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What is the intuition behind the equations for the parametric equation applications below? Rather than memorize the formulas for a quiz i'd like to gain a deep understanding of them. enter image description here

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I'm not sure what your question actually is. There are often several ways to describe mathematical objects, in your case curves in two dimensions (strictly speaking three dimensions, but the text you are citing deliberately ignores the third dimension). A curve in the plane can be either described parametrically, this means by looking at a map $$c = (c_1,c_2):(a,b)\rightarrow \mathbb{R}^2, \, t \mapsto c(t) = (c_1(t), c_2(t))$$ from some interval $(a,b)$ to the plane (often people write $x$ instead of $c_1$ and $y$ instead of $c_2$). For a given $t$, $c(t)$ is a point in the plane, and $c$ kind of describes how the curve moves through the plane as $t$ varies. Note that one may reparametrise the curve by using a mapping $\phi:(\bar{a}, \bar{b})\rightarrow (a,b)$ and then looking at $c(\phi(t))$. This corresponds to moving along the curve with different speed, as people say, thinking of the parameter as time (which need not imply a direct relation to physics, it's just common wording).

An alternate description is often possible, namely by looking at the set where some function $f:\mathbb{R}^2\rightarrow \mathbb{R}$ is constant. Taking the example from your posting one could look at the set where the function $f(x,y)=y-x+(32/\nu^2) x^2 $ is equal to zero, which is just a reformuluation of your $y = x-(32/\nu^2) x^2$. In this second representation the information about how the curve moves through the plane is lost. It is usually possible to swich the type of representaion, this is what happens in your example 5.

What you are looking at is actually an example from phyics. In physics the locus of a particle in time is usually described by a parametric curve which says at which place in space the particle is located at a given time $t$ -- so in this case $t$ really means time. The specific equations in your posting are equations of motion of a ball or bullet with some initial velocity and direction. They reflect the facts that, i) if a particle moves and no forces act on it, then it moves at constant speed $v$ -- resulting in a curve $c = (c_1t, c_2t)$ -- and, ii) that if a force acts on them, like gravity, then the speed increases and the locus of the particle changes quadratically in direction of the force, and, finally, iii) that the motion is a superposition of both the constant speed and accelerated motion.

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