Key takeaway: Exponential decay means a quantity loses the same percent of its current size in every period, so each period’s loss is smaller than the one before.
Grade 9 – Algebra 1F-LE.A.1cFunctions and Graphs
Definition of Exponential Decay
Exponential decay describes something that shrinks by a fixed percent of what is left. An e-bike bought for \$3,200 that loses 15% of its value a year drops \$480 in the first year but only \$408 in the second, because the second loss is 15% of a smaller number.
With the rate \(r\) written as a decimal, the multiplier is the decay factor \(1 – r\), which lies between 0 and 1. The graph falls steeply at first and then flattens out, getting close to zero without ever reaching it. Medicine leaving the bloodstream, a hot drink cooling toward room temperature, and radioactive material follow this pattern.
Formula and notation
For a starting amount \(a\), a rate \(r\) per period and \(t\) periods:
\[y = a(1 – r)^{t}\]
A 15% rate means \(r = 0.15\) and a factor of 0.85. Half-life problems use the factor \(\frac{1}{2}\) over each half-life period instead.
Key facts
- The decay factor is 1 minus the rate, so it lies between 0 and 1.
- The amount lost each period gets smaller, even though the percent is the same.
- The graph approaches the x-axis as a horizontal asymptote.
- For exponential decay, never subtract the percent repeatedly from the original; multiply by the factor.
Exponential Decay example
Dara buys an e-bike for \$3,200, and its resale value falls 15% each year. Estimate the value after 4 years to the nearest cent.
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Decay factor: \(1 – 0.15 = 0.85\).
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Model: \(y = 3200(0.85)^{t}\), so \(t = 4\) gives \(3200(0.85)^{4}\).
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Compute \(0.85^{4} \approx 0.522006\), then \(3200 \times 0.522006 \approx 1670.42\).
After 4 years the e-bike is worth about \$1,670.42.
Common mistake
Watch out
Jonah removes 15% of the original price four times : \(3200(1 – 0.15 \cdot 4) = 1{,}280\).
Why it is wrong
That takes the same \$480 away every year and treats the loss as 60% in total. Real value falls by 15% of the current value, so multiply by 0.85 four times : \(3200(0.85)^{4} \approx 1{,}670.42\), more than Jonah’s \$1,280.
Related terms and lessons
Browse the glossary: A–Z list · Algebra 1 vocabulary · Functions and Graphs