Use the table to answer the question.

Question

x
f(x)=a⋅bx
Growth Factor
x0
a⋅bx0

x0+Δx
a⋅b(x0+Δx)
a⋅b(x0+Δx)a⋅bx0=bx0+Δx−x0=bΔx

x0+2Δx
a⋅b(x0+2Δx)
a⋅b(x0+2Δx)a⋅b(x0+Δx)=bx0+2Δx−x0−Δx=bΔx

What could the table be used to prove?

Option #1: Over equal intervals, linear functions grow by equal differences.

Option #2: Over equal intervals, linear functions grow by equal factors.

Option #3: Over equal intervals, exponential functions grow by equal differences.

Option #4: Over equal intervals, exponential functions grow by equal factors.

(1 point)

Answer 1

The table demonstrates that when evaluating the function \(f(x) = a \cdot b^x\) at equal intervals, the growth factor remains consistent. Specifically, it shows how the ratios between consecutive evaluations (i.e., \(f(x_0 + \Delta x)\) and \(f(x_0)\), and \(f(x_0 + 2\Delta x)\) and \(f(x_0 + \Delta x)\)) lead to a consistent growth rate determined by the base \(b\).

Given this information, the correct option is:

Option #4: Over equal intervals, exponential functions grow by equal factors.

This is because the growth factor \(b^{\Delta x}\) remains constant, showing that the growth is proportional (by equal factors) rather than additive (as would be the case for linear functions).