Graphing Quadratic Inequality D2Dff2

1. **State the problem:** We need to graph the quadratic inequality $$y \leq 3x^2 - 6x + 1$$. 2. **Understand the quadratic inequality:** The inequality involves a quadratic expression on the right side, and the inequality symbol is \(\leq\), meaning the graph includes points where $$y$$ is less than or equal to the quadratic expression. 3. **Analyze the quadratic function:** The quadratic function is $$f(x) = 3x^2 - 6x + 1$$. - The coefficient of $$x^2$$ is positive ($$3 > 0$$), so the parabola opens upward. 4. **Graphing the parabola:** - The parabola is solid because the inequality is \(\leq\), which includes equality. - The vertex can be found using the formula $$x = -\frac{b}{2a}$$ where $$a=3$$ and $$b=-6$$. Calculate the vertex x-coordinate: $$x = -\frac{-6}{2 \times 3} = \frac{6}{6} = 1$$ Calculate the vertex y-coordinate: $$y = 3(1)^2 - 6(1) + 1 = 3 - 6 + 1 = -2$$ So the vertex is at $$(1, -2)$$. 5. **Shading the region:** - Since the inequality is $$y \leq 3x^2 - 6x + 1$$, the shaded region is **below** the parabola. 6. **Summary:** - The graph is a solid parabola opening upward with vertex at $$(1, -2)$$. - The shaded region is the area below the parabola. This matches the description given: a solid parabola opening upward with shading below it. **Final answer:** The graph representing $$y \leq 3x^2 - 6x + 1$$ is a solid upward-opening parabola with vertex at $$(1, -2)$$ and shading below the curve.