Question

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.

Answer 1

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.