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Using Sage's decent-but-needs-an-overhaul piecewise class:

sage: L = [[(1234, 1244), 0.0156672069165316*x + 19.333333335], [(1244,
1254), 0.015541264738745977*x + 19.333333335], [(1254, 1264),
0.015417331208133968*x + 19.333333335], [(1264, 1274),
0.015295358651107592*x + 19.333333335], [(1274, 1312),
0.015175300890894815*x + 19.333333335]]
sage: P = Piecewise(L,x)
sage: P
Piecewise defined function with 5 parts, [[(1234, 1244), x |--> 0.0156672069165316*x + 19.3333333350000], [(1244, 1254), x |--> 0.01554126473874598*x + 19.3333333350000], [(1254, 1264), x |--> 0.01541733120813397*x + 19.3333333350000], [(1264, 1274), x |--> 0.01529535865110759*x + 19.3333333350000], [(1274, 1312), x |--> 0.01517530089089482*x + 19.3333333350000]]
sage: plot(P)