This video is really cool, you have a super simple derivation for the transformation too. One thing I think you should have included was why using a rule to rotate the point counterclockwise ended up rotating the graph clockwise. It was because plugging the new expressions in for x and y was basically saying "the counterclockwise rotations of these points satisfy the equation." So the said points would be a rotation in the clockwise direction from the original equation (very similar to how replacing x with x+2 in a function actually moves the graph to left by 2, instead of the right). To go in the standard counterclockwise direction, you can plug in negative theta and simplify with sin and cos rules. Also, it explains why the parametric equations still rotated counterclockwise, because you replaced the functions with the expressions rather than x and y, so the values that were equal to the new x and y rotated counterclockwise, instead of the counterclockwise rotations of x and y satisfying an equation. Again, great video. You definitely deserve more subs for this quality of video and explanation.

"All I ask for is infinite precision, is that so much?" Every mathematician ever

For those people who don't want to watch the whole 16 minutes: 1) Replace all the X's in your function with "x cos(Θ) - y sin(Θ)" 2) Replace all the Y's in your function with "x sin(Θ) + y cos(Θ)" 3) Set the "Θ" parameter to whatever angle you want your graph to be rotated by And that's it!

Finally, I can rotate the line y = x I've always wanted to model the values of y where it is twice and much as x, but never knew how to rotate it, I can finally live in peace

I really like that he basically taught the polar coordinate plane and system without actually using or saying that it is. Props to this person.

One thing to keep in mind is that whenever you rotate a graph, it likely is no longer a function (if it was one to begin with). Some exceptions I can think of are straight lines, and sin and cos (rotated no more than 45 degrees). Otherwise,the curve will "bend over itself" and the same 'x' value can result in 2 different 'y' values. In other words, functions when rotated will, with some exceptions, always become implicit equations.

Desmos can use degrees if you open the menu (wrench in upper right corner) and change from radians to degrees. You can also changes axis limits, ticks, polar, and more.

If my math teacher had shown the movement and number changes like you did in the first 30 seconds here, I could have avoided so much pain. Why they expected everyone to be able to just look at the numbers and automatically understand I'll never know.

I actually found out how to do this quite recently. I was playing around and noticed if I change the x in y=x² to x+y and y to x-y I would get a rotated parabola. Then by changing the ratios to like 5x+3y I'd get different rotations but they'd also always get scaled by some factor. So I also added complicated scaling factors until I tried using trig functions to scale the axes and it became so much easier. And after having taken a linear algera class it also makes much more sense because it is basically just applying a rotation matrix to [x,y]

So, basically you apply a 2D rotation matrix on the curve of the function. I find it interesting the inverse of x^2 (which is an even function) is sqrt(x) which is rotated 90 degree clockwise, and the inverse of x^3 (which is an odd function) is cuberoot(x), which is rotated 90 degrees (either sides), and flipped horizontally. While the inverse of 1/x is 1/x itself. Cool!!

Thanks for the demonstration, wish you didn’t pitch shift vocals.

YOU DON'T KNOW HOW MUCH I WANTED THIS PARTICULAR VIDEO FOR DECADES BUT NO ONE MADE IT I WAS SO DISAPPOINTED... FINALLY I CAN NOW DIE IN PEACE ☮️❤️ This is what we learnt in Electromagnetic Field Theory course in details, it's called "Tensor", Tensors let you do this! This guy just derived it in a simple way, if u add one more axis it will become the tensor rotation formula.

I couldn't help myself but laugh when you added this modified voice that said "Shut up and tell us already." You did a great job!

This questions, of rotating the graph, have been in my interest for long time. I always though a general procedure exist - glad I found your channel. Great work.

I need to thank you so much for this, I've been working on a video game for a while now and decided that I would spawn things as I go rather than hand build in the editor. You have bestowed the power of rotating graphs upon me and now I can build using arrays and rotate after, simply amazing!!!!

i couldnt listen to this voice for 1 minute

Really cool content but that voice changer is really annoying, like borderline unbearable

never have i ever thought i would want to know how to rotate graphs like this. 10/10 gonna send it to my friends now

No pitch shift = 7x better video

cool video. why the pitch shift

You blew my mind with this video. The visual presentation makes it extremely clear that it indeed seems to work for all sorts of equations.

Just how much energy can be felt in a simple, yet neat video like this one? Keep it up, man!

I love how well you explain how the rotation matrix works!

I really enjoyed this! Great job! Such fun to watch.

