Problem 100 - College Algebra

Question

Graphing utilities can be used to shade regions in the rectangular coordinate system, thereby graphing an inequality in two variables Read the section of the user's manual for your graphing utility that describes how to shade a region. Then use your graphing utility to graph the inequalities in Exercises 97-102. $$y \geq \frac{1}{2} x^{2}-2$$

Step-by-Step Solution

Answer

The graph of the inequality \(y \geq \frac{1}{2} x^{2}-2\) is a region above and including the parabola defined by \(y=\frac{1}{2}x^{2} -2\)

Step 1: Understand the inequality

The inequality to be graphed is \(y \geq \frac{1}{2} x^{2}-2\). It represents a parabola since the graph of \(y = \frac{1}{2} x^{2}-2\) is a parabola. The symbol \(\geq\) indicates that the region to be shaded includes the boundary of the parabola.

Step 2: Plot the parabolic boundary

Plot the curve given by the function \(y = \frac{1}{2} x^{2}-2\). This serves as the boundary for your area. Set up your graphing utility to graph this function.

Step 3: Shade the region

Since the inequality is \(y \geq \frac{1}{2}x^{2} -2\), the area above the curve and including the curve itself will satisfy the inequality. Therefore, you have to shade this area. Use the shading tool in your graphing utility for this. Make sure you shade the region correctly, i.e., everything above and including the curve.

Key Concepts

Rectangular Coordinate System

The rectangular coordinate system, sometimes referred to as the Cartesian coordinate system, is a two-dimensional plane defined by two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). Each point in this plane can be identified by an ordered pair of numbers \(x, y\). These numbers represent coordinates, measured by their distance from the intersection point of the two axes, known as the origin. Use this system to plot functions and inequalities easily. It forms the basis for graphing equations and inequalities in algebra and calculus.

• The x-coordinate tells you how far to move horizontally from the origin.
• The y-coordinate tells you how far to move vertically from the origin.

Understanding this system well is crucial for successfully graphing equations and inequalities.

Parabolic Boundary

In mathematics, a parabola is a symmetrical open plane curve formed by all points at an equal distance from a fixed point, called the focus, and a fixed straight line, called the directrix. When graphing inequalities involving parabolas, the parabolic boundary is defined by the equation of the parabola, in this case, \(y = \frac{1}{2}x^2 - 2\). This equation forms the boundary line or curve on the graph in the rectangular coordinate system.

• Identify key points such as the vertex and axis of symmetry to sketch the parabola accurately.
• The boundary divides the plane into two regions: one that satisfies the inequality and one that does not.

It’s important to plot this boundary correctly to shade the appropriate region according to the inequality.

Graphing Utilities

Graphing utilities are tools that help you visualize equations and inequalities. They range from simple graph paper and pencils to advanced graphing calculators and software packages. In this exercise, using a graphing utility makes plotting complex equations, like parabolas, much easier.

• Graphing calculators allow you to input equations directly and plot their graphs automatically.
• Many graphing programs offer additional features, such as the ability to shade regions above, below, or between curves.

Knowing how to use these utilities effectively can save you time and reduce errors when graphing inequalities or other complex functions.

Shading Regions

When graphing an inequality, shading is used to indicate the set of points that satisfy the inequality. In the case of \(y \geq \frac{1}{2}x^2 - 2\), you will shade the region above the parabola, including the curve itself. This indicates that the y-values in this region satisfy the inequality.

• Start with the parabolic boundary plotted on your graph.
• By using the shading feature in a graphing utility, highlight the region above the curve.

It's essential to interpret and shade correctly to ensure your graph represents the solution to the inequality accurately. This visual representation can make it easier to understand complex mathematical relationships.