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 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
for any x below 30 that's also greater than
18 I have a linear function which is the rule
connecting the x’s to the y’s is that
the y=-1/5x+12 sending me right to the princess.
You guys ready? Here we go. And *kissing noises*
Success. I did use a quadratic here. Well
done, you've saved the princess. Your quest
is over and I feel so satisfied because I
did use a quadratic in there but I had a lot
of fun. I learned a lot about how rules that
connect numbers to other numbers can be shown
graphically and I learned a lot about myself
today. And I hope you learned a lot about
me too and as always, thanks for watching.
