1. **State the problem:** We are given a table of values for $x$ and $y$: $$\begin{array}{c|ccccc} x & 0 & 1 & 2 & 3 & 4 \\ y & 2.5 & 5 & 10 & 20 & 40 \\\end{array}$$ We want to determine which function best models the data based on whether the first difference, second difference, or ratio is constant. 2. **Recall the rules:** - If the first differences (differences between consecutive $y$ values) are constant, the data is modeled by a linear function. - If the second differences (differences of the first differences) are constant, the data is modeled by a quadratic function. - If the ratio of consecutive $y$ values is constant, the data is modeled by an exponential function. 3. **Calculate the first differences:** $$5 - 2.5 = 2.5$$ $$10 - 5 = 5$$ $$20 - 10 = 10$$ $$40 - 20 = 20$$ The first differences are $2.5, 5, 10, 20$, which are not constant. 4. **Calculate the second differences:** $$5 - 2.5 = 2.5$$ $$10 - 5 = 5$$ $$20 - 10 = 10$$ The second differences are $2.5, 5, 10$, which are not constant. 5. **Calculate the ratios:** $$\frac{5}{2.5} = 2$$ $$\frac{10}{5} = 2$$ $$\frac{20}{10} = 2$$ $$\frac{40}{20} = 2$$ The ratios are all equal to $2$, which is constant. 6. **Conclusion:** Since the ratio of consecutive $y$ values is constant, the data is best modeled by an exponential function. **Final answer:** The function that best models the data is an exponential function because there is a constant ratio between consecutive $y$ values.