Problem 35 - Algebra and Trigonometry

Question

21-46 . Graph the solution of the system of inequalities. Find the coordinates of all vertices, and determine whether the solution set is bounded. $$ \left\\{\begin{array}{c}{x^{2}-y \leq 0} \\ {2 x^{2}+y \leq 12}\end{array}\right. $$

Step-by-Step Solution

Answer

Vertices at \((2, 4), (-2, 4), (0, 0)\); solution set is bounded.

Step 1: Convert Inequalities into Equalities

To graph the solution set of the inequalities, begin by converting each inequality into an equality. For the first inequality, convert \(x^2 - y \leq 0\) to \(x^2 = y\). For the second inequality, convert \(2x^2 + y \leq 12\) to \(2x^2 + y = 12\).

Step 2: Graph the Parabola from the First Equation

The first equality \(y = x^2\) represents a parabola that opens upward with its vertex at the origin \((0,0)\). Plot this parabola on a coordinate plane as the boundary line of the first inequality.

Step 3: Graph the Second Equation as a Line

The second equality \(y = 12 - 2x^2\) represents a downward opening parabola with its vertex at the point \((0, 12)\). Draw this parabola on the coordinate plane as the boundary line for the second inequality.

Step 4: Determine Shaded Regions

Assess which side of each boundary line to shade by testing a point. For \(x^2 - y \leq 0\), test \((0, 1)\) which results in \(0^2 - 1 = -1 \leq 0\), showing that the region below the parabola \(y = x^2\) should be shaded. For \(2x^2 + y \leq 12\), test \((0, 0)\) which results in \(0 + 0 = 0 \leq 12\), thus the region below the parabola \(y = 12 - 2x^2\) should be shaded.

Step 5: Find Points of Intersection

Solve the system of equations \(x^2 = y\) and \(2x^2 + y = 12\) together. Substitute \(y = x^2\) from the first equation into the second to get \(2x^2 + x^2 = 12\) or \(3x^2 = 12\). Solving gives \(x^2 = 4\), thus \(x = 2\) or \(x = -2\). The corresponding \(y\) values are \(4\), giving intersection points \((2, 4)\) and \((-2, 4)\).

Step 6: Determine if the Solution Set is Bounded

The solution set is the overlapping shaded region from beneath both parabolas. It is entirely enclosed by the parabolas themselves and therefore is bounded. The vertices of this region are at \((2, 4), (-2, 4), (0, 0)\) and \((0, 12)\) where the lines intersect the axes.

Key Concepts

Systems of Inequalities

A system of inequalities consists of multiple inequalities that are considered simultaneously. Unlike equations, which have exact solutions, inequalities usually define a range of values. When graphing systems of inequalities, the solution is the region where all individual inequalities in the system overlap. In this exercise, we have two inequalities:

• \( x^2 - y \leq 0 \)
• \( 2x^2 + y \leq 12 \)

These inequalities describe regions on the coordinate plane. The solution to the system is the area where both regions overlap, satisfying both inequalities. Before finding the solution set, each inequality is often converted to an equality, which helps in plotting the boundary lines. By shading the region that satisfies each inequality, you can visually identify the solution set where these regions intersect.

Graphing Parabolas

Graphing parabolas involves plotting curves on a coordinate plane, generated by quadratic functions. Parabolas are symmetric and have distinctive U-shaped curves. In the given system, we need to graph two parabolas:

• For the first inequality \( x^2 - y \leq 0 \), convert it to the equation \( y = x^2 \). This describes a parabola that opens upwards with its vertex at the origin \((0, 0)\).
• For the second inequality \( 2x^2 + y \leq 12 \), convert it to the equation \( y = 12 - 2x^2 \). This represents a downward-opening parabola with its vertex at \((0, 12)\).

Each parabola marks the boundary of its respective inequality. To graph them, determine several points by substituting different values for \( x \) and finding corresponding \( y \) values. Connect these points smoothly, reflecting the parabola's curve shape.

Intersection Points

Intersection points are where the graphs of two equations meet, indicating solutions that satisfy both equations simultaneously. Finding these points is crucial in systems of inequalities, as they help outline the solution region. To find intersection points for the given system, solve the equations:

• \( y = x^2 \)
• \( y = 12 - 2x^2 \)

By substituting \( y = x^2 \) into \( y = 12 - 2x^2 \), we get \( 3x^2 = 12 \). Solving gives \( x = \pm 2 \), with corresponding \( y \) values of 4. This provides the intersection points \((2, 4)\) and \((-2, 4)\). These points are crucial for describing the boundary of the solution region.

Bounded Solution Set

A bounded solution set is an area enclosed within finite limits on a graph. When working with systems of inequalities, determining if the solution set is bounded is key to understanding the region's extent and confinement.In our exercise, the solution set lies where the shaded regions from both parabolas overlap.

• The parabola \( y = x^2 \) and \( y = 12 - 2x^2 \) enclose a region shaped somewhat like a lens or football.
• This region is bounded as it does not extend indefinitely in any direction and is enclosed by the intersecting curves.
• The vertices, or corner points of this bounded region, are \((2, 4)\), \((-2, 4)\), \((0, 0)\), and \((0, 12)\).

These vertices indicate the outermost boundaries of the solution area, showing that the solution is fully contained within the parabolas' curves.