Question Use technology to find points and then graph the function \( y=-2|x+3|-2 \), following the instructions below. Plot at least five points that fit on the axes below. Click a point to deleteit

Sure! Let's graph the function \( y = -2|x + 3| - 2 \) by following these steps: ### 1. Understand the Function The function \( y = -2|x + 3| - 2 \) is a **V-shaped** absolute value function that has been transformed in the following ways: - **Horizontal Shift:** The graph is shifted **3 units to the left** because of the \( +3 \) inside the absolute value. - **Vertical Stretch and Reflection:** The graph is **vertically stretched by a factor of 2** and **reflected over the x-axis** due to the coefficient \( -2 \). - **Vertical Shift:** The entire graph is shifted **2 units downward** because of the \( -2 \) at the end. ### 2. Create a Table of Points To accurately plot the graph, let's calculate the value of \( y \) for various \( x \)-values. We'll choose \( x \)-values around the vertex to ensure we capture the shape of the graph. | \( x \) | Calculation | \( y \) | |--------|---------------------------------------------|---------| | -5 | \( y = -2|{-5} + 3| - 2 = -2|{-2}| - 2 = -4 - 2 \) | -6 | | -4 | \( y = -2|{-4} + 3| - 2 = -2|{-1}| - 2 = -2 - 2 \) | -4 | | -3 | \( y = -2|{-3} + 3| - 2 = -2|{0}| - 2 = 0 - 2 \) | -2 | | -2 | \( y = -2|{-2} + 3| - 2 = -2|{1}| - 2 = -2 - 2 \) | -4 | | -1 | \( y = -2|{-1} + 3| - 2 = -2|{2}| - 2 = -4 - 2 \) | -6 | ### 3. Plot the Points Using the table above, plot the following points on your coordinate plane: 1. **(-5, -6)** 2. **(-4, -4)** 3. **(-3, -2)** *(This is the vertex of the graph)* 4. **(-2, -4)** 5. **(-1, -6)** ### 4. Draw the Graph 1. **Mark the Vertex:** Start by plotting the vertex at **(-3, -2)**. 2. **Plot the Other Points:** Add the points **(-5, -6)**, **(-4, -4)**, **(-2, -4)**, and **(-1, -6)**. 3. **Connect the Points:** Draw straight lines connecting the points to form a "V" shape. Since the coefficient is negative (\(-2\)), the graph opens downward. ### 5. Interactive Features If you're using graphing software or an online tool (like Desmos or GeoGebra), you can: - **Plot the Points:** Enter the points to see them on the graph. - **Click to Delete Points:** Most interactive graphing tools allow you to click on a plotted point to remove it. Simply click on any point you wish to delete. ### 6. Visual Representation Here's a rough sketch of how the graph should look: ``` y | | * | * * | * * | * * | * * |------------------------- x ``` *(The vertex is at the lowest point of the "V.")* ### 7. Verification To ensure accuracy, you can plug additional \( x \)-values into the function and verify that they lie on the graph you've drawn. ### 8. Using Technology For a more precise graph, consider using online graphing tools: - **[Desmos](https://www.desmos.com/calculator):** A user-friendly graphing calculator. - **[GeoGebra](https://www.geogebra.org/graphing):** Offers dynamic graphing features. These tools allow you to input the function directly, visualize it instantly, and interact with the graph by adding or removing points as needed. --- Feel free to reach out if you need further assistance with graphing or understanding the function!