How to rotate any graph by any angle

Without matrices!

[uncleben]

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.

[OrangeC]

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

[pkmnhx_]

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

[IlTechnoDashlI]

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!

[stealthgamer]

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.

[jacobbaer]

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.

[withjoe]

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.

[justinelliott]

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.

[WhyneedanAlias]

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]

[int_t]

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!!

[trippstreehouse]

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

[shmuelalexis]

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.

[DoctressCalibrator]

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!

[igxniisan]

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.

[sander_bouwhuis]

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.

[magnusalferes]

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!!!!

[ergenarkenk]

I never realised this video was 16 minutes until the end! Fascinating presentation, really loved it!

[pseudonym]

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

[mrmaaf]

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

[jungtaemin]

You would have changed my life 10 years ago

And you just made maths 10 times cooler for me, a mechanical engineering student