Problem 26 - Precalculus: Mathematical for Calculus

Question

Graph the solution of the system of inequalities. Find the coordinates of all vertices, and determine whether the solution set is bounded. $$\left\\{\begin{aligned}y & \geq x^{2} \\\x+y & \geq 6\end{aligned}\right.$$

Step-by-Step Solution

Answer

Vertices: (2, 4), (-3, 9), and (6, 0); Solution set is bounded.

Step 1: Graph the Parabola

The inequality \( y \geq x^2 \) represents the region above or on the parabola \( y = x^2 \). To graph it, first sketch the parabola by plotting points such as (0,0), (1,1), (-1,1), (2,4), and (-2,4). Shade the region above the parabola as it represents all points \( y \) greater than or equal to \( x^2 \).

Step 2: Graph the Line

The inequality \( x + y \geq 6 \) represents the region above or on the line \( x + y = 6 \). This is a straight line with a slope of -1 and a y-intercept of 6. To graph it, pick two points, for example, (0,6) and (6,0). Draw the line connecting these points and shade the region above it to represent \( x + y \geq 6 \).

Step 3: Find the Intersection Points

To find potential vertices of the solution region, solve for the intersection of \( y = x^2 \) and \( x + y = 6 \). Substitute \( y = x^2 \) into \( x + y = 6 \), giving \( x + x^2 = 6 \), which simplifies to \( x^2 + x - 6 = 0 \). Solve this quadratic equation using the quadratic formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a=1, b=1, c=-6 \). This gives two solutions: \( x = 2 \) and \( x = -3 \). Substitute back to find \( y \ (2, 4) \) and \(( -3, 9)\).

Step 4: Identify Solution Area and Vertices

Consider the shaded regions from Step 1 and Step 2. The solution set is the overlap, which is bounded by the parabola and the line. The vertices or intersection points obtained are (2, 4), (-3, 9) and another point from the line \((6, 0)\) when \(x + y = 6\).

Step 5: Check Boundedness of the Solution Set

Since the region determined by the intersection of the parabola and the line forms a closed area with intersection points for the boundaries, the solution set is indeed bounded. In this context, it means the solution area is contained within specific boundaries and does not extend to infinity.

Key Concepts

Parabola

A parabola is a U-shaped curve that can open either upwards or downwards depending on its equation. In this problem, the equation of the parabola given is \( y = x^2 \), which opens upwards. The parabola represents all the points \( (x, y) \) where \( y \) is equal to the square of \( x \). To graph this parabola, you should start by plotting key points such as \( (0, 0) \), \( (1, 1) \), and \( (-1, 1) \). As \( x \) increases or decreases, \( y \) gets larger, so you could also include points like \( (2, 4) \) and \( (-2, 4) \) to better sketch the curve.

When the inequality is \( y \geq x^2 \), it indicates that the solution is not just the curve itself but the entire region above the parabola. This means that every point above and including the parabola will satisfy the inequality. Hence, shade this region when graphing to show all possible solutions.

Linear Inequality

A linear inequality involves a straight line and shades a region on one side of the line. In this case, the inequality \( x + y \geq 6 \) deals with a line on the coordinate plane. The line \( x + y = 6 \) is a linear equation that you can rewrite in slope-intercept form as \( y = -x + 6 \), showing a slope of -1 and a y-intercept at (0, 6).
To graph this line, identify two points it passes through, such as (0, 6) and (6, 0). Connect these points with a straight line. The inequality \( x + y \geq 6 \) means that we consider not just the line but the region above it. Therefore, when graphing, it's crucial to shade the area that includes the points where \( x + y \) is greater than or equal to 6.

This shaded region complements the parabola's shading. Identifying the overlap between the shaded parabola region and the linear inequality's shaded region is critical when solving the system of inequalities.

Bounded Region

A bounded region is an area in the coordinate plane enclosed within certain boundaries, preventing the solution from extending indefinitely. In this exercise, the solution set is enclosed by the constraints set by the parabola \( y = x^2 \) and the line \( x + y = 6 \). Finding the intersections between these constraints helps determine if the region is indeed bounded.

The parabola and line intersect at points (2, 4) and (-3, 9). The line itself provides another critical boundary point at (6, 0). These intersection points serve as vertices, plotting a triangle-like region on the graph. Since all points of the solution set must lie within this triangular area, it is indeed a bounded region. This "boundedness" suggests that the solution doesn't stretch indefinitely but is contained within these vertices, which is essential in many applied contexts. The graphing of this entire system reveals a triangle-shaped region complete with defined boundaries, illustrating a complete and bounded solution.