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)
Option #
is correct.

Answer 1

The table describes the growth of a function of the form \( f(x) = a \cdot b^x \), which is an exponential function. The derivation shows that the ratio of the function values over equal intervals (specifically \( \Delta x \)) is constant and equal to the growth factor \( b^{\Delta x} \).

This indicates that as \( x \) increases by equal increments (specifically \( \Delta x \)), the function values grow by a constant multiplicative factor.

Therefore, the correct option is:

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