desmos.com, a free online graphing calculator by a math teacher named John Rowe and somehow

it go retweeted into my timeline and I couldn't be happier that it was because I love it but

before we play let's fuel up on gamer knowledge. When Mario jumps, in fact whenever anything

jumps, whenever anything is tossed into the air here in Earth's uniform gravitational

field, the path it follows is the shape of a parabola which means it can be described

by a quadratic function. A function is just a mapping between an input set of numbers

and an output set of numbers such that every number in the input set connects to one number

in the output set. Now if there's a rule that allows me to easily when given an input

number tell you which output number it's connected to that's awesome. And that rule

might be for example, take the input number and double it so if you give me two I know

that the two in the input set connects to, what is two doubled, four. It connects to

four in the output set. Uhh ten in the input set connects to twenty in the output set and

so on. Every single one of those pairs, two four, ten twenty, one hundred two hundred,

that entire collection of pairs is the function itself. Here on my screen I have an equation

y=x. If y represents numbers in our output set and x represents our input set then we

can actually create two axes. This horizontal one is an x axis and this vertical one is

a y axis and we can plot all of those pairs of points and create a beautiful line in the

case of y=x. Y=x means that if you give me an x, an input value of say 3, right here,

I know what the output value will be. Since y=x the x value being 3 gives us a y value

of 3 right there, boom! Whoops I hit 3.01. Right there, boom! That was 2.99. Right there

(3,3). That ordered pair is in our function but look we're not here to talk about just

any old function. We're here to talk about quadratic functions. So let's raise x to

the power of 2. Boom! A parabola. Here every value on the x axis is paired with the value

on the y axis that equals that x value squared. So what is two squared? Four! (2,4). Boom!

Almost made it. There it is! But what is -2 squared? Well that's just 4 as well so (-2,4)

is also in the function. You might notice a problem here. This does not look like a

path Mario would follow when he jumps. We might need to transform this graph a little

bit and that is exactly what Super Mario Quadratics is all about. So let's talk about how to

transform the shape and position of a parabola on a graph. Let me first of all try to shift

the parabola up and down. Right now it's vertex is at (0,0) where x=0 and y=0. But

if I wanted that to be higher it really wouldn't be that difficult. I would just add a number

here because y=x^2. But if it equals x^2+1, hey! I've just shifted the whole parabola

up one unit. You give me an x, say 0, I square it, still 0 but then I add 1 and we've got

(0,1). Okay but what if I wanted to move it down I think you're probably way ahead of

me, just subtract. Now I can over it down. I can move it down 1. I can move it down 8.

I can move it down just half a unit. Not too shabby. But what if I wanted to move it left

and right. Well now I have to change something about the x value that I'm putting in. Before

I square it would be ideal. So take a look at this. If I don't just make y=x^2 but

instead decide to do something like adding, whoops! Adding, 1. Hey! Look at that! If I

add one to x before it's squared the whole parabola shifts to the left. So it goes, it's

vertex, in fact every point moves in the negative direction and that's because now whatever

x value I give you doesn't just gets squared. Instead the value one above it gets squared

and that becomes the y value. So to get for instance y=0 I need an input value, an x value

that is such that when you add 1, it's square is 0. And that would be -1 for x. -1+1 is

0 squared is 0. Okay so adding to the x brings us to the left which means let's just check

to make sure, subtracting from x brings us to the right. Beautiful. Perfect. But now

let's make the parabola fatter or thinner. If I multiply x^2 by a number it will become

larger. So imagine this point right here, (2,4) and by point I mean ordered pair. If

I say that y is equal not to x^2 in this case, in this case x is 2, but instead to say x^2x2

then this value wouldn't be 4, it would be 8. Every value is going to be larger which

means the parabola will become thinner. It will grow a lot more rapidly. So let me just

do that. Let me put a 2 in there. Ooh! Let me put a 3, yikes! Whoa! Actually let me have

a lot of fun. I'm gonna make a variable. I'm gonna call it c. I'm gonna add a slider

and now I can choose whatever value I wanna multiply x^2 by. Here it's just 1. So it's

just x^2. But if I make c larger the parabola gets thinner and thinner and thinner. And

if I make c smaller, less than 1, it gets fatter. Whoa! Did you see what just happened.

