1. Let's start by understanding what a variation table is. It shows how a function's value changes as the input changes, often indicating increasing or decreasing behavior. 2. Suppose we have a function $f(x)$ and a variation table with some known values and one missing value. 3. For example, consider the table: | $x$ | 1 | 2 | 3 | 4 | |-----|---|---|---|---| | $f(x)$ | 2 | ? | 6 | 8 | 4. We want to find the missing value $f(2)$. 5. If the function is linear (constant rate of change), the difference between consecutive $f(x)$ values is constant. 6. Calculate the differences where possible: $f(3) - f(4) = 6 - 8 = -2$ 7. Assuming constant difference, $f(2) - f(3) = -2$ so $f(2) = f(3) - 2 = 6 - 2 = 4$. 8. The completed table is: | $x$ | 1 | 2 | 3 | 4 | |-----|---|---|---|---| | $f(x)$ | 2 | 4 | 6 | 8 | 9. This method works if the function is linear or the variation is arithmetic. 10. If the variation is not linear, other methods like ratios for geometric variation or given formulas are needed. Final answer: The missing value $f(2)$ is 4.