Equal Differences or Equal Factors

In this lesson you will: Test a table for a constant first difference over equal input steps; Test a table for a constant ratio between consecutive outputs; Decide from a description whether a quantity changes by a constant amount or a constant factor.

Two rows under the table

Given a table, do not guess. Write two rows underneath it. The first holds the differences between consecutive outputs; the second holds the ratios.

A constant difference row means the function is linear: it grows by equal differences over equal intervals. A constant ratio row means it is exponential: it grows by equal factors over equal intervals. If neither row settles down, the pattern is neither, and there are plenty of those.

The input column has to step evenly

Both tests say over equal intervals, and that phrase is doing real work. A difference of 12 means nothing until you know whether it happened over one unit of input or over four.

When the input steps unevenly, divide each output change by its own input change. If every one of those quotients agrees, the function is still linear. Inputs 1,3,6,10 with outputs 4,10,19,31 give changes 6,9,12 over steps 2,3,4 — and 6/2=9/3=12/4=3, so the rate is 3 per unit throughout.

The same decision, in words

Descriptions carry the same signal. “Adds 40 members each month” is a constant amount, so it is linear. “Doubles each month” or “triples every day” is a constant factor, so it is exponential.

Chapter 14 builds exponential functions properly — their graphs, their algebra, their behavior over long stretches. Here you only have to name which family a pattern belongs to, and say what the table or the sentence told you.

Common mistake — Differencing a table whose inputs jump unevenly

A table gives inputs 0, 3, 4, 8 with outputs 50, 38, 34, 18. The differences are -12, -4, -16. A student concludes the function is not linear.

The conclusion follows the rule as it is usually stated, and on a table stepping by one it would be right. The input column simply never got a second look.

Those inputs step by 3, 1, and 4. Dividing gives -12/3=-4, -4/1=-4, and -16/4=-4. The rate is -4 per unit all the way down, so the function is linear: f(x)=50-4x, and f(8)=50-32=18 checks.

The mirror-image error is just as common: calling a table exponential because the outputs climb fast. Fast is not the test. Constant ratios are the test.

How to catch it: Read the input column first and write its step size in each gap. Only when every step is the same size may you compare raw differences.