91Ó°ÊÓ

A parabola is a smooth, symmetrical curve, which is the graph of a quadratic function. In this context, we have the function given by the equation \( y = x^2 \). This particular parabola opens upwards and has its vertex at the origin, \(0, 0\). The vertex is the lowest point on this graph.

The inequality \( y \geq x^2 \) adjusts this graph by indicating that we should shade the area above the parabola. This means we include all points where the y-coordinate is greater than or equal to the value of \( x^2 \) at any given x. When you graph this, you're visualizing not just the curve itself, but also all possible solutions that lie in the region above it. This shaded region helps in determining the feasible region once other inequalities are overlaid on the same graph.

Intersection points occur where two graphs, in this case a line and a parabola, cross each other. To find these points in the exercise, we solve the equation \( y = x^2 \) and the equation \( x + y = 6 \) simultaneously.

Setting \( x^2 = 6 - x \) gives us a quadratic equation \( x^2 + x - 6 = 0 \). We solve this by factoring it into \( (x-2)(x+3) = 0 \), giving us solutions for x as \( x=2 \) and \( x=-3 \). We then substitute these x-values back into either equation to find their corresponding y-values. For \( x=2 \), \( y = 4 \), and for \( x=-3 \), \( y = 9 \).

• Therefore, the intersection points are at \( (2, 4) \) and \( (-3, 9) \).

Understanding and finding these intersection points is crucial as they typically serve as vertices for the feasible region.

The feasible region is the area of the graph that satisfies all inequalities in a system. In this exercise, the feasible region is where the shaded areas from multiple inequalities overlap.

For \( y \geq x^2 \) and \( x + y \geq 6 \), the shaded regions overlap above the parabola and above the line. To determine the actual vertices of this feasible region, consider the intersection points and the behavior of the graphs. Vertices can be found at points like \( (2, 4) \) and \( (-3, 9) \) as well as along parts of the boundaries extending to infinity.

• The feasible region will typically be bounded by some combination of the line and parabola segments, especially near these key intersection points.

By confirming that these points satisfy all the system's inequalities, you ensure that you've correctly outlined the region within which all solutions lie.

A bounded solution is where the feasible region is contained within a closed and finite area of the graph. In other words, it's surrounded and contained on all sides without extending to infinity in any direction.

In this problem, even though it seems like the points between these inequalities create a feasible area, the overall region isn't actually bounded. The parabola \( y = x^2 \) extends upwards indefinitely, and the linear inequality \( x + y \geq 6 \) does not close off this space.

• Thus, the solution set is considered unbounded, extending infinitely towards the upper region of the graph.

Recognizing whether a solution is bounded or unbounded is crucial for understanding the scope of potential solutions in graphical representations.