If c becomes negative every value reflects around the x axis and the parabola is now

upside down. This is a beautiful discovery because this looks a lot more like Mario's

jumps. Mario doesn't jump like that. He jumps like this so we want this inverted parabola

and we can do that by just making the y value when given an x value negative whatever x^2

would be. I think we are ready to Super Mario Quadratic. So here's the tweet where John

Rowe announced the game and I was actually waiting a long time for this to come out.

I had seen little teasers of how it might work and I was like alright I gotta play this.

As you know I'm kinda a gamer. And when it finally came out I was all over it. So

here's the game. Let's start with just World 1-1. Beautiful. So here's Mario. We

want him to catch that coin and the rules of the game are that Mario just runs along

the ground until he reaches this dotted line which is our parabola. And at that point he

will jump and follow that path until it intersects again with the ground and then he'll continue

on with the ground. So let's say I do nothing and I keep this equation the way it is, if

I have Mario jump he's going to not make it all the way to the coin. Alright so let's

reset and think about this a bit more. I want this parabola to go up a little bit higher.

And as you can see it's vertex is already at 1. So I need that to be at 6. Oh look!

All we need to do of course is add a number so let me instead of adding 1, add 6. Piece

of cake! So I don't wanna make Mario jump just yet. I want his jump to be even more

radical which mean for me I want his jump to be steeper and as you can see right now

we're multiplying, we have a coefficient in front of the x of -1 but if we make that

number bigger the parabola will be skinner, it'll be steeper, the numbers will change

more quickly as we run through choices of x. So let me make that negative, should I

make it -10? Okay. I don't care, my mom isn't gonna watch this, -20. Yes! Okay,

do it Mar-io. I call him Mar. As you can see the coin is spinning and I get a next coins

button. What I love is that there isn't like a specific equation you have to put in

as long as you reach the coin you get to move on. Alright. Ay caramba! This is gonna need

a fatter parabola. Do you think -1 is enou….ohh. No we need, we need, something a little fatter.

Is 2 enough? Oh well no, 2 makes it thinner. I need smaller than 1. How about .5. Psh!

Uhh some call it luck, I call it the game zone. My brain's always in it. Jump. Let's

see if this works. Nailed it! Nailed it. No big surprise there. Oooh! Try to throw me

a curveball by moving the coins to the left. I know how to move a parabola. I need to add

to x before it's squared. So instead of subtracting by 10, let's only subtract by

5. Actually let's, it looks like 7. There's a convenient, there are units along the axes.

So let me try 7. Ohhhh! Somebody's a genius and it's all of us because when you learn

we're all winners. Let's move on to a different world. Here's…ooh! World 1-2.

Ahh now here we need to jump over the lava and the parabola as it already exists is,

well it needs to be inverted so we already know that we need to make this coefficient

in front of the x negative. Now we need to raise it up to it looks like about 10 and

we raise it up by adding 10. Perfect! And then I need it to be shifted to the right

10 which means subtracting 10. Hey! Oh it's not fat enough is it. Mario's gonna fall

right in the lava so to do that we need to multiply by a smaller number so the coefficient

should be maybe -.1. I seriously have only played the first level. I wanted this all

to be a surprise but when you're as good as I am no game is a match. Okay. Fifth World,

sixth world, give me some kind of new challenge. Ooh! These are looking…wait…Bowser has

the princess locked in the room. You need to defeat him to unlock the door. There's

a button “Defeat Bowser.” Yeah okay I'll just click the button. Oh that just launched

the puzzle. Enter a quadratic below to jump on to Bowser's head until you have defeated

him. Okay so. I wonder if I like land on him with more force if I jump higher. Well let's

try y=, I know that I want this to be negative because I wanna jump up and come down so I'll

do a negative, oops, -x^2 okay. But I need to shift this on the x axis to the right so

that means subtracting but I'll subtract like 10 perfect. Okay now I wanna jump up

nice and high so let's add, I mean let's add 8 so I wind up at the top of the page

and then I need this to be fat enough that I actually land on Bowser so let me try making

this fatter but using a coefficient of .5. Not really enough. How about .3. Yeah! Let's

see what happens. Oh my gosh! Now I knew math was fun but I did't know it was….wait did

