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  • Today I wanna play my new favorite game, Super Mario Quadratics. The game was created inside

  • 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 lookingwaitBowser has

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

  • a buttonDefeat 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 justdid 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