Converting Between Growth Rate and Growth Factor
Growth rate and growth factor are two equivalent ways to describe exponential growth, connected by the formula: Growth Factor = 1 + Growth Rate. In Grade 11 enVision Algebra 1 (Chapter 6: Exponents and Exponential Functions), students learn that a 20% annual growth rate corresponds to a growth factor of 1.20 — the multiplier applied each period. Converting between the two is essential for writing correct exponential functions of the form f(x) = a(1 + r)ˣ from percentage descriptions and for interpreting real-world data.
Key Concepts
For exponential growth functions, the growth rate $r$ and growth factor $(1+r)$ are related by: $$\text{Growth Factor} = 1 + \text{Growth Rate}$$ $$\text{Growth Rate} = \text{Growth Factor} 1$$.
Common Questions
What is the relationship between growth rate and growth factor?
Growth Factor = 1 + Growth Rate, and Growth Rate = Growth Factor − 1.
If a population grows at 5% per year, what is the growth factor?
Growth factor = 1 + 0.05 = 1.05. The population is multiplied by 1.05 each year.
If the growth factor is 1.35, what is the growth rate?
Growth rate = 1.35 − 1 = 0.35 = 35% per period.
How do you write an exponential function from a percentage growth rate?
Use f(x) = a(1 + r)ˣ, where a is the initial value and r is the decimal growth rate. For 8% growth: f(x) = a(1.08)ˣ.
What does a growth factor less than 1 indicate?
A growth factor between 0 and 1 indicates exponential decay rather than growth — the quantity decreases each period.
Why do exponential functions use growth factor rather than growth rate in the base?
Because the function multiplies by the growth factor each time period, not by the growth rate. Using 1 + r directly in the base makes this multiplicative process clear.