As part of a typical introduction to derivatives, distance as a function of time and velocity as a function of time are each depicted as a line on a 2D graph. My question is how conceptualizing these three quantities in this way is reasonable, given their relation to each other.
The core algebraic formula at play is $y/t = v$, where $y$ represents distance, $t$ represents time, and $v$ represents velocity. Since this formula includes three variables, my first intuition would be to draw its graph in 3D space, assigning one variable per axis. Another apparent possibility, motivated by looking ahead and seeing that both distance and velocity are treated as functions of time, would be to draw them as a parametric curve in 2D space. These approaches seem fine. However, what's not clear is why we're able to selectively graph distance as a function of time, plucking $y$ and $t$ from the $y/t = v$ formula, without having already taken the impact of $v$ into account.
As an analogy, I don't think it's possible to graph $2a = 3b + 4c$ as a line on a 2D plane (whether the $ab$, $bc$, or $ac$ plane). Unless one of the variables happens to be $0$, it's guaranteed to affect the relation between the other two. This problem should hold regardless the variables' function statuses.
As for graphing velocity as a function of time, when both $y$ and $t$ in the formula $y/t = v$ are very small, we end up with $dy/dt = v$, where $dy/dt$ is understood to be a single variable. There is no ultimate value of $dy$ or $dt$; the distance and time values in the earlier formula are discarded when we shrink them an unspecified amount, leaving only their ratio in its place. However, while reducing the number of variables is exactly what we needed to graph the formula 2-dimensionally, our new formula, $dy/dt = v$, no longer includes time as a variable; there is only a ratio $y'$, and a velocity $v$. So we can't graph velocity as a function of time from this formula alone without referencing the prior formula, but if we do reference the prior formula, then the initial problem of dealing with an extra variable returns.
Generally, in a situation concerning velocity or distance, both will be expressed in terms of a variable time. That is, you will have that distance is a function of time, and velocity is a function of time. For example, you will have an expression for distance in terms of time that's something like $$ d(t) = t^2+5. $$ This can easily be plotted onto a 2D graph since we can simply treat distance as the $t^2+5$ as the $y$ value and $t$ as the $x$ value. We can then construct an expression for the velocity by deriving this expression with respect to time, giving us that $$ v(t) = 2t. $$ These expressions then give us two 2D plots that gives us a visual representation of how the two quantities are related at each point $t$.
As for why we can "pluck" $y$ from the expression $y/t=v$, it is usually not actually that simple. In fact this formula is somewhat misleading as user amd pointed out, as it only applies in the cases he mentioned. The expression we actually use is $$ \frac{dy}{dt} = v $$ where $y$ is distance. In words, this translates to velocity being the "rate of change of distance with respect to time". Thus to "pluck" $y$ out of this equation we would need to integrate both sides with respect to $t$ and solve for some constant, ideally using an initial condition for $y$ given in the context of the problem. This will still lead to a two-dimensional graph with respect to time for distance. Hope this helps!