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

To plot the points and 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. For \( g(x) = -3(x + 3)^2 + 2 \): - \( h = -3 \) - \( k = 2 \) So, the vertex is at the point \((-3, 2)\). ### Step 2: Determine the Direction and Width Since \( a = -3 \) (which is negative), the parabola opens downwards. The value of \( |a| = 3 \) indicates that the parabola is relatively narrow. ### Step 3: Calculate Additional Points To plot the parabola, we can calculate a few additional points by choosing values for \( x \): 1. **For \( x = -4 \)**: \[ g(-4) = -3(-4 + 3)^2 + 2 = -3(-1)^2 + 2 = -3 + 2 = -1 \] Point: \((-4, -1)\) 2. **For \( x = -2 \)**: \[ g(-2) = -3(-2 + 3)^2 + 2 = -3(1)^2 + 2 = -3 + 2 = -1 \] Point: \((-2, -1)\) 3. **For \( x = -1 \)**: \[ g(-1) = -3(-1 + 3)^2 + 2 = -3(2)^2 + 2 = -3(4) + 2 = -12 + 2 = -10 \] Point: \((-1, -10)\) 4. **For \( x = 0 \)**: \[ g(0) = -3(0 + 3)^2 + 2 = -3(3)^2 + 2 = -3(9) + 2 = -27 + 2 = -25 \] Point: \((0, -25)\) 5. **For \( x = 1 \)**: \[ g(1) = -3(1 + 3)^2 + 2 = -3(4)^2 + 2 = -3(16) + 2 = -48 + 2 = -46 \] Point: \((1, -46)\) ### Step 4: Summary of Points We have the following points to plot: - Vertex: \((-3, 2)\) - Additional points: - \((-4, -1)\) - \((-2, -1)\) - \((-1, -10)\) - \((0, -25)\) - \((1, -46)\) ### Step 5: Graphing the Parabola To graph the parabola, plot the points on a coordinate system and draw a smooth curve through them, ensuring that the curve opens downwards. ### Visualization While I can't create visual graphs directly, you can use graphing software or graph paper to plot these points and sketch the parabola. The vertex will be the highest point, and the curve will extend downwards from there.