I know that derivation of sin theta cos theta for rotation is very confusing, and I avoid doing it that way because of this. I advocate for the usage of *complex numbers* for this purpose. Complex numbers make the concept of rotational transform much easier to grasp, but you need to learn complex numbers before doing a rotational transform with it. Math is fun. If you learn something as intricate as complex numbers, you will find other harder things getting easier for you (such as finding rotated coordinates in this case). Rotation is just a special case of complex number multiplication.

You would have changed my life 10 years ago And you just made maths 10 times cooler for me, a mechanical engineering student

I've been fighting with this for about 30 years ... thank you!

20 years ago I was a high school student. And I ask exactly same question to my math teacher, but he even didn't understand the question. And after 20 years now finally I get an answer to this. Thanks a lot!

I wrote this equation into a KZfaq post ages ago! I was wondering when I would finally see a KZfaq video for this... Thanks!

This was freaking beautiful.

This is too good, your videos are not boring at all

🙌 you have my praise from all the math holic kids and myself for creating this video

This is amazing, beautiful and such a perfect explanation!!

What a profound and clear explanation! Thank you!

Oh my god, this is amazing, i'm almost a decade after my education, but this made me want to study maths again and i'm not even joking.

I've always wondered how you could rotate a graph, this video answered that question!

This is a nice intuiative demo, great work

Thanks, I didn't know this was possible. I've tried before and failed miserably so thank you for giving me the answer that I thought did not exist.

Tip: If your using desmos if you wanted it based on degree angles set x degree = x*pi/180 when in radian mode on the trig functions.

You’re voice sounds so cool!

Wow!! Thanks for a great video. Love your voice !!!

You can also derive the formula using the complex domain. Define a complex function of a Real variable as Z(x) = x + i*f(x), then since rotation in the complex plane is just multiplication you multiply Z(x) by e^(it), where t is your angle. When you do that, the Real and Imaginary parts of your new rotated function are identical to your transformation rules.

This is definitely the best math video i have seen in a while

Great job Dexter! You've a new subscriber!

Keep this amazing content up and you'll gain the subs you deserve

Loved the video, I went through the same process back in grade 9 when we were doing trigonometry. Here’s a calculus and trig question which I was asked in my last weeks (after the exam so we weren’t wholly wasting time) in Math C as a grade 12 student as a practical application of the calculus we’d been taught: ‘Given a continuous and differentiable function f(x), describe a general method to find all points whose distance to the nearest point(s) on f(x) is equal to a variable k’. I cant remember what the answer to this was but it was certainly a journey to get there, spent a week of my spare time on desmos :p

This video was what I was searching for weeks ago before I came up with my own way.

it’s beautiful!

It blew my mind. Totally amazing

Awesome! This helped me answer one of my student's questions. Thanks!

It's beautiful I've looked at it for 5 hours now

You can change desmos to use degrees btw Also, for graphs it’s much more efficient (and in my opinion simpler) to represent the graph as a parametric, not to mention it’s actually doable to graph it by hand

This is pretty cool and a great explanation

I had this same thought experiment back in high school, I think I googled if it was possible or not and then forgot about it. Now I'm kicking myself for not trying to figure it out because it's so simple. Thank you for this video, my high school self is ecstatic right now (and current self too).

Amazing video, i really wondered that but i abandoned the idea! Thanks man!

I recently learned stuff about rotation matrix in college, so really informative video for me ^^

If we use a slide to move the "diagonal sine" diagonally, wouldn't that make it look like a moving escalator?

Wow thats really fun! Great video!

bruh. This is so cool, dude. Like, really freaking cool.

So cool Love the video

You can also just use some functional equations and inequalities. Suppose we want to define cosine in terms of sine, or vice versa. Set c^2 = 1-s^2 Now if we have a good value of s, we don't need c to appear anywhere in our rotations. what are some valid values? well s should vary between -1 and 1. We can see that the coordinate transform is just giving us a linear combination of x and y. The scalar coefficients might as well be c and s. Let u = (1-s^2)x - (s^2)y . v= (s^2)x+(1-s^2)y. As it is, this will only cover one quadrant. But if you flip the sign of the first term in both u and v, you get the corresponding quadrant across the y-axis. Flipping the sign on the second term in each will put us below the x-axis. By the two combinations of two possible sign changes we get 4 possible quadrants. Great. Now just keep s between 1 and -1. Substitute u for x and v for y. if it's easier, just rewrite your original expression as a level curve. let y=f(x)=x^2. Now let g(x,y)=x^2-y. you can graph this by entering 0=g(x,y) in most software. If so then 0=g(u,v) will rotate the figure smoothly as s varies. again, just flip the signs to cover the other quadrants. [edit] I should add that the constraint on s isn't arbitrary. if |s| > 1 then the rotation is no longer rigid...that is, the shape is not preserved without deformation. As you can probably guess from the graph of y=sin(x), this is going to cover all possible rotations before you run out of s values. If you only let s vary from 0 to 1 then stitching together all the various sign permutations should cover the entire range of angles. [clarification] To get a mathematically positive rotation, let s vary from 1 to 0, 0 to 1, 1 to 0, then finally 0 to 1, bringing you back to y=f(x). Just switch to the next quadrant when s reaches either 0 or 1.

