Constant Moving Speed I've made this graph:
https://www.desmos.com/calculator/czk3ylyokj
As you can see, the purple point is slowing down as it approaces the extreme point.
How can I make this point move with constant speed along the f function graph?
 A: Your point is moving at a constant speed with respect to the $x$ axis since that's how you parametrized it. Since the function you are using $x^2$ grows much faster then $x$ the point travels a further distance along the curve when it's further from the origin.
To fix this you would need to parametrize the curve in such a way that the point travels a constant distance at any given time. You should consider computing the arclength traveled and making that linear in the parameter $t$.
To be honest I would have to write that down before I can give you a formula.
Actually an easy partial solution is to plug in $\sqrt{|t|}$ instead of $t$ everywhere. You need to fix the fact that now you're only going up down one side of the parabola though. Which can be done by clever usage of the sign function.
A: Using your parametrization (time t), the goal is to get the coordinate $(v(t), f(v(t)))$ that has constant velocity as follows, 
\begin{align}
v'(t)^2 + f'(v(t))^2 v'(t)^2 = c^2.
\end{align}
This is a differential equation, and you can solve it in terms of $v(t)$ with $f(x) = x^2 - 5$. But it needs some classification by the sign of $t$ and might be difficult to visualize in the web application you used.
