1. **State the problem:** We are given a table of values for a function $k(r)$ at points $r = 1, 3, 5, 7, 9$ and need to determine if the function is linear, quadratic, or neither. 2. **Recall the method:** To identify the type of function from discrete values, we check the first and second differences. - For a linear function, the first differences (changes in $k(r)$) are constant. - For a quadratic function, the second differences (changes in first differences) are constant. 3. **Calculate first differences:** $$\Delta k = k(r_{i+1}) - k(r_i)$$ - $k(3)-k(1) = 3 - 2 = 1$ - $k(5)-k(3) = 6 - 3 = 3$ - $k(7)-k(5) = 7 - 6 = 1$ - $k(9)-k(7) = 10 - 7 = 3$ First differences: $1, 3, 1, 3$ 4. **Calculate second differences:** $$\Delta^2 k = \Delta k_{i+1} - \Delta k_i$$ - $3 - 1 = 2$ - $1 - 3 = -2$ - $3 - 1 = 2$ Second differences: $2, -2, 2$ 5. **Analyze differences:** - First differences are not constant. - Second differences are not constant either (they alternate between 2 and -2). 6. **Conclusion:** Since neither first nor second differences are constant, the function is neither linear nor quadratic based on the given data. **Note:** The original answer states the function is quadratic because the second differences are constant, but the calculated second differences are not constant. **Final answer:** The function is neither linear nor quadratic based on the given values.