2.5 Exponential Function Context and Data Modeling

Modeling growth and decay

Real-world scenarios like population growth, radioactive decay, and compound interest can be modeled with exponential functions of the form f(t) = a·b^t. A growth factor b > 1 indicates increase; 0 < b < 1 indicates decay. The constant a represents the initial value at t = 0.

Constructing models from data

When given data with a constant percent change over equal intervals, an exponential model fits best (unlike arithmetic sequences, which have constant differences). To build the model, find a (initial value) and b (1 + percent change as decimal, or ratio between consecutive outputs) using two data points or a ratio test.

Comparing model types

Exponential functions grow/decay by a constant ratio per unit interval, while linear functions change by a constant difference. When comparing tables, check if consecutive y-values share a common ratio (exponential) or common difference (linear) to decide which model is appropriate for the given context.

Key terms

General exponential model

f(t) = a·b^t, where a is the initial value and b is the growth/decay factor.

Growth vs. decay factor

b > 1 indicates exponential growth; 0 < b < 1 indicates exponential decay.

Percent change to growth factor

Convert percent change r to growth factor using b = 1 + r (growth) or b = 1 - r (decay).

Identifying exponential data

Data is exponential if consecutive y-values (for equally spaced x-values) share a constant ratio.

Finding b from two points

Given (t1, y1) and (t2, y2) on f(t)=a·b^t, solve y2/y1 = b^(t2-t1) for b.

Try an AP-style question

A town of 500 grows 4% per year. Which models the population after t years?

1. P = 500(0.04)ᵗ
2. P = 500(1.4)ᵗ
3. P = 500 + 0.04t
4. P = 500(1.04)ᵗ