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Graphing inequalities involves plotting a graph based on an inequality equation and then shading a region to represent all possible solutions. In our example, we have two inequalities: \(y \, \geq \, x^2 \,+\, 4x\, + \, 4\) and \(y \,<\, -x^{2}\). To graph these:

1. **Plot the Parabolas:** Start by plotting each quadratic equation just as you would plot any function with an equal sign. For the first inequality, plot the parabola \(y = x^2 + 4x + 4\), which opens upwards. For the second, plot \(y = -x^2\), which opens downwards.

2. **Shading the Regions:** After plotting the parabolas, shade the area above the parabola for \(y \, \geq \, x^2 \,+\, 4x\, + \, 4\) because it includes all points where y is greater than the parabola. For \(y \,<\, -x^{2}\), shade the area below the parabola, including all points where y is less than the parabola.

By graphing and shading, you visually show the solutions that satisfy each inequality.

Understanding parabolas is key when working with quadratic inequalities. A parabola is a symmetric curve that forms the graph of a second-degree polynomial function. Key properties of parabolas include:

1. **Vertex:** The highest or lowest point on the parabola. For \(y = x^2 + 4x + 4\), completing the square or using the vertex formula \(x = -b/2a\), we find the vertex is at (-2,0). Similarly, the vertex of \(y = -x^2\) is at (0,0).

2. **Direction:** Depending on the coefficient of \(x^2\), the parabola can open upwards (positive coefficient) or downwards (negative coefficient). The equality \(y\,=x^2+4x+4\) has a positive coefficient, so the parabola opens upwards. Conversely, \(y\,=-x^2\) opens downwards.

Understanding these properties helps in sketching the parabolas accurately, enabling you to solve quadratic inequalities efficiently.

The solution set for a system of inequalities is the region where the shaded areas from individual inequalities overlap. This overlap represents all points (x, y) that satisfy all inequalities in the system simultaneously.

1. **Identifying the Overlap:** Once you have shaded the regions for each inequality, look for the area where the shaded regions intersect. This intersection is where both inequalities are true at the same time.

2. **Graph Details:** For our example, where \(y \, \geq \, x^2 \,+\, 4x\, + \, 4\) and \(y \,<\, -x^{2}\), the solution set is the region between the two parabolas. The boundaries are crucial: use a solid line for inclusive inequalities (\(y \, \geq \, x^2 \,+\, 4x\, + \, 4\)) and a dashed line for strict inequalities (\(y \,<\, -x^{2}\)).

This helps in visually confirming the set of all solutions that satisfy the given inequalities, making it easier to understand the overall solution.