Family Overview

In Plain Language

Topic 2.5 is about building and using exponential models from contexts and data.

• Proportional change: over equal input intervals, outputs multiply by a constant factor $b$ — that is exponential structure (2.5.A.1).
• Build the model: from an initial value and growth factor, from two data points (system), from transformations of $f(x)=ab^x$, or from technology/regression (2.5.A.3–5).
• Offsets: sometimes add a constant to outputs first so the remaining pattern is proportional (2.5.A.2).
• Natural base: $e\approx 2.718$ often appears in continuous-growth models (2.5.A.6).
• Apply and interpret: $b$ is the per-unit growth factor; relate to percent change; equivalent forms can show daily vs weekly growth (2.5.B.1–2).
• Predict: use the model within reasonable domain constraints (2.5.B.3).

Why It Matters

This is where exponentials meet real data: population, finance, decay, and science contexts. AP questions ask students to construct models, interpret parameters, rewrite for a new time unit, and make justified predictions.

How Families Can Help

• Practice ratio tables: successive outputs should have constant ratio.
• Translate words: "doubles every 3 hours" $\rightarrow$ $b=2^{1/3}$ per hour.
• Compare equivalent forms: $2^d$ vs $(2^7)^{d/7}$.
• Discuss when predictions break down outside the data window.

What Not to Do

• Do not fit a line when ratios are nearly constant.
• Do not ignore a horizontal asymptote / baseline when data level off.
• Do not extrapolate far beyond the observed domain without caution.
• Do not confuse percent change with percent of (growth factor $1+r$).