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 you provided focuses on the growth of the function \( f(x) = a \cdot b^{x} \), which is an exponential function. The calculations in the table show how the function changes over equal intervals of \( \Delta x \).

The key findings from the table can be summarized as follows:

1. The growth factor from \( x_0 \) to \( x_0 + \Delta x \) is \( b^{\Delta x} \).
2. The growth factor from \( x_0 + \Delta x \) to \( x_0 + 2\Delta x \) is also \( b^{\Delta x} \).

This indicates that over equal intervals (\( \Delta x \)), the function grows by equal factors (specifically \( b^{\Delta x} \)).

Based on the options provided:

- **Option #1:** Over equal intervals, linear functions grow by equal differences. (This is true for linear functions, but not related to your table.)
- **Option #2:** Over equal intervals, linear functions grow by equal factors. (This is also true for linear functions but not relevant here.)
- **Option #3:** Over equal intervals, exponential functions grow by equal differences. (This is not true; exponential functions grow by factors, not differences.)
- **Option #4:** Over equal intervals, exponential functions grow by equal factors. (This is the correct statement that aligns with the information in your table.)

Thus, the correct answer is **Option #4**: Over equal intervals, exponential functions grow by equal factors.