Change Origin of Shapes in Graph

Question

This might seem pretty simple, but keep in mind I'm in middle school.

I graphed a spiral using $r = \theta$. The problem is, the origin is at $(0, 0)$. How can I change the origin to something like $(0, 5)$?

I have the same problem with a semicircle. I have a semicircle graphed with $p=-\sqrt{4-y^2}$. I can change the $X$ axis by adding a $+$ sign after the square root, but I also want to change the $Y$ axis.

Thanks in Advance!

Answer 1

Your first question is hard. There is no simple way to transform the simple polar coordinate ($r,\theta$) equation for a spiral originating at the origin, to a spiral originating elsewhere. It can be done but not with things you know about in middle school.

The second question is easier. Try something like

$$p = -\sqrt{4-(y-1)^2}$$ which forces $y$ to behave like $y+1$ used to behave. Fool around with different valuews instead of that $-1$ to see what it does.

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The ugly answer to the second question, if you insist on expressing the shifted spiral in polar coordinates is to temporarily work in $(x,y)$ coordinates, where the shifted spiral will be

$$\sqrt{x^2+(y-5)^2} = \tan^{-1}\frac{ y-5}{x} $$ If you then take $$r = \sqrt{x^2+y^2}$$ and $$\theta = \tan^{-1}\frac{y}{x}$$

you would have your curve.

Told you you didn't want to do that...