Choosing a Model — Linear, Exponential, or Quadratic

In this lesson you will: Test a table with equally spaced inputs for a constant first difference, a constant ratio, or a constant second difference; Match a description in words to the family of function it forces; Explain why a quantity growing by a constant factor eventually passes one growing by a constant amount.

Equal steps in, then look at what comes out

Given a table, you do not have to guess. There is a decision procedure, it is three lines long, and it settles the question.

Every one of the three tests assumes the inputs are equally spaced. Check that first, always, because a test run on uneven spacing returns a confident wrong answer.

The second-difference test, worked

Take the outputs 2, 5, 14, 29, 50 at x=0,1,2,3,4. The first differences are 3, 9, 15, 21 — not constant, so not linear. The ratios are 2.5, 2.8, 2.07, 1.72 — nowhere near constant, so not exponential.

Now difference the differences: 9-3=6, 15-9=6, 21-15=6. Constant. The model is quadratic, and the second difference even hands you the leading coefficient.

Reading the family out of a sentence

When there is no table, the words carry the same information. Anything that adds or subtracts a fixed amount per step is linear. Anything that multiplies by a fixed factor per step — doubles, triples, halves, grows 8 percent, loses 12 percent — is exponential.

Quadratic is the one students miss, because English rarely says \"quadratic.\" Look instead for a product of two changing quantities, an area as a length grows, or anything thrown or dropped.

Fast for a while is not the same as exponential

A model can grow quickly and still be linear, and a model can start slowly and still be exponential. The two properties are unrelated, which is exactly why \"it is growing fast\" is not a diagnosis.

Compare y=10x with y=2^x. For the first five inputs the line is well ahead — at x=5, 50 against 32. At x=6 the exponential passes it, 64 against 60, and from there the gap widens without limit. That is the content of F.LE.3: a quantity growing by a constant factor eventually exceeds one growing by a constant amount, no matter how large that amount is.

Common mistake — Running a difference test on unequally spaced inputs

A table gives y=2, 5, 11, 14 at x=0, 1, 3, 4. The first differences come out 3, 6, 3. Not constant, so the student rules out linear and starts hunting for a quadratic.

It is a fair mistake: every table in the textbook has inputs 0,1,2,3,4, so the eye stops checking. And the output column really does step unevenly.

The inputs here step by 1, 2, 1. Divide each change in y by its change in x: 3 / 1=3, 6 / 2=3, 3 / 1=3. The rate is a flat 3 throughout. The data is linear, y=3x+2.

There is no quadratic anywhere in this problem. The student is about to compute second differences of numbers that were never comparable in the first place.

How to catch it: Difference the x-column before you touch the y-column. If those gaps are not all equal, no difference test is valid yet — either fill in the missing rows or divide each change in y by its own change in x.