plot five points, two points po the lete fot the verty and two points to the right Graph the parabola \( g(x)=-3(x+3)^{2}+2 \)

To plot the points and graph the parabola given by the equation \( g(x) = -3(x + 3)^2 + 2 \), we can follow these steps: ### Step 1: Identify the Vertex The equation is in vertex form \( g(x) = a(x - h)^2 + k \), where \((h, k)\) is the vertex of the parabola. From the equation \( g(x) = -3(x + 3)^2 + 2 \): - \( h = -3 \) - \( k = 2 \) So, the vertex is at the point \((-3, 2)\). ### Step 2: Plot the Vertex We can start by plotting the vertex: - Vertex: \((-3, 2)\) ### Step 3: Plot Additional Points Next, we need to plot two points to the left of the vertex and two points to the right. 1. **Left of the vertex**: - For \( x = -4 \): \[ g(-4) = -3(-4 + 3)^2 + 2 = -3(-1)^2 + 2 = -3 + 2 = -1 \] Point: \((-4, -1)\) - For \( x = -5 \): \[ g(-5) = -3(-5 + 3)^2 + 2 = -3(-2)^2 + 2 = -3(4) + 2 = -12 + 2 = -10 \] Point: \((-5, -10)\) 2. **Right of the vertex**: - For \( x = -2 \): \[ g(-2) = -3(-2 + 3)^2 + 2 = -3(1)^2 + 2 = -3 + 2 = -1 \] Point: \((-2, -1)\) - For \( x = -1 \): \[ g(-1) = -3(-1 + 3)^2 + 2 = -3(2)^2 + 2 = -3(4) + 2 = -12 + 2 = -10 \] Point: \((-1, -10)\) ### Step 4: Summary of Points Now we have the following points to plot: - Vertex: \((-3, 2)\) - Left Points: \((-4, -1)\), \((-5, -10)\) - Right Points: \((-2, -1)\), \((-1, -10)\) ### Step 5: Graph the Parabola To graph the parabola, plot the points on a coordinate plane and draw a smooth curve through them, opening downwards since the coefficient of the squared term is negative. ### Graphing Here’s a rough sketch of how the graph would look: ``` y | | * | * * | * * | * * | * * | * * | * * | * * | * * | * * |* * +---------------------- x -5 -4 -3 -2 -1 0 1 ``` ### Conclusion The parabola opens downwards with the vertex at \((-3, 2)\) and the plotted points showing the shape of the parabola.