What is the value of r from these examples?

Question

1. f(x) = 2(1.25)^x
2. f(x) = 1.25(0.85)^t
3. f(x) = 5(2.24)^t

Answer 1

The base number in these equations, the number inside the parentheses, represents (1 + r). So, in # 1 for example, 1 + r = 1.25, and r = .25 which is frequently expressed as a percentage, r = 25%. For # 2, r will be a negative number, which means this exponential function would model exponential depreciation, at an annual rate of - 15%. Example # 3 has a whopping growth rate of 124%.

Answer 2

1) f(x) = 2(1.25)^x

It's a growth because the factor 1.25 > 1, then:

1 + r = 1.25

r = 1.25 - 1

r = 0.25

r = 25 % growth

2) f(x) = 1.25(0.85)^t

It's a decay because the factor 0.85 < 1, then:

1 - r = 0.85

-r = 0.85 - 1

-r = -0.15

r = 0.15

r = 15% decay

3) f(x) = 5(2.24)^t

It's a growth because the factor 2.24 > 1, then:

1 + r = 2.24

r = 2.24 - 1

r = 1.24

r = 124% growth