Quadratic Inequality 6D3D9F

1. **State the problem:** We need to identify the graph that represents the inequality $$y < -2x^2 + 8x - 4$$. 2. **Understand the quadratic function:** The quadratic function is $$y = -2x^2 + 8x - 4$$. 3. **Analyze the parabola:** Since the coefficient of $$x^2$$ is $$-2$$ (negative), the parabola opens downward. 4. **Find the vertex:** The vertex of a parabola $$y = ax^2 + bx + c$$ is at $$x = -\frac{b}{2a}$$. Calculate: $$x = -\frac{8}{2 \times (-2)} = -\frac{8}{-4} = 2$$ Find $$y$$ at $$x=2$$: $$y = -2(2)^2 + 8(2) - 4 = -2(4) + 16 - 4 = -8 + 16 - 4 = 4$$ So the vertex is at $$(2, 4)$$. 5. **Find the y-intercept:** Set $$x=0$$: $$y = -2(0)^2 + 8(0) - 4 = -4$$ So the parabola crosses the y-axis at $$(0, -4)$$. 6. **Interpret the inequality:** The inequality $$y < -2x^2 + 8x - 4$$ means the solution region is all points **below** the parabola. 7. **Summary:** The correct graph should show a downward-opening parabola with vertex at $$(2,4)$$, crossing the y-axis at $$(0,-4)$$, and the shaded region below the curve. **Final answer:** The graph representing $$y < -2x^2 + 8x - 4$$ is the downward-opening parabola with vertex $$(2,4)$$ and y-intercept $$(0,-4)$$, with the region below the curve shaded.