Let's break down each part of the problem step by step, referencing the graph and the given functions: Given: - **Position function:** \( s(t) = -16(t-3)^2 + 150 \) (solid line) - **Velocity function:** \( v(t) \) (dashed line) ### **a. When the velocity is positive, what do you notice about the position function?** #### **Step-by-Step Solution:** 1. **Velocity and Position Relationship:** Velocity is the derivative of the position function. If velocity is **positive**, the position function is **increasing** (moving upward). 2. **Graph Observation:** On the graph, when the dashed velocity line is **above the t-axis** (positive values), the solid position curve is **rising**. #### **Final Answer:** **When the velocity is positive, the position function is increasing.** --- ### **b. When the velocity is negative, what do you notice about the position function?** #### **Step-by-Step Solution:** 1. **Velocity and Position Relationship:** If velocity is **negative**, the position function is **decreasing** (moving downward). 2. **Graph Observation:** When the dashed velocity line is **below the t-axis** (negative values), the solid position curve is **falling**. #### **Final Answer:** **When the velocity is negative, the position function is decreasing.** --- ### **c. What is the special point on the position function for which the velocity is equal to 0?** #### **Step-by-Step Solution:** 1. **Find Velocity Function:** The velocity is the derivative of the position function: \[ s(t) = -16(t-3)^2 + 150 \] \[ v(t) = s'(t) = -16 \cdot 2(t-3) = -32(t-3) \] 2. **Set velocity to zero to find the special point:** \[ v(t) = 0 \implies -32(t-3) = 0 \implies t = 3 \] 3. **Find the corresponding position:** \[ s(3) = -16(3-3)^2 + 150 = 150 \] 4. **Interpretation:** At \( t = 3 \), the velocity is zero. On the graph, this is the **vertex** (maximum point) of the position curve. #### **Final Answer:** **The special point is the vertex (maximum point) of the position function at \( t = 3 \), \( s(3) = 150 \), where the velocity is zero.** --- ## **Summary Table** | Part | Explanation | Final Answer | |------|-------------|--------------| | (a) | Velocity positive ⟹ position increasing | Position function is increasing | | (b) | Velocity negative ⟹ position decreasing | Position function is decreasing | | (c) | Velocity zero ⟹ vertex of position function | \( t = 3, s(3) = 150 \); maximum point |