STOP its too beautiful

lol this reminds me of when i was in year 7 randomly asking my maths teacher if there was a formula for graphing an ellipse. i actually DID learn it a few years later which was pretty cool!

my calculus book gave a nice answer to this, you can write the function in polar coordinates which makes it easier to rotate then switch back.

You deserve so many more subscribers

I found an easy way to put it into Desmos, have different equations: defining the original function before rotation (“f(x)=…”), defining a variable (“b=…”) and the function after rotation. And preferably, put it on “degree” mode. For the parabola at a 45º angle, it would be: f(x)=x^2 b=45 x*sin(b)+y*cos(b) = f(x*cos(b) - y*sin(b)) Change variable “b” to however many degrees rotated you want and change the “f(x)=…” equation to whatever you need to change it to and the rotated equation will update with it. Setting it up like this will also make it so u only have to update less things whenever you want to make a change

Yes! I finally have a vague concept for how this works

JC!! THAT'S VERY NICE!!. THANKS TO SHOW IT.

The little wrench in the top right is settings, you can swap between radians and degrees

this has the same vibe as being lost at walmart and not being able to find your parents.

I figured this thing a long time ago, partially by myself when I was in school (currently in University, making games) Was really curious how it works back then. Thank you for the explanation to others who are as curious as I was! Good luck with the channel

finally, I can spin x^2 + y^2 = 1 after I'm struggling for years

baby's first rotation matrix

Quite informative & humorous

I was thinking about and in autumn I gave up, and finally... Thank you!!

The rotating cubic looks hypnotic.

As an Algebra 2 student I am flabbergasted by the ludicrous graphs that were previewed.

Thank you! I have been asking myself how to do this since Year 8!! Thank you for a great explanation of it and with cool looking functions too (what is MOB??) Looking forward to amazing my students and your part 2 video! Looks very interesting and fun

I did not know I needed this. Thank u so much :D))))

I think what you derived is the inverse of what we want, transforming P’ to P.. since we get points that follow x’=(y’)^2 _after_ (rather than before) rotation. We should instead have solved for x,y in terms of x’,y’, substituting these expressions in the original equation y=x^2 Makes sense as the graphs are later shown rotating in the negative direction. If we want to rotate by +theta, we could just negate all the thetas in the expression Nice video, in any case

Based on the voice I thought that this video was from 12 years ago

all three videos are fantastic. thanks much

Oh my god. Years ago I pondered with a classmate of mine in algebra 2 whether there was a way to rotate a graph like a parabola and they were like “probably not.” I finally have the method.

Simply Beautiful!

God I Wish I saw this vid when I tried to rotate points on a grid great video

Interesting video. What I really liked was in the beginning, with then chalkboard background how you had the x and y coordinates in the lower part of then board changing as the parabolas etc on the actual graph changed. But when you moved on to the Trigonometric functions they just stayed the same on the bottom as the drawing rotated would be great if that could be made visual too

This reminded me, that in computer graphics, coordinates are represented using matrices and you can compute rotations very easily. If rotation is in angles however, this method is required.

Great video. Thank you for making it.

Absolutely wonderful

it's like i ask myself a question and somehow a couple weeks later someone delivers. this has happened four times in a row now

Thank you, I Always wondered about something like that, very intereseting yet simple

Crazy crazy crazy how I was JUST looking at my old writeup of this proof seconds before finding this video. Didn't say anything or look up anything even remotely related to this. Just looked at a physical paper in my notebook with this proof on it, then this video is reccomended to me. That's absolutely nuts.

Actually in my alg 2 in highschool, we rotated to remove the xy term in the ax^2 + bxy+cy^2+dx+ey+f=0 equation and showed all eq of that form was one of the conics. This was pre-trig. and of course this was 'fun' especially showing 1/xy=1 (written as 1=xy) is a hyperbola

Melting your FPU for science 😂Thanks, that was cool stuff!

Thanks for your insight.

I love math how this guy explains it

"without matrices!" what you described is exactly the 2d rotation matrix but just formatted differently

You can also multiply a rotation matrice with the vectors of the graph (x,f(x))

This is a dream come true!

IT'S SO BEAUTIFUL