How similar are circular and parabolic paths? I'd for a long time thought of parabolas as semi-circles.

However, if you take half of a circle the ends will - look - parallel, where as parabolas continue to extend horizontally and infinitely.

Are the only similarities then the fact that they're curved?

Are sinusoidal graphs connected semi-circles or parabolas?

If they were connected semi-circles, wouldn't the gradient at $x=0$ be $\infty$ or $\frac10$
Notice: I hope the pictures explain themselves and that you enjoy pondering over this post :-)

Circles and parabolas are both second degree equations, meaning that they can be created by cutting open a cone (conic sections): (You could also make these by shining a flashlight)



This is related to eccentricity.



You could imagine eccentricity to be how much these graphs are stretched: ($e=$ eccentricity)

circle: $e=0$

ellipse: $0&lt;e&lt;1$

parabola: $e=1$

hyperbola: $e&gt;1$



Clearly sinusoidal graphs are related to circles (see here for original post):



while probably not what you were thinking of, sinusoidal functions look parabolic locally:



which is due to Taylor's theorem.
Sinusoidal graphs can be approximated by parabolas in some regions ... depending on the type of sinusoid, the graph may be exactly replicated by linking parabolas together.

You are correct in saying that parabolas are linked to circles, but not in a simple way, as circles have tangents that rotate about completely as you rotate around the circle; parabolas do not.

Parabolas also are onto functions, whereas circles cannot be.