponents first, 1² is 1
Other questions on this? Okay.
So this, we agreed, was -7
(student: So that's our point, are we done?)
No, we're more than halfway done though
This is the vertex, let's graph this point
(student: Does it help to use graph paper?)
It helps to use graph paper, yeah
Where do we graph it? Let's graph it here
(1, -7) is here
So this is our starting point
We agreed that we are going to have this kind of parabola coming up here
We don't know where exactly
So let's pick a point to the left of this,
and a point to the right of this
And then we will graph it
So what would be a good point, say, to plug in for x?
An easy one to plug in for x?
(student: -1?)
Even easier than plugging in -1 (students: 0?)
0 is always a very good choice for plugging in
Always try to plug in 0, it's so easy
So we plug in 0 for x
(student: Wait, why?)
Because we want to find other points on the parabola
So we can graph it
"Why" is a great question, so we plug in 0 now
So when x is 0, y is what?
(student: -5?)
-5
Questions?
(student: uh yeah... why?)
What is 2*0²? 0.
(student: So it's all going to be 0?)
This is 0, minus, what is 4*0?
So you have 0 - 0, what's that?
(student: And 0 - 5 is -5)
That's why it's -5
(student: I get it, I get it.)
If you don't, that is okay, you can ask
Ask if you have a follow-up, please
So (0,-5) is our next point, let's graph it
There it is, here is (1,-7)
And we said (0,-5) is here
(0,-5) we have, and we start the parabola
(student: How do you know... never mind)
(student: Where do you draw the line?)
(caption note: once again, no pun intended)
What line?
The parabola line?
You mean this one?
Over here? We don't know this
(student: Why do you need the vertex)
Well we need a starting point,
We got that, that's our vertex
And we need another point so we know where to go
If our other point was here,
our parabola would be a lot wider
If it was closer our parabola would be thinner
In this case, the point that we got was (0,-5)
Alright, so we need points, now we need to know where to go on this side
So here is my question to you;
put on your thinking caps
Star Trekians
*cool sound*
Sound effects rule
Wouldn't it make sense that this parabola would be perfectly symmetric 
As in, whatever is on this side
it would look exactly like on this side?
(students: Yeah)
So wouldn't it make sense that
you can never have something like that?
You can never have something like that
because this is not even
Right?
(student: Right) Not even
Where do we go to get (0,-5)?
How far to the left [do] we go, and how far up do we go?
(student: We went up 2, and 1 to the left)
So we went this way 1, and we went up 2
Wouldn't you agree it would make sense
that to get this point
That we go this way 1 and up 2?
That's the idea
That would make this perfectly even
So we go to the right 1, and up 2, again
Notice how that's going to be,
hopefully if I do it okay
That's going to be perfectly even,
watch this point here
What is the value of this point?
(student: It is (5,2))
Almost
(2,-5)
Now, observe...
(student: How did you...)
How do we get this?
So, from 1 here, this is the value for x, 
from 1 we went to the 0
we went 1 this way and up 2
Correct?
So, to make this perfectly even we should go 1 this way
and up 2
That way the parabola will be perfectly symmetric
So it's like, if I'm the vertex now,
looking up at the parabola
This arm happens when I go this way 1, and up 2
So that's this arm, that's that point
So wouldn't it from the vertex, to make this one
I should go this way 1 and up 2?
That will make a perfectly symmetric bow
And that's what we did, yes?
(student: Is it always going to be perfectly symmetric?)
Yes, parabolas are always symmetric
(student: And you always go from the vertex?)
Always from the vertex
And, observe this!
What do you think would happen 
if we plugged in 2 into the equation?
What would result?
If we plugged in 2 here?
-5
If we plugged in 2 we would get -5
Can you try that for us?
So plug in 2 and 2 
Let's find out, I'll help you out
(student: 2*2 is 4; 4² is 16)
Okay, so you're doing 2(2) first,
you want to do the 2² first
2² is?
(student: So 4*2 is 8)
This is 8, so I'm going to write 8 here
Minus 4*2? so minus...?
(student: 8)
What's 8 - 8?
(student: 0)
0-5?
(student: -5) Here it is, -5
Few more things with this
There is a line that we can draw within this parabola
that will cut it perfectly down the middle
Will this line be horizontal or vertical, maybe?
(student: Wait, why are you making a line?)
So what's going to... aahh... so many thoughts, right?
What the book will ask you is for what is called
axis of symmetry
So the axis of symmetry is the line that we can draw
that will cut this perfectly in half
What will the line look like?
(student: Vertical?)
Vertical line, where?
(student: On the vertex)
On the vertex, correct
This line cuts the parabola perfectly in half
What's the equation of a vertical line always?
What does it start with?
Equation of a vertical line?
y equals? x equals?
x equals
x=
x =; what value for x here?
1, x = 1
Here's a big hint; the axis of symmetry is always the x value of the vertex
I'll say that again, the axis of symmetry is always the x value of the vertex
So there you go,
x=1 is the axis of symmetry
And, umm, that's it
That's your parabola
This is it, the problem would say graph this
You say "Okay, here's the graph"
And they say "What is the vertex?"
You say "Vertex: (1,-7)"
They would say what is the axis of symmetry?
(student: 1)
You would say "x=1" that's a common mistake
It's a line
So the equation of a line...
(student: oh, (1,0)) 
No, that's a point, (0,1) is a point
This is the equation of a line
"x=1" is the equation of a line
And one more thing, it will ask you
"will this parabola open up or down?"
In this case we graphed it, oh good,
we can tell by the graph that the parabola opens up
There's another way to tell if the parabola opens up,
only by looking at it
And that has to do with 'a'
If a is positive, it opens up
if a is negative, it opens down
Let's write that down
a postive, opens up
a negative, opens down
So you can look at this equation right here,
And if I said "Yo! Does this open up or down"
immediately you would say...?
