1. **Problem Statement:** Determine whether each function (A, B, and C) is linear or nonlinear based on their representations (table, graph, equation) and justify the answers. 2. **Function A (Table):** - Given points: (3,5), (6,0), (9,-5), (12,-10). - To check linearity, calculate the rate of change (slope) between consecutive points: $$\text{slope} = \frac{\Delta y}{\Delta x}$$ - Between (3,5) and (6,0): $$\frac{0 - 5}{6 - 3} = \frac{-5}{3} = -\frac{5}{3}$$ - Between (6,0) and (9,-5): $$\frac{-5 - 0}{9 - 6} = \frac{-5}{3} = -\frac{5}{3}$$ - Between (9,-5) and (12,-10): $$\frac{-10 - (-5)}{12 - 9} = \frac{-5}{3} = -\frac{5}{3}$$ - Since the slope is constant between all points, Function A is **linear**. 3. **Function B (Graph):** - The graph is described as a downward sloping S-shaped curve. - An S-shaped curve indicates the slope changes at different points (non-constant rate of change). - Linear functions have straight-line graphs with constant slope. - Therefore, Function B is **nonlinear**. 4. **Function C (Equation):** - Given equation: $$y = 2x^2 + 3x - 8$$ - The presence of the $$x^2$$ term means the function is quadratic. - Quadratic functions are nonlinear because their rate of change is not constant. - Hence, Function C is **nonlinear**. **Final answers:** - Function A: Linear - Function B: Nonlinear - Function C: Nonlinear