I just…did I win? Is he gonna stop spinning? Change equation. Oh okay now it's time to

go again. I'm gonna do what's called a pro-gamer move. I'm not even gonna make

a quadratic equation. I hope it doesn't break the game. I hope it doesn't mean that I

forfeit because I'm cheating or something but watch this. Instead of squaring I'm

going to raise x-10 to the power of 3. Oooh! Look at that. That's a pretty cool dive

bomb move. I'm gonna shift this thing back a little because I wanna hit Bowser. Plus

1. I think that's gonna be good enough. But you might say you know Michael what I

don't understand is that Mario, wait if I start from here what happens? Oh it doesn't….oh

dive bomb! So Mario just starts the beginning of your function. Oh wow! So I could choose

all kinds of things. Let me do y=sin(x) yeah. Okay. But I want, I wanna actually hit him.

So let me, I don't really know how to change the sin function. But let me obviously if

I just subtract 1 it moves it down. Okay perfect. So let's, wait what if I did, what about

the sin of x^2 what does that look like? Ohh! Yes! Okay this is gonna be good. I'd like

to rotate it. I wonder is there a way to rotate this, what if I multiply it by another sin

function. Oh that looks pretty sweet. But it's not hitting Bowser so let's move

it down a little bit. I'm gonna just take this whole equation and I'm going to subtract

3. Yeah! Start. Oh go through the door! Okay so not using a quadratic was kind of a super

move. I think I'm ready for the ninth challenge, Mario, Mario Help Me. Save the Princess. Now

look okay this game obeys mathematical rules. Like I could literally, I could just put in

y=6 and I get a straight line and Mario just follows the straight line and reaches the

princess and like I've won. But that's not fun is it? No not at all. What we want

is to have some fun and collect that star or do we? Let's try another weird function.

And by weird I just mean not a quadratic. What if I just do tangent of x. That's gonna

give me, oh I wrote this all wrong. Tangent of x. One thing this helps you do is you really

learn how to properly use notation because you have to and you see the results immediately.

If I just have Mario follow tangent of x, oohhh! Hey! That worked! That was actually

Luigi's voice. What if I did this. What if I had y= okay x. But I could do a piecewise

function so I can say that from, for x is less than 6, y=x. So now you see there's this

little dotted line there, that means that the output value, the y value is just equal

to x up until we reach x=6. I might go a little further than that. Let's try 8. oops not

89. Okay beautiful. Now let's say for x is less than uhhh 14 I want the function to

simply equal 8. So now it'll be a straight line. Then for x is, let's do, I'll actually

do a parabola. I'll do a quadratic right there. For x is less than 14, y=, well let's

just put in x^2 and see what that looks like. To put a little parabola here we need it to

be shifted 16 over so I need to subtract 16. And then I need to raise it up 10. So that

means plus 10. Now where is it? Oh that's squared should not be on the outside. I'm

telling you this is like the best way to learn how to do proper notation. Okay so for x less

than, not less than 14, less than like let's say 20. There it is! Oops! I forgot to invert

it. Negative sign. Beautiful. Now I wonder what happens at this discontinuity. Well I

can make that work. I can make it more pleasant by raising this up a little bit higher. I

don't want 20, I want like 19. Or no I want like 18. Yeah! Beautiful. Okay now let's

fatten this parabola out by making that coefficient smaller. If I do like .3, ahh that's pretty

good. .2? Nah. .4? That looks really nice okay great! And then finally for x less than

ya know 30, I want the equation to be like umm well I wanna go down. So I'm going to

have it equal, let me make it like .2x. Ahh there it is okay great! So I wanna make this

-.2x but then I wanna raise it up so let me add like 10. Not enough? How about I add 13.

Ooh too much. 12? That's beautiful. Alright this is my grand finale, for x values that

are less than 8, y just equals x. So we get this 45 degree angle line right here. Where

we go up to this value on y which is 8 when x=8. But then for input values between 8 and

14 y just equals a constant and that constant is right there its 8. So we just ride right

along y=8 until we reach 14 and for those values between and 18 I've got a parabolic

shape, a nice little quadratic function so I hop up and get the star and then after 18