Lesson 9.3 WS Determine if the graph is a minimum or a maximum after identifying the a. \( \begin{array}{ll}\text { 1) } y=-2(x+4)^{2}-4 & \text { 2) } y=3(x+1)^{2}+1\end{array} \)

Sure, let's analyze each quadratic function to determine the value of \( a \) and whether the graph has a **minimum** or a **maximum** point. --- ### 1) \( y = -2(x + 4)^2 - 4 \) **Step 1: Identify \( a \)** - The quadratic function is in the form \( y = a(x - h)^2 + k \). - Comparing, \( a = -2 \). **Step 2: Determine the Orientation** - If \( a > 0 \), the parabola opens **upward** (has a **minimum**). - If \( a < 0 \), the parabola opens **downward** (has a **maximum**). **Here, \( a = -2 \) which is less than 0.** **Conclusion:** - **Maximum Point** - **Vertex (Maximum Point):** \( (-4, -4) \) **Graph Behavior:** - The parabola opens **downward**, reaching its highest point at the vertex \( (-4, -4) \). --- ### 2) \( y = 3(x + 1)^2 + 1 \) **Step 1: Identify \( a \)** - The quadratic function is in the form \( y = a(x - h)^2 + k \). - Comparing, \( a = 3 \). **Step 2: Determine the Orientation** - If \( a > 0 \), the parabola opens **upward** (has a **minimum**). - If \( a < 0 \), the parabola opens **downward** (has a **maximum**). **Here, \( a = 3 \) which is greater than 0.** **Conclusion:** - **Minimum Point** - **Vertex (Minimum Point):** \( (-1, 1) \) **Graph Behavior:** - The parabola opens **upward**, reaching its lowest point at the vertex \( (-1, 1) \). --- ### **Summary** 1. **\( y = -2(x + 4)^2 - 4 \)** - **\( a = -2 \)** - **Maximum Point at \( (-4, -4) \)** - Parabola opens **downward**. 2. **\( y = 3(x + 1)^2 + 1 \)** - **\( a = 3 \)** - **Minimum Point at \( (-1, 1) \)** - Parabola opens **upward**. --- **Visual Representation:** 1. **First Parabola (\( a = -2 \))** - **Vertex:** Low point since it opens downward. - **Shape:** Inverted U. 2. **Second Parabola (\( a = 3 \))** - **Vertex:** Low point since it opens upward. - **Shape:** Regular U. Feel free to plot these equations on a graphing tool to visualize their shapes and verify the locations of their minimum or maximum points!