Problem 30 - Precalculus Mathematics 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. $$\left\\{\begin{array}{l} y \geq x^{2} \\ y \leq 4 \\ x \geq 0 \end{array}\right.$$

Step-by-Step Solution

Answer

Vertices are (0,0) and (2,4); the solution set is bounded.

Step 1: Graph the Inequality

First, graph the inequality \( y \geq x^2 \). This represents the region above or on the parabola \( y = x^2 \). Use a solid line for the parabola since points on the line are included in the solution.

Step 2: Graph the Second Inequality

Next, graph the inequality \( y \leq 4 \). This represents the region below or on the horizontal line \( y = 4 \). Use a solid line for this boundary as well, as points on the line are part of the solution.

Step 3: Graph the Third Inequality

Then, graph the inequality \( x \geq 0 \). This is the region to the right of or on the vertical line \( x = 0 \). The points on the line are included, so use a solid line.

Step 4: Find the Intersection Points

Find the intersection points of the lines with each other. The intersection of \( y = x^2 \) and \( y = 4 \) occurs where \( x^2 = 4 \). Solving yields \( x = 2 \) (since \( x \geq 0 \)). Thus, the point is \((2, 4)\). Also, consider the intersection of \( x = 0 \) and \( y = x^2 \), giving the point \((0, 0)\).

Step 5: Identify the Bounded Region

Shade the region common to all three inequalities. The shaded region will be above the parabola \( y = x^2 \), below the line \( y = 4 \), and to the right of the line \( x = 0 \). This region is bounded by the points \((0,0)\), \((2,4)\), and the segment of the vertically infinite line \( x=0 \) up to \( y = 4 \).

Step 6: Determine Boundedness

Check if the solution set forms a closed area. Since all constraints form a closed and finite region between the vertex \((2,4)\) and the axis, the solution set is bounded.

Key Concepts

Graphing Inequalities

When graphing inequalities, you are essentially showcasing regions on a graph that satisfy particular conditions. Let's explore how you can do this by taking it step by step. First, consider an inequality like \( y \geq x^2 \). This means you want the region that is either on or above the parabola \( y = x^2 \). Use a solid curve to represent this because points on the line are valid solutions.
A similar method is used for graphing horizontal or vertical lines, where you graph \( y \leq 4 \) as a solid horizontal line representing points on or below \( y = 4 \). Lastly, for \( x \geq 0 \), use a vertical line to indicate that you are considering points on or to the right of \( x = 0 \). With these lines, you're framing the region where all inequalities are true.
Remember to always use a solid line when the inequality allows it (like \( \geq \) or \( \leq \)). With dashed lines, you would exclude the line itself if you have a \( > \) or \( < \) symbol.

Intersection Points

Finding intersection points is crucial when solving systems of inequalities. These points help to identify the vertices of the region that satisfy all inequalities. For our case, you first need to find where the inequalities meet on the graph.
Consider the intersection of \( y = x^2 \) and \( y = 4 \). Set these equations equal to each other: \( x^2 = 4 \). By solving this, you get \( x = 2 \) since we are only interested in values where \( x \geq 0 \). Therefore, the intersection point is \((2, 4)\).

• Another important intersection is where the parabola \( y = x^2 \) meets the y-axis, which occurs at the origin \((0, 0)\).

By identifying these intersections, you outline the vertices of the region where the inequalities hold true.

Bounded Regions

A bounded region is where the system of inequalities encloses a finite area on the graph. It's like drawing a shape in the graph that closes off within finite limits. Once the intersection points are found, you can determine the overall shape that encompasses these regions.
Consider the region that lies:

• Above the parabola \( y = x^2 \)
• Below the line \( y = 4 \)
• Right of the line \( x = 0 \)

This area is "bounded" because it doesn't extend infinitely in any direction. It's enclosed by the points \((0,0)\) and \((2,4)\). Understanding this is essential as it clarifies the limits of where all conditions are satisfied.

Inequality Solutions

The solution to a system of inequalities is the region or area where all individual inequality conditions hold true simultaneously. This solution set often occurs in a form of a shaded region on the graph.
Our specific inequalities form a bounded region between a curve, a horizontal line, and a vertical axis, making it easy to visually assess which points belong to the solution set. In this case, the solution encompasses all points in the purple shaded region, whether directly on the lines or in the area they enclose.

• Points like \((1, 1)\), \((0.5, 3)\), or even \((2,4)\) itself are all valid solutions.

Working with inequalities requires checking if your calculated or estimated point satisfies each inequality part of the system. Once comprehended, finding and verifying inequality solutions becomes a more intuitive process.