Question

Use the table to answer the question.

This table displays the exponential growth of a function over equal intervals.
x
𝑥
f(x)=a⋅bx
𝑓
(
𝑥
)
=
𝑎
⋅
𝑏
𝑥
x0
𝑥
0
a⋅bx0
𝑎
⋅
𝑏
𝑥
0
x0+Δx
𝑥
0
+
Δ
𝑥
a⋅b(x0+Δx)
𝑎
⋅
𝑏
(
𝑥
0
+
Δ
𝑥
)
x0+2Δx
𝑥
0
+
2
Δ
𝑥
a⋅b(x0+2Δx)
𝑎
⋅
𝑏
(
𝑥
0
+
2

Answer 1

From f(x) = a·b^x and the x-values x0, x0+Δx, x0+2Δx,...

- f(x0+Δx) = a·b^{x0+Δx} = (a·b^{x0})·b^{Δx} = f(x0)·b^{Δx}.
- f(x0+2Δx) = a·b^{x0+2Δx} = f(x0)·b^{2Δx} = f(x0)·(b^{Δx})^2.
- In general, for integer k ≥ 0: f(x0 + kΔx) = f(x0)·(b^{Δx})^k.

So the values form a geometric sequence with common multiplicative factor b^{Δx} for each equal interval Δx. The percent growth per interval is 100·(b^{Δx} − 1)%.