Investigating Growth & Decay — Visual Lab
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.
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 \(1\)?
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).
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?
Key Vocabulary & Standards
The precise words a mathematician uses for what the explorer shows — and the TEKS this lab builds.
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).
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.
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.