Change in linear and exponential functions

Take any two input values a fixed distance apart. A linear function changes by the same amount across every such interval: rent up 50 dollars every year, water dropping 3 cm every hour. An exponential function changes by the same factor: a population multiplied by 1.04 every year, a dose multiplied by 0.85 every hour.

On a table with equally spaced inputs, the diagnosis is mechanical. Constant differences between consecutive outputs mean linear. Constant ratios mean exponential. The words in a story diagnose the same way: "grows by 200 each year" is an amount, "grows by 2 percent each year" is a factor in disguise. The single most reliable habit in this unit is to run both tests before naming the family.

The exponential form $f(x) = a \cdot b^x$ has two dials, and each answers exactly one question.

1. a is the initial value: $f(0) = a \cdot b^0 = a$. It is where the function starts, never how fast it grows.
2. b is the growth factor: each 1-unit step of x multiplies the output by b. The percent change per step is $b - 1$: a factor of 1.04 is 4% growth, a factor of 0.85 is a 15% loss. The factor and the rate are related but not interchangeable.

Over an input interval of length k, the multiplications compound: the output is multiplied by $b^k$, not by $b \cdot k$. A factor of 2 applied over an interval of length 3 multiplies the output by $2^3 = 8$, because doubling three times is eight times, not six.

Two clean points determine each family. For a linear model, the slope is the output change divided by the input change. For an exponential model, take the ratio of outputs and un-compound it: if $f(0) = 6$ and $f(2) = 24$, then $b^2 = 24/6 = 4$, so $b = 2$ and $f(x) = 6 \cdot 2^x$. Note what did not happen: nobody subtracted 6 from 24 and divided by 2. That computes a slope, and slopes belong to the additive family.

When the given point is not at x = 0, the same discipline holds: ratios give $b^k$ across a gap of length k, and the k-th root recovers b. The exponent always counts how many steps of compounding separate the two values.

Pit $g(x) = 1000x$ against $f(x) = 1.01^x$. For a long stretch the line is laughably ahead: at x = 100 the line reads 100{,}000 while the exponential reads about 2.7. But the line adds a flat 1000 per step forever, while the exponential multiplies by 1.01, and its additions grow with its size. Eventually each exponential step adds more than 1000, then more than a million, and the line never leads again.

This is a theorem, not an anecdote: an increasing exponential function eventually exceeds any linear function, and in fact any polynomial. Multiplicative growth compounds; additive growth does not. When a problem asks about the long run, the family decides the answer before any arithmetic does.