Problem 97 - College Algebra Essentials

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. $$y \leq 4 x+4$$

Step-by-Step Solution

Answer

The graph of the inequality \(y \leq 4x + 4\) would be a line with slope 4 and y-intercept 4 and the region below the line and the line itself are shaded.

Step 1: Rewrite the Inequality

Rewrite the inequality \(y \leq 4x + 4\) in slope-intercept form, which is already done. This form of the equation is \(y = mx + b\), where \(m\) is the slope (in this case, 4) and \(b\) is the y-intercept (in this case, 4).

Step 2: Input the Inequality into the Graphing Utility

Input the inequality \(y \leq 4x + 4\) into the graphing utility. This is typically done by typing in the equation exactly as it is and then specifying that \(y\) is less than or equal to \(4x + 4\). The graphing utility should then graph the line \(y = 4x + 4\).

Step 3: Shade the Appropriate Region

The inequality specifies that \(y\) is less than or equal to \(4x + 4\). This means the area that satisfies the inequality is the area below the line or the line itself. Use the graphing utility's shading option to shade this region. Be sure to check if the line is included in the shading area (indicating 'equal to').

Key Concepts

Rectangular Coordinate System

The rectangular coordinate system is a mathematical framework that allows us to plot points, lines, and curves based on two axes: the x-axis (horizontal) and the y-axis (vertical).

This system is sometimes referred to as the Cartesian coordinate system. It provides a visual way to represent algebraic equations and inequalities.

• The origin, where the x-axis and y-axis intersect, is at point (0,0).
• Each point in the plane is defined by a pair of coordinates (x, y), representing its horizontal and vertical position respectively.
• Quadrants divide the plane into four sections, where each quadrant corresponds to a different sign configuration of the coordinates.

When graphing inequalities like \( y \leq 4x + 4 \), we use this system to identify which areas satisfy the inequality. The inequality is represented by a region in the coordinate plane, typically shaded to distinguish it from other areas that do not satisfy the inequality.

Graphing Utilities

Graphing utilities are tools, either digital (software) or hardware (devices), designed to help visualize mathematical equations and inequalities.

They play a crucial role in identifying relationships between variables, especially when graphing has become a more complex task.

• Graphing calculators are a popular form of graphing utility. These devices allow you to input equations and immediately see a visual representation.
• Software options like Desmos or GeoGebra offer similar functionalities, often with interactive and advanced graphing capabilities.
• Most graphing utilities provide features to input equations in various forms and explore the graph by zooming, shifting, or even analyzing points of interest.

To graph an inequality like \( y \leq 4x + 4 \), graphing utilities highlight not only the line represented by the equation, but also the region that shows which values of \( y \) meet the inequality criteria. Many utilities enable shading options to clearly mark this area, helping visualize solutions easily.

Slope-Intercept Form

The slope-intercept form of a line is one of the most straightforward ways to express linear equations. It is written as \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept of the line.

Understanding this form is key to analyzing and graphing lines efficiently.

• The slope \( m \) indicates the steepness and direction of the line. A positive slope means the line rises from left to right, while a negative slope means it falls.
• The y-intercept \( b \) represents the point where the line crosses the y-axis.
• Converting equations to this form allows for easier interpretation of how changes in \( x \) affect \( y \).

In the context of the inequality \( y \leq 4x + 4 \), the equation \( y = 4x + 4 \) is already in slope-intercept form. Here, the slope \( m \) is 4, indicating a relatively steep ascent, and the y-intercept \( b \) is 4, showing that the line crosses the y-axis at (0,4). This form aids in graphing the line and subsequently determining the appropriate shading for the inequality.