Topic 2.2 Change in Linear and Exponential Functions: contrast linear functions, which change by a constant amount, with exponential functions, which change by a constant percentage or factor over equal-length input intervals.

What this topic is asking

The College Board (Topic 2.2) wants you to contrast linear change with exponential change. A linear function adds a constant amount over each equal-length input interval; an exponential function multiplies by a constant factor (equivalently, grows by a constant percentage). Recognizing which behavior a data set or context shows, and understanding why exponential eventually outpaces linear, is the core idea.

Linear: constant difference

This is the continuous version of an arithmetic sequence: equal steps in input give equal changes in output.

Exponential: constant ratio

This is the continuous version of a geometric sequence. A growth factor $b > 1$ gives growth; $0 < b < 1$ gives decay. A growth of $r$ percent per step corresponds to $b = 1 + \frac{r}{100}$.

The difference-versus-ratio test

Given a table with equally spaced inputs, the test is mechanical. Compute first differences: constant means linear. Compute ratios: constant means exponential. A data set fitting neither needs a different model. This pairs with the difference tests from Topic 1.3, now extended to include the ratio test for exponential behavior.

Why exponential overtakes linear

A central exam idea is that exponential growth always wins in the long run. A linear function adds a fixed amount each step, so its growth per step is constant. An exponential function multiplies by a fixed factor, so its growth per step is proportional to its current size and keeps getting larger. No matter how steep the line or how gentle the exponential, repeated multiplication eventually produces increments that dwarf any constant addition, so the exponential curve crosses and then permanently exceeds the line. This is why "linear at first, exponential later" describes so many real situations, and why the exam asks you to reason about long-run dominance rather than just early values.

A clarifying point is that a constant percentage and a constant amount feel similar over a short window but diverge sharply over a long one. Eight percent of a thousand is eighty, so an $8\%$ exponential and an $80$-per-year line look identical in the first year. But the percentage applies to a growing base, so next year $8\%$ is more than eighty, and the gap widens forever. Keeping "percentage of a changing amount" distinct from "fixed amount" is the conceptual key to this topic.

Try this

Q1. A population grows by $5\%$ per year. Is this linear or exponential, and what is the growth factor? [1 point]

• Cue. Exponential; the growth factor is $b = 1 + 0.05 = 1.05$.

Q2. A table has constant first differences of $7$. Linear or exponential? [1 point]

• Cue. Linear, because the differences (not the ratios) are constant.