One parameter family of functions describing the curve

"one parameter family of functions" describing the curve

Homework Statement

Let a cube of unit length be represented by the set of vectors such that three of the cube's sides are aligned with the three orthonormal basis set vectors e1 e2 and e3, and one of the cube vertices lies on the origin of the coordinate system. Let Ω be the deformed configuration such that f:X belonging to Ω0 -> x=f(X) belonging to Ω is smooth, differentiable and bijective defined as:
(X1,X2 and X3 are three original positions)

x1=1.1X1+0.02X1^2+0.01X2+0.03X3
x2=0.001X1+0.9X2+0.003X3
x3=0.001X1+0.005X2+0.009X3^2+0.9X3

a- The displacement function u
b- The new position of the 8 vertices of the cube
c- The deformed curves of three sides of the cube that are aligned with the basis vectors e1, e2 and e3.

The Attempt at a Solution

a- It's super easy: u=x-X so we already have both and u can be obtained as a function of X1 and X2 and X3

b- Well we already have the original positions of the vertices (X) so by substituting them into the x equations the new positions are obtained.

c- I don't understand it. I know that the answer wants the description of the curve created after the edge deforms. The instructor wants the "one parameter family of functions" describing the curve.
But I don't understand what this "one parameter family of functions" means! Can anyone please help me? I'm in urgent need of help as usual :(
But I don't understand what it means

Even if you have any sort of understanding or interpretation please please let me know. Thanks

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