Problem 39 - Precalculus: Mathematics for Calculus

Question

Graph the solution set of the system of inequalities. Find the coordinates of all vertices, and determine whether the solution set is bounded. $$\left\\{\begin{array}{l} y \leq 9-x^{2} \\ x \geq 0, \quad y \geq 0 \end{array}\right.$$

Step-by-Step Solution

Answer

Vertices: (0, 0), (3, 0), and (0, 9). The solution set is bounded.

Step 1: Understand the Inequalities

The system consists of the inequalities \( y \leq 9 - x^2 \), \( x \geq 0 \), and \( y \geq 0 \). The first inequality represents a parabola that opens downward with a vertex at \((0,9)\). The other two represent the constraints that the solution must be in the first quadrant.

Step 2: Plot the Parabola

To graph \( y = 9 - x^2 \), start by plotting the vertex at \((0,9)\) and using the fact that the parabola opens downward with symmetry about the y-axis. Calculate some additional points, like \( (1, 8) \) and \( (2, 5) \). Draw the curve considering \( y \leq 9 - x^2 \), which includes the area below the parabola including the curve itself.

Step 3: Plot Additional Constraints

Graph the line \( x = 0 \), which is the y-axis, and the line \( y = 0 \), which is the x-axis. These indicate that the feasible region is the part of the plane in the first quadrant below or on the parabola.

Step 4: Identify and Find Vertices

Vertices of the solution set occur at intersections of the boundary lines. The parabola intersects \( y = 0 \) only where \( 9 - x^2 = 0 \). Solving \( 9 - x^2 = 0 \) gives \( x = \pm 3 \), but only \( x = 3 \) is in the feasible region since \( x \geq 0 \). Thus, two vertices are at \((0,0)\) and \((3,0)\). The parabola at \( x = 0 \) gives the vertex \((0, 9)\) in the feasible set.

Step 5: Determine If the Solution Set is Bounded

The assumptions \( x \geq 0 \) and \( y \geq 0 \) limit the solution to the first quadrant, forming a closed region bounded by \( y = 0 \), \( x = 0 \), and \( y = 9 - x^2 \). The solution is bounded because it is enclosed within a finite area.

Key Concepts

Graphing Inequalities

When graphing inequalities like the system given, we start by understanding each inequality separately. The inequality \( y \leq 9 - x^2 \) is a bit more complex because it involves a quadratic term. This inequality describes all the points below the parabola \( y = 9 - x^2 \), which opens downward with a peak at \( (0,9) \). To graph this, plot the parabola first by marking the vertex and other nearby points such as \( (1,8) \) and \( (2,5) \). These points help guide how the curve bends. Remember to shade below the curve, as the inequality indicates \( y \) values that are less than or equal to the expression.

The lines \( x \geq 0 \) and \( y \geq 0 \) are simpler. They correspond to the boundaries set by the x- and y-axes. Simply, they mean only consider points to the right of the y-axis and above the x-axis. When combined, these three boundaries form a region within the first quadrant of the coordinate plane where all the inequalities hold true. Ensure that each part of the inequality system is considered to accurately draw the feasible region.

Solution Set

The solution set of a system of inequalities represents all points that satisfy every inequality simultaneously. In this context, it means the area on the graph that meets all the conditions described by the inequalities.

For the given inequalities, the solution set is the region in the first quadrant of the coordinate plane that lies below the curve \( y = 9 - x^2 \). The boundaries defined by \( x \geq 0 \) (right of the y-axis) and \( y \geq 0 \) (above the x-axis) further limit this region, creating a specific area where all conditions are valid.

• The solution set is the visual representation where "everything works" - all inequalities are true.
• This area will be shaded on the graph to make it stand out, making it easy to see the solution in the context of the entire coordinate plane.

Recognizing this solution area helps in solving inequalities graphically, and appreciating how different lines and curves interact.

Bounded Region

A bounded region in a graph means that the solution set is enclosed within finite boundaries.

In this exercise, the region formed by the inequalities is bounded because it is closed off by the parabola \( y = 9 - x^2 \) at the top, and the axes at the other two edges. Specifically, the horizon line made by \( y=0 \), vertically by \( x=0 \), and the downward curve of \( y=9-x^2 \) form a triangular shape.

• A bounded region implies that there is a finite area where the solution holds true.
• Evaluating whether a solution set is bounded can help determine if there are limits to the possible solution values.

This bounded area provides insights into constraint satisfaction, ensuring solutions conform to specific spatial or geometric limits.

Vertices of Inequalities

Vertices in the context of graphing inequalities refer to the corner points where the boundaries of the solution set intersect. Finding these vertices is essential because they often represent extreme or important values where the conditions of the inequalities meet.

For the problem at hand, the vertices can be identified by solving the intersections of the boundary curves. The key vertices for this solution set are:

• \((0,0)\): where \( x = 0 \) and \( y=0 \) meet at the origin, indicating the bottom left corner of the region.
• \((3,0)\): found by solving \( 9 - x^2 = 0 \), which gives solutions \( x = \pm3 \). Since \( x \geq 0 \) is a condition, only \( x = 3 \) remains valid inside the solution set.
• \((0,9)\): the vertex of the parabola that sits at the top of the solution space.

Recognizing these points allows us to understand the full extent and the exact boundaries of the feasible region defined by the system.