Desmos Graphing Calculator
Graph y = 10000(1.03)^x to watch an exponential model curve upward.
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1 · Exponential models
An exponential model changes by a constant percentage each period, so it multiplies by a fixed factor:
y = P · (1 + r)t growth | y = P · (1 − r)t decay
Here P is the starting value, r is the percent change (as a decimal), and t is time. The factor (1 + r) is the growth factor; (1 − r) is the decay factor. Its graph curves rather than following a straight line.
Learn the Concepts
2 · Linear or exponential?
Choosing the right model comes down to one question: how does the quantity change?
- Changes by a fixed amount each step → linear (y = mx + b).
- Changes by a fixed percentage each step → exponential (y = P(1 + r)t).
Over time, exponential growth always overtakes linear growth, even if it starts slower — the curve eventually outruns any straight line.
Worked Examples
See it done, step by step
Example 1 — A town of 10,000 grows 3% per year. Population after 5 years?
Growth factor is 1.03.
P(5) = 10000 (1.03)5 ≈ 10000 × 1.15927 ≈ 11,593 people.
Example 2 — A $20,000 car loses 15% of its value each year. Value after 3 years?
Decay factor is 1 − 0.15 = 0.85.
V(3) = 20000 (0.85)3 = 20000 × 0.614125 ≈ $12,282.50.
Example 3 — Linear +$1,000/yr vs. exponential +10%/yr, both from $10,000. After 10 years?
Compute each model at t = 10.
Linear: 10000 + 1000(10) = $20,000.
Exponential: 10000 (1.10)10 ≈ $25,937.
The exponential model is far ahead — and the gap only widens.
Watch & Review
Resources for this module
Growth models quiz
Revisit the linear-vs-exponential comparisons from the growth quiz.
Open quizLecture materials
Module 14 slides and any posted videos are in your Canvas course.
Open CanvasCheck Yourself
Quick self-check
1. A quantity decreases 20% per year. What is its decay factor?
1 − 0.20 = 0.80.
2. Is y = 500(1.04)t growth or decay, and at what rate?
Growth at 4% per period (the factor 1.04 = 1 + 0.04).
3. Salary rising $2,000 every year — linear or exponential?
Linear — a fixed amount is added each year, not a fixed percentage.
Ready to turn it in?
When you can model with exponentials and choose the right model, complete the Module 14 homework and quiz in Canvas. Next stop: the cumulative final review.
Submit in Canvas