NOT.6:5.1 - Combine a formula, sample table and plot without freezing old values

A mathematical account defines f(x) = 2x + b for real x. A table stores its values at x = 0, 1 and 2, and a plot shows those same sample points with x horizontally and f(x) vertically. With b = 1, the table values are 1, 3 and 5.

The formula supports a value outside the table: f(3) = 7. The table supplies direct sample lookup; the plot exposes the relative positions of the samples. Connect each row’s x to the formula input and each plotted point to that row’s pair. The plot is generated from the table. Its agreement is not additional evidence that the formula describes some physical phenomenon.

Now the user requests f(1) = 4 while keeping the coefficient 2 fixed. The relation 2*1 + b = 4 gives b = 2. Recompute the table as 2, 4, 6 and move the plotted points accordingly; the formula now gives f(3) = 8. The old values at 0 and 2 were earlier derived values, not constraints freezing those points.

If the user instead requires f(0) = 1 and f(2) = 5 to remain, the three requested values cannot belong to a line of this form. f(0) fixes b = 1 and hence f(1) = 3. Expose the conflict. Choosing a different model or dropping a requirement changes the mathematical task; silently moving one representation would only hide it.

Finally, the three samples alone do not establish the formula for other inputs. Their generating formula is supplied here. If the table instead contains measurements, recover that different status before treating interpolation as a derivation.