I'm interested in what shapes you can plot on a graph (cartesian co-ordinates) using an equation (1 line) like y=x......
I'm want to know if you can plot a cycloid on a graph with an equation....
A graph of the form y= f(x), where f is a function of f, cannot "turn back" on itself since by definition of "function", we cannot have more than one value of y for any x.
But the graph of $\displaystyle x^2+ y^2= R^2$ is a circle. The cycloid can be graphed using parametric equations x=a(t- sin(t)), y= a(1- cos(t)) or by using the single Cartesian equation $\displaystyle x= cos^{-1}(1- y/a)- \sqrt{y(2a- y)}$.