But it can also be viewed as translating the quadratic expression along the “X-axis” so that (at its new origin) it is left/right symmetric.
That is,
Q(x) = ax^2 + b x + c
With the right substitution, x’ = (x - B), the linear term vanishes. So when you re-write in terms of “x”, you get:
Q (x) = a (x - B)^2 + C
So the intuition is that the linear term in the original quadratic is the thing that shifts the “symmetry axis” of the quadratic.
I have found this helpful when “X” is a vector and you have a quadratic form. In this case, the coordinate shift centers the quadratic “bowl” about some point in R^n.
*
The chain rule for differentiation is another one with simple geometry but cumbersome notation. It’s like: we know[*] that
f(x) = g(h(j(k(x))))
must have a linear approximation about some point x0. The only possible thing it could be is the product of all the little local curve slopes of k, j, h, and g, at the “correct” point in each.
Thinking about little slopes also clarifies derivatives like
f(x) = g(x^3, x^2)
where g is an arbitrary function of two variables.
[*] Because we read it in baby Rudin, ofc
Geometric “proofs” like this are neat, but are no real substitute for the algebraic ones. I’d argue that in cases like the present one they also don’t provide any deeper insights. You’re just moving geometric shapes around instead of algebraic symbols. They might give you the feeling that the theorem isn’t as arbitrary as you thought, but it isn’t arbitrary in algebra either.
I’m putting “proof” in quotes here because there are many examples of incorrect geometric “proofs”, and there is generally no formal geometric way to verify their correctness.
There are formal models of synthetic (i.e. axiom-and-proof based) Euclidean geometry where proofs can in fact be verified. This is accomplished by rigorously defining the set of allowed "moves" in the proof and their semantics, much like one would define allowed steps in an algebraic computation.