2.2 Change in Linear and Exponential Functions

Linear vs. Exponential Change

Linear functions change by equal differences over equal intervals — they have a constant rate of change (slope). Exponential functions change by equal ratios over equal intervals — for equal-length input intervals, the output values are multiplied by a constant factor. This is why exponential growth eventually outpaces any linear growth.

Identifying Patterns from Data

Given a table with evenly spaced x-values, check successive differences of y-values: if they're constant, the data is linear. If instead successive ratios of y-values are constant, the data is exponential. This lets you classify and model real data sets without seeing an explicit equation first.

Long-Run Behavior

Because exponential functions multiply by a constant factor repeatedly, their growth accelerates (or decays toward zero) faster than any linear function's steady addition. Over a long domain, an increasing exponential function will always eventually exceed an increasing linear function, no matter the starting values or rates.

Key terms

Linear function change pattern

Equal differences in output for equal-length intervals in input (constant additive rate).

Exponential function change pattern

Equal ratios in output for equal-length intervals in input (constant multiplicative factor).

How to test if data is exponential

Check consecutive ratios of y-values (for evenly spaced x-values) — if constant, it's exponential.

How to test if data is linear

Check consecutive differences of y-values (for evenly spaced x-values) — if constant, it's linear.

Long-run comparison of linear vs. exponential growth

An increasing exponential function will always eventually exceed any increasing linear function, regardless of initial values.

Try an AP-style question

Which model fits?

1. Quadratic, with a = 0.5
2. Linear, with slope 2
3. Linear, with slope 1.5
4. Exponential, with growth factor 1.5