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:
- The growth factor from \( x_0 \) to \( x_0 + \Delta x \) is \( b^{\Delta x} \).
- 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.