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Investigating Growth & Decay — Module Hub & Visuals
Module 4. When change is a constant multiplier instead of a constant difference, growth bends. Build it, bend it, and watch the curve outrun the line.
Lesson Slides & PDFs
Student-facing lesson slide PDFs are grouped by topic below for review, catch-up, and family support.
📖 Student Edition for this module (PDF) ✍ Skills Practice — Student Edition (PDF)
Graphic Organizers & Activities
Foldable notes and activities for each lesson, with the one to start with listed first. Sign in with your alias and PIN to open them.
Start Here — What Module 4 Is About
What this module is for, and what you will be able to do by the end of it.
This lab puts the exponential function \(y = a\,b^{x}\) in your hands. Slide the initial value \(a\) and the base \(b\), flip on a straight-line comparison, and the graph, the value table, and a real scenario all redraw in real time. The single most important question: is \(b\) bigger or smaller than \(1\)?
Key Vocabulary
The precise words for what you just investigated. The standards this module carries are in the next section.
Vocabulary
- Exponential function — a function of the form \(y = a\,b^{x}\), where the variable is in the exponent.
- Initial value (\(a\)) — the output when \(x = 0\); where the curve meets the \(y\)-axis.
- Base / growth factor (\(b\)) — the constant multiplier applied at each step.
- Growth — the case \(b > 1\); the quantity increases by a fixed percent each step.
- Decay — the case \(0 < b < 1\); the quantity decreases by a fixed percent each step.
- Doubling time / half-life — how long until the quantity multiplies by 2 (growth) or by \(\tfrac12\) (decay).
Standards in This Module
The Texas Essential Knowledge and Skills this module is built to teach. RDY marks a STAAR readiness standard.
TEKS in this lab
This lab targets the Module 4 exponential standards — describing and graphing \(y = a\,b^{x}\), distinguishing growth from decay, and building exponential models for real situations. Readiness standards (RDY) are weighted heaviest on the STAAR EOC.
What You're Seeing & What To Try
A quick orientation, then four experiments to run in the explorer above.
What you're seeing
The curve plots \(y = a\,b^{x}\). The value \(a\) is the starting amount at \(x = 0\) — move the \(a\) slider and the whole curve slides up or down. The base \(b\) is the per-step multiplier: every time \(x\) goes up by one, \(y\) is multiplied by \(b\).
When \(b > 1\) each step is bigger than the last, so the curve bends upward — growth. When \(0 < b < 1\) each step keeps only a fraction, so it decays toward zero. The dashed line \(y = m x + a\) adds the same amount every step; compare the two and you'll see why a multiplier eventually beats any constant rate.
Try this
- Set \(b\) just above 1 (say 1.10). The curve barely bends — now push it to 2.00 and watch it explode past the line.
- Find the doubling time: keep \(b > 1\) and read the amber marker. Then double \(b\); does the doubling time get shorter or longer?
- Switch to the Medicine · decay context and drag \(b\) to 0.50. Where does the half-life land? Why does it sit exactly at one step?
- Turn on the comparison line and match its slope \(m\) to the curve's first jump. Does the line ever catch back up to the exponential?
Growth vs Decay Explorer
Drag the sliders. The accent curve is the exponential model; the dashed steel line adds the same amount every step. The amber marker finds the doubling time (growth) or half-life (decay).
Want to See This in 3D?
Optional, and always second. The lab above is where you measure and calculate; this is the same idea as an everyday object you can turn over and inspect. Open it when the lab already makes sense — or skip it entirely.
Done exploring? Head back to the course or check where Module 4 lands on the calendar.
Module and topic structure follow the TEA Bluebonnet Learning — Secondary Mathematics open curriculum (Edition 1, adapted from Carnegie Learning), licensed CC BY-NC 4.0. Classroom use is non-commercial.