Check out this dialogue between two calculus students (based on a true story):

Devyn

Riley, I think we’ve been too explicit with each other. We should try to be more implicit.

Riley

I. Um. Don’t really…

Devyn

I mean when plotting things!

Riley

Okay, but I still have no idea what you are talking about.

Devyn

Remember when we first learned the equation of a line, and the “standard form” was

\[ ax+by = c \]

or something, which is totally useless for graphing. Also a circle is

\[ x^2 + y^2 = r^2 \]

or something, and here \(y\) isn’t even a function of \(x\).

Riley

Ah, I’m starting to remember. We can write the same thing in two ways. For example, if you write

\[ y = mx + b, \]

then \(y\) is explicity a function of \(x\) but if you write

\[ ax + by = c, \]

then \(y\) is implicitly a function of \(x\).

Devyn

What I’m trying to say is that we need to learn how to work with these “implicit” functions.