1. **State the problem:** Determine whether the data sets in problems 14 and 15 represent linear, quadratic, or neither functions. 2. **Recall definitions:** - A linear function has a constant first difference in $y$ values when $x$ increases by 1. - A quadratic function has a constant second difference in $y$ values. 3. **Problem 14:** - Given points: $(-3,48), (-2,22), (-1,6), (0,0), (1,4), (2,18), (3,42)$ - Calculate first differences of $y$: $$22-48 = -26, \quad 6-22 = -16, \quad 0-6 = -6, \quad 4-0 = 4, \quad 18-4 = 14, \quad 42-18 = 24$$ - First differences: $-26, -16, -6, 4, 14, 24$ - Calculate second differences: $$-16 - (-26) = 10, \quad -6 - (-16) = 10, \quad 4 - (-6) = 10, \quad 14 - 4 = 10, \quad 24 - 14 = 10$$ - Second differences are constant at 10, so the function is quadratic. 4. **Problem 15:** - Given points: $(-3,4), (-2,1), (-1,0), (0,1), (1,4), (2,9), (3,16)$ - Calculate first differences of $y$: $$1-4 = -3, \quad 0-1 = -1, \quad 1-0 = 1, \quad 4-1 = 3, \quad 9-4 = 5, \quad 16-9 = 7$$ - First differences: $-3, -1, 1, 3, 5, 7$ - Calculate second differences: $$-1 - (-3) = 2, \quad 1 - (-1) = 2, \quad 3 - 1 = 2, \quad 5 - 3 = 2, \quad 7 - 5 = 2$$ - Second differences are constant at 2, so the function is quadratic. **Final answers:** - Problem 14: B. quadratic - Problem 15: B. quadratic