Graph the parabola. \[ y=-3(x+4)^{2}-1 \] Plot five points on the parabola:

To graph the parabola given by the equation \[ y = -3(x + 4)^2 - 1, \] we can start by identifying the vertex and then plotting some additional points. ### Step 1: Identify the Vertex The equation is in vertex form \( y = a(x - h)^2 + k \), where \((h, k)\) is the vertex. Here, we have: - \( a = -3 \) - \( h = -4 \) - \( k = -1 \) Thus, the vertex of the parabola is at the point \((-4, -1)\). ### Step 2: Determine the Direction Since \( a = -3 \) is negative, the parabola opens downwards. ### Step 3: Calculate Additional Points We can choose values for \( x \) around the vertex to find corresponding \( y \) values. Let's calculate \( y \) for \( x = -6, -5, -4, -3, -2 \). 1. **For \( x = -6 \)**: \[ y = -3(-6 + 4)^2 - 1 = -3(-2)^2 - 1 = -3(4) - 1 = -12 - 1 = -13. \] Point: \((-6, -13)\) 2. **For \( x = -5 \)**: \[ y = -3(-5 + 4)^2 - 1 = -3(-1)^2 - 1 = -3(1) - 1 = -3 - 1 = -4. \] Point: \((-5, -4)\) 3. **For \( x = -4 \)** (the vertex): \[ y = -3(-4 + 4)^2 - 1 = -3(0)^2 - 1 = -1. \] Point: \((-4, -1)\) 4. **For \( x = -3 \)**: \[ y = -3(-3 + 4)^2 - 1 = -3(1)^2 - 1 = -3(1) - 1 = -3 - 1 = -4. \] Point: \((-3, -4)\) 5. **For \( x = -2 \)**: \[ y = -3(-2 + 4)^2 - 1 = -3(2)^2 - 1 = -3(4) - 1 = -12 - 1 = -13. \] Point: \((-2, -13)\) ### Step 4: Summary of Points The five points we have calculated are: 1. \((-6, -13)\) 2. \((-5, -4)\) 3. \((-4, -1)\) (vertex) 4. \((-3, -4)\) 5. \((-2, -13)\) ### Step 5: Plotting the Points Now, we can plot these points on a coordinate plane: - \((-6, -13)\) - \((-5, -4)\) - \((-4, -1)\) - \((-3, -4)\) - \((-2, -13)\) ### Step 6: Draw the Parabola Connect the points smoothly to form the shape of the parabola, which opens downwards. ### Final Graph The graph of the parabola will look like this: ``` y | | * | * * | * * | * * | * * | * * | * * | * * | * * | * * |* * +---------------------- x -6 -5 -4 -3 -2 -1 0 ``` This is a rough sketch of the parabola based on the calculated points.