I don't understand why we represent functions $f:I \subseteq \Bbb R \to \Bbb R^2$ the way we do. I don't understand why we represent functions $f:I \subseteq \Bbb R \to \Bbb R^2$ the way we do: doing an analogy with how we represent functions from $\Bbb R$ to $\Bbb R$ or from $\Bbb R^2 \to \Bbb R$, my impression would be to represent functions like $f$ in the format: $$(t,f_1(t),f_2(t))$$
But instead we grab all the points $(f_1(t),f_2(t))$, that is, the image of $f$ and we plot that in $\Bbb R^2$, and we call $f$ a parametrization. 
As an example of this, let $f(t)=(\cos t, \sin t)$. If we represented this by plotting the points $(t,\cos t, \sin t)$ we'd get a helix. But instead we 'splash' that helix into the wall and we get the parametrization of a circle.

Because it's a useful simplification sometimes. Explicit time dependence is really scary, actually. Imagine I'm watching a system that shouldn't depend explicitly on time (earth's gravitational field shouldn't fluctuate based on time, for example). But I'm watching a ball travel and its position does depend on time. If the ball is at position $(x,y)$ at time $t_1$ and again at time $t_2$ I want to say that the ball is orbiting.
But if I imagine the ball's coordinates as having a time component, the it will never hit the same point twice. That is $(t_1,x,y) \neq (t_2,x,y)$. So the ball would never really be in orbit. 
This idea is strongly reflected in the field of differential equations, where systems that do not explicitly depend on time are much better understand than those that do. Things like poincare-bendixon theorem (which are awesome) depend heavily on the notion of orbits. "If a ball hits the same point twice its path must be periodic" is a critical lemma for the theorem. 
