Problem 13 - Precalculus

Question

Graph the given relation. $$ \\{(x, 2) \mid-2 \leq x<3\\} $$

Step-by-Step Solution

Answer

Graph a horizontal line from (-2, 2) to just before (3, 2).

Step 1: Understand the Set of Points

The given relation \(\{(x, 2) \mid -2 \leq x < 3\}\) represents a set of points where the y-coordinate is always 2, and the x-coordinate varies between -2 and 3. This implies a horizontal line, since the y-value is constant.

Step 2: Identify the Range of x

The x-values range from -2 to 3, where -2 is included (closed interval at the start) and 3 is not included (open interval at the end). This means the line will start at x = -2 and go up to but not include x = 3.

Step 3: Plot the Start and Endpoint

Plot the point (-2, 2) on a graph, marking it as a solid point since -2 is part of the relation, and plot (3, 2) as an open circle since 3 is not included in the range.

Step 4: Draw the Horizontal Line

Connect the points (-2, 2) and (just before 3, 2) with a straight horizontal line. The open circle at (3, 2) indicates that the line does not include this point, but it approaches it closely.

Key Concepts

Understanding Horizontal Lines

In graphing, a horizontal line is a straight line that moves from left to right on the graph and is parallel to the x-axis. What differentiates a horizontal line is that its y-coordinate remains constant for all points along the line.

For instance, if a horizontal line is represented as \(y = c\), then every point on this line will have the same y-value \(c\). Thus, in our exercise where we have the relation \(\{(x, 2) \mid -2 \leq x < 3\}\), the y-coordinate for each point will be 2, resulting in a horizontal line at y = 2. Horizontal lines are key in understanding and visualizing graphical data, and learning to plot them is a fundamental skill in graphing relations.

What is a Set of Points?

A set of points, in the context of graphing, refers to a collection of coordinates that represent specific locations on a graph. Points have an x-coordinate and a y-coordinate, denoted as \((x, y)\).

In the exercise, the set of points is given by \(\{(x, 2) \mid -2 \leq x < 3\}\). This tells us that every point in the set has a y-coordinate of 2, but the x-coordinate changes, ranging from -2 up to but not including 3. Understanding a set of points is crucial because it essentially outlines the collection of locations that will be plotted on the graph, helping define the shape and location of the lines and curves you'll graph.

Plotting Points on a Graph

Plotting points involves placing dots at specific locations on a grid, based on their coordinates. The x-coordinate decides how far left or right the point is, while the y-coordinate determines its vertical position.

To plot the set of points from our example, start by marking the endpoint \((-2, 2)\) with a solid dot, showing that -2 is included in the set of points (a closed interval). Then plot \((3, 2)\) with an open circle because this endpoint is not included in the set (an open interval). Connect these two with a straight line, stopping just before the open circle. This line represents all the points within the bounds \(-2 \leq x < 3\), making sure not to include the open circle.

Exploring Open and Closed Intervals

Intervals tell us about which parts of a range are included or excluded. A closed interval means the endpoint is included, often noted with a bracket \([ \text{or} ]\), while an open interval means it is not included, shown with a parenthesis \(( \text{or} )\).

In the relation \(\{(x, 2) \mid -2 \leq x < 3\}\), the range \(-2 \leq x < 3\) shows us a mix of open and closed intervals. The \(x\) value of -2 is part of the graph (closed interval), hence the solid point at \((-2, 2)\). The \(x\) value of 3 is not part of the graph (open interval), hence the open circle at \((3, 2)\). Recognizing these helps us correctly plot points and lines on the graph, indicating which edges of the line are inclusive or exclusive. This understanding is crucial when defining relationships in both mathematics and applied fields like data analysis.