Problem 70 - 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. $$ y \geq x^{2}-4 $$

Step-by-Step Solution

Answer

The inequality \(y \geq x^{2}-4\) can be graphed as an upward facing parabola that has been shifted 4 units downwards, with the region above the parabola shaded to represent the values that satisfy the inequality.

Step 1: Understanding the Inequality

The inequality \(y \geq x^{2}-4\) represent a region in the rectangular coordinate system including the parabola \(y = x^{2}-4\) and the area above it. It's important to note the ‘greater than or equal to’ symbol - this means that the solutions are not only the line represented by the equation but also the area above it.

Step 2: Plotting the Parabola

To start off with the graphing, first graph the parabola \(y = x^{2}-4\). This is a standard upward opening parabola shifted 4 units down along the y-axis. The vertex of the parabola is at (0,-4).

Step 3: Shading the Region

Once the parabola is plotted, the next step is to figure out which side of the parabola is to be shaded. As the inequality symbol is 'greater than or equal to', we shade the region above the parabola, which includes the parabola itself.

Key Concepts

Coordinate System

The coordinate system, often referred to as the Cartesian coordinate system, is a fundamental element for graphing various types of equations, including inequalities. It consists of two perpendicular lines — one horizontal (the x-axis) and one vertical (the y-axis) — that intersect at a point called the origin. Every point in the plane can be identified by an ordered pair of numbers (x, y), known as its coordinates.

The coordinate system makes it easy to visualize relationships between numbers and can be used to graph lines, curves, and shaded regions that represent solutions to inequalities. In the case of quadratic inequalities, the coordinate system allows us to plot the curve of the quadratic function and determine the region that satisfies the inequality.

Quadratic Inequalities

Quadratic inequalities involve expressions of the form \( ax^2 + bx + c \), and solving them means finding the range of values that make the inequality true. Unlike linear inequalities that are graphed as straight lines, quadratic inequalities are represented by parabolas.

When graphing a quadratic inequality, such as \( y \geq x^2 - 4 \) on the coordinate system, you start by graphing the associated quadratic equation \( y = x^2 - 4 \), which is a parabola. To find the solution to the inequality, you determine which side of the parabola satisfies the inequality — in this case, the region above the parabola, because \( y \), the output value, is greater than or equal to the expression \( x^2 - 4 \).

Graphing Utility

A graphing utility is an essential tool for efficiently plotting functions and inequalities. It can quickly provide a visual representation of complex equations, such as quadratics, on a coordinate system. To use a graphing utility to solve an inequality, follow these steps:

• Enter the equation of the boundary of the inequality (in the case of quadratic inequalities, the quadratic equation) into the graphing utility.
• Consult the user manual to learn how to apply shading to the graph for inequalities. Shading helps to visually represent the area of all possible solutions.
• Use the utility to shade the correct region that satisfies the inequality — above or below the curve, depending on the direction of the inequality symbol.

Graphing utilities often offer additional features such as zooming and tracing, which can further aid in understanding the graph and the region it defines.