Use the table to answer the 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 answer

The table provided illustrates the behavior of the function \( f(x) = a \cdot b^x \) as \( x \) increases, specifically focusing on how the function changes over equal intervals of \( \Delta x \).

From the computations shown, the growth factor is expressed as \( b^{\Delta x} \), which demonstrates that when \( x \) increases by a fixed amount \( \Delta x \), the function's output is multiplied by a constant factor \( b^{\Delta x} \).

Given these details, the correct option that the table could be used to prove is:

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

This illustrates that for an exponential function, the growth rate (or growth factor) remains consistent over equal intervals, unlike linear functions which grow by equal differences.