Up, why?
(student: Because a is positive)
What is a?
(student: Positive)
What is the value of a?
2
2 is positive so it opens up
We are just going to do one more, that's it
You ready?
(student: No, one second) One second...
One more second?
(student: Like, three)
HA HA
(student: That's your superpower)
Oh my energy water!
Oh, I'm actually advertising falsely, never mind
(student: That was the longest single problem that we have ever done)
Congratulations
(student: Thanks)
*Robert buzzes*
Which way does this parabola open? Up or down?
(students: Down!)
Everybody is like, halfway through the problem
and it's like "what's the vertex?" "Down!"
Aren't we on that problem?
a equals?
(student: -3)
b equals?
(student: -6)
What's c?
(student: 4)
(student: Isn't b 6?)
-6, include the sign
Oh, you're talking about the...
No here
Good looking out, friends
(student: We love you!)
Love you too
(student: Why do you have to put so much on there?)
Because we didn't see it over here
In the frame
Okay, so great! What's the value of the vertex?
How do we get the vertex?
(student: x is -b/2x...) Okay, let's do it 
Vertex; (-b/2a, something)
What is -b?
(student: it's 3)
(students: 6)
-b is 6
What is 2 times what the value of a is?
(student: Is it always going to equal 1?)
No, no, coincidence 
I'm staying close to home, you know
(student: I like it)
The vertex can be anything
I'm just making it...
(student: Oh, I have a question) Yes?
(student: So if you get 3/4 as your vertex
you just plug in 3/4?)
So the vertex is a little less then 1, something
Question?
(student: Wouldn't it be -6 and not positive 6?) 
So what is 6/-6?
(students: -1)
And so, -1
(student: Wait, I thought it would be positive 6 
on the bottom too?)
2a, right? What's a?
(student: -3)
So what's 2*(-3)?
So, -1
What do we do with our -1 to get the y?
(student: You plug it in)
Plug it in, any questions about that
or have we picked that up?
That was important
We're going to plug it in sooo carefully with the negatives
y = -3( )² - 6( ) + 4
So, we are going to plug -1 now
Volunteer, you got it? Cool!
(student: Could you just wait a second)
Yeah, wait, wait, wait
Yeah, wait for a sec
And go!
How do we do it?
(student: (-1)*(-1) which is 1)
Good, exponents first
(student: Then you do (-3)*1 which is -3)
Right, (-3)*1 is -3, good, keep going
(student: Then you do that, which is +6)
(student: Then +4 and you get 7)
Seven
(student: That's ironic) I know
It's just coincidence
Irony would be completely unexpected,
(student: I didn't expect that)
No, irony would be amusingly
and humoristically unexpected)
Like, you work so hard to get all kinds of money; 
thinking you would be happy,
But then you work so hard that you don't stay healthy,
(student: That's not amusing)
I know, and then you have to pay all the money to medical bills, and so that's irony
(student: Thank you)
But this is just a point
(student: Isn't irony, like, when a fireman's house...)
Yes, a fireman's house
(student: Or a police commits a crime)
Or a police commits a crime
Yes, that is technically ironic, but maybe also very sad
So there we go, (-1,7)
Let's graph this point
(-1,7)
1, 2, 3, 4, ha, he, hennigg, ih
When I count I don't speak English
I just he, hee, hohoho
Now what do we do?
What do we do now?
(student: You plug in different points)
Okay, what's a good point to plug in?
(student: 0)
Let's plug in 0
I'm taking off the vertex, we're almost there
(-1,7), thank you, and now 0
When x is 0, you got it? Alright.
(student: 4)
(student: Wait... never mind, don't worry about me)
It's okay, I'm glad , if I am going too fast, say it
We're going to, together, WE!
So we plug in 0,
0is the easiest thing to plug in, you can plug in right here
0 - 0 + 4
And 4
(0,4) here
And notice, it's looking like it's going to open down now
Based on our 'a' being negative, remember?
a being negative, it's going to open down,
the graph confirms that
(student: What do you mean "open down"?)
(student: So, like an 'n'?)
like a head, yeah
(student: No!)
Like an alien head, ni ni niii
(student: Oh, it's perfect for that!)
Like a bowl and then you're like...
Like, what if this thing had been a 'U',
here, like this
It's my hat
*students laugh*
Superhero fashion bowl
Remember, we talked about how to make this symmetric?
From here, what's this point? (-1,7)
Where do we go to get the (0,4)?
How far which way and then... you said it in the back
That was great, let's get someone else
How do we go from here to here?
(student: Down...)
Down, how much?
(student: By 3)
So we went down 3, how far this way?
(student: 1)
(student: How do you know which one you went down and which one you went up?
y is going down, x is going left or right
We went down 3 and over to the right 1
So to make it symmetric, which way do we go?
(student: Left)
Left, how much? (student: 1)
And how much down?
(student: 3)
So when I put a point here, and that goes like that,
what's the value of that point?
(student: -2)
(-2,4)
(student: It's sort of like an 'A' without the bar)
It should be curvier, I just didn't draw it curvier
And what is one thing we haven't done in this problem?
Axis of symmetry
The axis of symmetry is what?
Ist it a line or a point or a...
(student: A line)
...or a bird
It's a line, right
(student: x = -1)
x = -1
The line that cuts this is half is the value of
x = -1
Alright. Questions?
It really amazes me how it's been,
what, 20 minutes or so
We can do a new topic and know
everything that's going on
Practise a little at home, you will be great
It's going to be fine
Says Captain Ahdooties
He does the Star Trek dance
More power, Captain!
Okay, thanks!
(students: Bye!)
