I'm looking for an equation that looks like the image below. I am comfortable constructing functions that pass the vertical line test, but as this is a relation, I am not really sure how to start. Would this need to be defined implicitly?
An explanation on how you constructed it would be great as well! I should note that the scale does not matter, just the general shape.
On
I would look at the trace as a bunch of circles with one side cut off, with the cut being along the $x$ axis. That means the centers are just less than one radius away from the axis. We need the horizontal spacing between two centers to be twice the length of the cutoff so the circles will meet at the axis.
It looks like your circles are three units wide, so the radius will be $1.5$ The cutoff is two units wide. if $A$ is the center of the first circle, we can draw $AB$ which is $1.5$ and note that point $B$ is $(1,0)$, so the height of $A$ must be $\sqrt{1.5^2-1^2}=\sqrt 1.25\approx 1.12$. Point $A$ is $(0,\sqrt {1.25})$, the centers above the axis are $(4k,\sqrt {1.25})$, the centers below the axis are $(2+4k,-\sqrt {1.25})$ Then the part of circle $A$ you want is $x^2+(y-\sqrt{1.25})^2=1.5^2, y \ge 0$ and you can write a similar relation for each circle.
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There are many ways to achieve this. For example, start with the equation $y=\sin x$. Now perturb $y$ slightly. Some trial and error shows that subtracting $\frac32\sin y$ achieves the desired effect. This yields the implicit equation $$y-\frac32\sin y-\sin x=0.$$ You can plot this using Wolfram|Alpha.
Try a parametric representation with a constant speed drift term in the $x$-direction, like so: