How can I do exactly as the title asks? I'm tearing my hair out trying to figure this out. I watched this video, and I now understand how to rotate about the origin, but I tried using the same steps to find the equation for any point, using h and k to represent the coordinates of the "new origin", and I got lost. I got to this point, and have no idea how to simplify, or if its even possible at all.
Any help at all will be greatly appreciated. Thank you.
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To use this method one should convert $y=F(x)$ form into two separate parametric equations of form $x=f(t), y = g(t) $. For example $$ y^2 = 4 f x ; \rightarrow x= ft^2, y=2 ft $$
Next, a translation and a rotation through angle $\theta$ can be done as you know$ (x,y)$ shifts $(h,k)$ between required points.
$$ x_1= x \cos \theta - y \sin \theta+ h\, ; y_1= x \sin \theta + y \cos \theta + k. $$
Next, you can see yourself if eliminating $\theta$,( which can be done) is desirable or worthwhile.
Do you have a function you're working with? i.e $f(x)$ = something?
If so, you can translate the point you're rotating about to the origin, rotate and then translate back.
To translate vertically by $\pm k$, simply look at $f(x) \pm k$. To translate horizontally by $\pm h$, simply look at $f(x \mp h)$
As an example of translating, say we have the function $f(x) = (x-2)^2 + 4$ and we want to translate it so the point $(2, 4)$ coincides with the origin. Using the notation in the previous paragraph, $k = -4$ and $h = -2$. To take care of the vertical translation we look at $g(x) = f(x) - 4 = (x-2)^2$. To take care of the horizontal translation we look at $h(x) = g(x+2) = x^2$. Now our function has been translated to the origin. You can rotate $h(x)$ and then translate its rotated version vertically by $+4$ and horizontally by $+2$