Lesson 7 — Exponential Functions and Their Features

Summary

Learners contrast exponential change with linear change — steady multiply versus steady add — and read the features of y = a·bˣ: the initial value a, the growth factor b, the growth rate r = b − 1, and the horizontal asymptote. They model a salary or savings scenario and explain the effect of the base.

Objectives

• Model exponential growth and decay and explain the effect of the rate — reading the features of the exponential function. (D05.S2.10.02)

Connection

Two jobs are offered. One pays a steady raise of 1,000 a year; the other pays a 5% raise each year. At first the steady raise looks better. But the 5% job grows on top of its own growth — each year’s raise is bigger than the last — and before long it passes the flat one and keeps climbing. That is the difference between linear change (add a fixed amount) and exponential change (multiply by a fixed factor). Savings, populations, and many natural processes are exponential; reading the base and the rate tells you which curve you are on and where it is headed.

Materials

• Linear-vs-exponential chart
• Graph paper
• Math journal

Preparation

• Copy or draw the linear-vs-exponential chart.
• Retrieval: from Lesson 3, the growth factor b = 1 + r; from Grade 9, linear and exponential functions in context (D05.S2.09.01). Today we formalize the exponential function and its features.
• Prepare the two-offers worked example.

Facilitator note

This lesson is written to the learner (“you”). The idea to land: the exponential function y = a·bˣ grows by a fixed factor each step — its features are the initial value a (the y-intercept), the base b (the growth or decay factor), the rate r = b − 1, and a horizontal asymptote — and the base, not the starting value, decides how fast it runs. Teach with worked examples and guided practice (S-011), and let the linear-vs-exponential contrast do the motivating.

The intellectual lens: “multiply by a fixed factor” is the whole story; every feature is a restatement of it. The critical-thinking lens: ask learners to test which offer is better at year 1, 5, and 20 — a small difference in rate becomes a huge difference over time, so the time horizon matters. The environment lens: exponential growth is the shape behind compounding problems — a warning, not just a number — and it is why small rates over long times deserve respect. The egalitarian lens: two people offered 4% and 6% returns live different futures; who gets which rate is often about access, not merit — a fairness question, not only a math one. Preview: Lesson 8 applies this function to population, finance, and the environment, and to the limit of growth.

Procedure

1. Recall (5 min). From Lesson 3, what is the growth factor for a 5% increase? For a 10% decrease? Name the difference between “add 3 each time” and “multiply by 3 each time.”
2. Contrast the two shapes (10 min). A linear function adds a fixed amount each step; an exponential function multiplies by a fixed factor each step. On the chart, compare tables: linear (100, 110, 120, …) vs exponential (100, 105, 110.25, …). At first they are close; the exponential curve then bends upward and pulls away.
3. Read the features (15 min). Worked example: y = 100 · (1.05)ˣ.
   • Initial value a = 100: the y-intercept, the value at x = 0.
   • Growth factor b = 1.05: multiply by this each step.
   • Growth rate r = b − 1 = 0.05, or 5% per step.
   • Asymptote: for decay (0 < b < 1), the curve levels toward y = 0; for growth (b > 1) it climbs without bound.
4. Model the two offers (15 min). Offer A: start 100 and add 10 a year (linear). Offer B: start 100 and grow 10% a year (exponential). Build a table for years 0–6, and graph both. Find the year Offer B passes Offer A. State what the base (1.10) and the rate (10%) each mean in words.
5. Guided practice (5 min). With a partner, write the exponential function for a quantity that starts at 200 and decays by 20% each step, and state its asymptote.
6. Close (5 min). Say in one sentence how the base and the starting value each shape an exponential function’s future.

Differentiation

• Support: Build only the table first and describe the trend in words before graphing.
• Extension: Find the doubling time by trial or reasoning (for b = 1.10, how many steps until roughly 200?), and explain why the base sets it.

Assessment

• Formative (peer + self): Can the learner write an exponential function from a scenario, identify a, b, and r, and explain the effect of changing the rate?
• Portfolio artifact (unit): The linear-vs-exponential chart with the two-offers model, added to the functions toolkit.

Home connection

Ask an adult: if you could have a flat raise or a percentage raise each year, which would you choose and why? Use the two-curves picture to explain what the percentage raise does over many years.

Resources

• On worked examples and guided practice: Kirschner, Sweller & Clark (2006), https://doi.org/10.1207/s15326985ep4102_1 (S-011).