Problem 71 - 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. $$ y \geq \frac{1}{2} x^{2}-2 $$

Step-by-Step Solution

Answer

Firstly, understand the inequality to know that we are dealing with an upward-opening parabola. Identify the vertex (0, -2). Use a graphing tool to plot the graph with the vertex and any other points by choosing arbitrary x values and finding the corresponding y values. Finally, shade the region above the parabola (including the parabola) since the inequality indicates the y values are greater than or equal to \(\frac{1}{2}x^{2} - 2\).

Step 1: Understanding the Inequality

The inequality is written in the format \(y \geq \frac{1}{2}x^{2}-2\). It is a quadratic inequality, and the graph will be a parabola opening upwards because the coefficient of \(x^{2}\) is positive.

Step 2: Identify the Vertex

The standard form of the quadratic function is \(y = ax^{2} + bx + c\), where the coordinates of the vertex is \((-b/2a , c - b^{2}/4a)\). In this case, \(a = 1/2\) and \(c = -2\), so the vertex is (0, -2).

Step 3: Plot the Graph

Use the graphing utility tool to plot the graph. Start by plotting the vertex. The parabola is symmetrical, so plot other points on either side of the vertex. Since there is no specific range for x given, you can choose any arbitrary values for x. When you replace x in the equation with these values, the corresponding y values will give the points on the graph.

Step 4: Shading the region

The inequality is \(y \geq \frac{1}{2}x^{2}-2\), which means we are interested in the y-values that are greater than or equal to \(\frac{1}{2}x^{2} - 2\). On the graph, shade the region above the parabola including the parabola to represent these y-values.

Key Concepts

Parabola

A parabola is a U-shaped curve that is graphically represented by a quadratic equation. It can open either upwards or downwards, depending on the sign of the quadratic term's coefficient. If the coefficient in front of the squared term is positive, as in our equation, the parabola will curve upward. When it's negative, the parabola curves downward.

Characteristics of a parabola include its vertex, which is the highest or lowest point depending on its orientation, and its axis of symmetry, which vertically divides the parabola into two mirror-image halves. parabolas are commonly found in physics due to their emergence in descriptions of projectile motion and optics.

Key characteristics also involve the direction and width of the parabola. The larger the absolute value of the coefficient of the squared term, the narrower the parabola. The smaller this value, the wider it opens. These qualities make parabolas versatile and important in understanding quadratic relationships.

Quadratic Inequality

A quadratic inequality involves a quadratic expression, often in the standard format such as \(y \geq ax^2 + bx + c\). In inequalities, the symbol represents a relationship that is not just an equation. These relationships can show greater than or equal to (≥), less than or equal to (≤), greater than (>), or less than (<).

Understanding a quadratic inequality involves graphing the associated parabola and then identifying which region of the graph satisfies the inequality. For example, in \(y \geq \frac{1}{2}x^2 - 2\), we are interested in all y-values that are above or on the curve of the graph formed by the parabola.

When dealing with such inequalities, it is important to consider:

• The direction of the inequality (whether it includes equal to or not)
• The portions of the graph which satisfy the inequality
• Visual representation of the solutions in the coordinate plane by shading regions

Solving quadratic inequalities thus involves evaluating points and determining the validity of solutions within a set graph area.

Vertex of a Parabola

The vertex of a parabola is an essential point that reflects where the curve changes direction. It can be identified conveniently in the standard quadratic function form: \(y = ax^2 + bx + c\). The vertex is found using the formula \(x = -\frac{b}{2a}\) for its x-coordinate and subsequently substituting back to find the y-coordinate.

In the problem \(y \geq \frac{1}{2}x^2 - 2\), there is no linear term, so \(b = 0\). This makes finding the vertex straightforward:

• x-coordinate: \(-\frac{b}{2a} = 0\)
• y-coordinate is simply \(c = -2\)

Thus, the vertex is at the point (0, -2). Hence, the graph of the parabola opens upward with its vertex as its lowest point. This offers a starting point when sketching the parabola and shading regions for inequalities since it provides a reference for symmetry and point plotting.

Graphing Utility

Graphing utilities are software tools used to visually analyze mathematical functions including plotting equations and inequalities. These tools can range from graphing calculators to advanced computer software.

Key benefits of using graphing utilities include:

• Quickly and accurately plotting complex functions such as quadratic equations
• Visualizing solutions to inequalities by shading the appropriate regions
• Experimenting with different parameters and observing the effects on the graph

When graphing the inequality \(y \geq \frac{1}{2}x^2 - 2\) using a graphing utility, you follow steps like plotting the parabola using its vertex and additional points, and then shading the region above it where the inequality holds true. Typically, this involves selecting specific features in the graphing utility to apply shading which aids in effectively communicating which parts of the coordinate plane satisfy the inequality.