Key takeaway: Exponential growth means a quantity gains the same percent in every period, so each period’s increase is larger than the one before it.
Grade 9 – Algebra 1F-LE.A.1cFunctions and Graphs
Definition of Exponential Growth
Exponential growth describes something that rises by a fixed percent each period rather than by a fixed amount. A herd of 850 elk growing 7% a year gains about 60 animals in the first year, about 64 in the second, and more every year afterward, because each gain is a percent of an ever larger herd.
Writing the rate \(r\) as a decimal, the herd is multiplied by the growth factor \(1 + r\) each year. The graph rises slowly at first and then steeply. Savings with interest, a spreading online video, and bacteria in a dish can all follow this pattern, at least for a while.
Formula and notation
For a starting amount \(a\), a rate \(r\) per period and \(t\) periods:
\[y = a(1 + r)^{t}\]
The factor \(1 + r\) must be greater than 1. A 7% rate is \(r = 0.07\), so the factor is 1.07. The time \(t\) and the rate must use the same period, such as years.
Key facts
- The growth factor is 1 plus the rate written as a decimal.
- The amount added each period increases, while the percent stays the same.
- A graph of exponential growth is a curve that bends upward.
- Doubling time is the number of periods needed for the quantity to double.
Exponential Growth example
A wildlife park counts 850 elk and expects the herd to grow 7% per year. Predict the herd size after 6 years, to the nearest animal.
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Growth factor: \(1 + 0.07 = 1.07\).
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Model: \(y = 850(1.07)^{t}\), so for \(t = 6\), \(y = 850(1.07)^{6}\).
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Compute \(1.07^{6} \approx 1.500730\), then \(850 \times 1.500730 \approx 1275.6\).
The herd will have about 1,276 elk.
Common mistake
Watch out
Lia calculates 7% growth for 6 years as \(850(1 + 0.07 \cdot 6) = 1{,}207\).
Why it is wrong
Multiplying the rate by 6 treats the gain as the same 60 elk each year, which is linear growth. In exponential growth the 7% applies to the new total every year, so the factor 1.07 is raised to the 6th power. That gives 1,276, noticeably more than 1,207.
Related terms and lessons
Browse the glossary: A–Z list · Algebra 1 vocabulary · Functions and Graphs