1. The problem is to sketch the graph of the function $$y = -2(x + 1)^2 - 3$$ by transforming the graph of $$y = x^2$$. 2. The base function is $$y = x^2$$, which is a parabola opening upwards with vertex at the origin (0,0). 3. The transformations applied are: - Horizontal translation left by 1 unit due to $$(x + 1)$$ inside the square. - Vertical stretch by a factor of 2 because of the coefficient 2 multiplying the squared term. - Reflection across the x-axis because of the negative sign in front of 2. - Vertical translation down by 3 units due to the $$-3$$ outside the squared term. 4. Step-by-step transformation: - Start with $$y = x^2$$. - Translate left by 1: $$y = (x + 1)^2$$. - Stretch vertically by 2: $$y = 2(x + 1)^2$$. - Reflect across x-axis: $$y = -2(x + 1)^2$$. - Translate down by 3: $$y = -2(x + 1)^2 - 3$$. 5. The vertex of the transformed parabola is at $$(-1, -3)$$. 6. The parabola opens downward (due to the negative coefficient) and is narrower than the base parabola (due to the stretch factor 2). 7. To confirm, use a graphing utility to plot $$y = -2(x + 1)^2 - 3$$ and verify the vertex and shape. Final answer: The graph is a parabola with vertex at $$(-1, -3)$$, opening downward, vertically stretched by 2, and shifted left and down as described.