The 42nd Period | STEM Education Hub

Use this as a quick reference for building models from ratios, pairs, and data, the growth factor, equivalent forms, and making predictions.

Exponential Function Context and Data Modeling infographic

🧭 Plot Summary

Lessons 2.3 and 2.4 gave you the general form and the tools to rewrite exponential expressions. Now we use them on real situations. Exponential functions model growth patterns where successive outputs over equal-length intervals are proportional. You will build models from a ratio and an initial value, from two input-output pairs, by adding a constant to reveal a hidden pattern, by transformations, and with exponential regression on a calculator. You will meet the natural base e, interpret the base as a growth factor tied to a percent change, and use predictions that respect the limits of the context.

What you'll do in this lesson

Decide whether a situation or data set is exponential, using proportional outputs over equal-length intervals.
Reveal a hidden exponential pattern by adding or subtracting a constant, and model it with ab^x + k.
Build a model from a ratio and initial value, from two input-output pairs, or with exponential regression.
Use transformations of ab^x and the natural base e in models.
Interpret the base as a growth factor tied to a percent change, and choose the equivalent form that answers your question.
Predict values, respect the contextual domain, and find the input that reaches a target output.

Why it matters

Populations, investments, cooling drinks, and medicine in the body are all modeled this way. The skill is not just computing a number. It is choosing a model that fits, saying what its base means in the situation, and deciding whether a prediction is believable. Topic 2.6 compares exponential models with linear and quadratic ones.

✅ Self-Check Before You Move On

Check off each item as you get there. These aren't grades — they're your own signal.

I can tell whether data or a situation is exponential by checking for proportional outputs over equal-length intervals.I can add or subtract a constant k to reveal a proportional pattern and write a model f(x) = ab^x + k.I can build an exponential model from an initial value and a ratio, or from two input-output pairs by solving for a and b.I can build a model by applying transformations to f(x) = ab^x that match a situation.I can construct an exponential model from a data set using exponential regression with technology.I can recognize the natural base e ≈ 2.718 and rewrite a·e^kx as a·(e^k)^x.I can interpret the base b as a growth factor and connect it to a percent change: b = 1 + r.I can use equivalent forms, such as a daily factor and a weekly factor, to reveal different properties of the same model.I can use a model to predict outputs, respect the domain of the context, and find the input that gives a target output.

🔬 Deep Dive

Your textbook for this topic: proportional growth, shifted data, two-pair models, transformations, regression, the natural base, growth factors, and predictions — with interactive tools.

Open Deep Dive →

🧭 Guided Practice

Step through worked examples for every essential knowledge statement in 2.5.A and 2.5.B. Two or three examples each, revealed one step at a time.

Start Guided Practice →

📝 Assignment

20 questions covering the whole lesson. Answer each one and get feedback right away, including the most likely reason behind a wrong choice.

Open Assignment →

📌 AP Exam Note

Expect modeling problems that give a table, a graph, or a verbal description and ask you to write an exponential model, interpret its base as a percent change, and make a prediction with units. Know how to use two input-output pairs, when to add a constant to the outputs, and how to use exponential regression on a calculator. State any limit the context puts on the input, and round sensibly.