Problem 22 - Algebra and Trigonometry

Question

\(19-32\) Sketch the region given by the set. $$ \\{(x, y) | y=-2\\} $$

Step-by-Step Solution

Answer

The region is a horizontal line at \( y = -2 \) extending infinitely along the x-axis.

Step 1: Identify the Equation

The given set is \( \{(x, y) | y = -2 \} \). This represents all the points \((x, y)\) where the y-coordinate is -2.

Step 2: Understand the Graph of the Equation

The equation \( y = -2 \) is a horizontal line. For any value of \( x \), the \( y \) coordinate remains constant at -2. Therefore, this line is parallel to the x-axis.

Step 3: Determine the Range of x-values

Since no restrictions are given for \( x \), \( x \) can be any real number. This implies that the line extends infinitely in both positive and negative x-directions.

Step 4: Sketch the Region

To sketch the line \( y = -2 \), draw a horizontal line that crosses the y-axis at -2. This line will extend left and right indefinitely, parallel to the x-axis.

Step 5: Verify the Sketch

Ensure the line is correctly positioned by checking that all points on the line have a y-coordinate of -2, confirming it represents the set \( \{ (x, y) | y = -2 \} \).

Key Concepts

Understanding Horizontal Lines

A horizontal line in coordinate geometry has a constant y-value across all points. This means for a horizontal line like our example, where the equation is \( y = -2 \), the y-coordinate of every point on the line doesn't change; it stays at -2 no matter what the x-coordinate is. This creates a perfectly straight line parallel to the x-axis.

Horizontal lines are unique because:

• They do not tilt or slope; they remain flat.
• They extend infinitely in the left and right directions along the x-axis.

These attributes make horizontal lines easy to identify and sketch. They simply require you to draw a straight line parallel to the x-axis at the given y-value.

The Y-axis in Graphs

The y-axis is one of the two main axes in a two-dimensional graph. It runs vertically and is used to determine the vertical position of a point. In the context of our problem, the y-axis helps to identify where the horizontal line \( y = -2 \) crosses.
It’s important to remember:

• The y-axis represents the set of possible y-values for points in the plane.
• The y-coordinate of any point on the y-axis is zero in a standard coordinate system, except where specified otherwise (as in the set {\( (x, y) | y = -2 \)}).

Identifying where a horizontal line crosses the y-axis can help visually confirm its placement on a graph.

The Role of Real Numbers

Real numbers in coordinate geometry define the values that coordinates can take. In our problem, when we say \( x \) can be any real number, it means there are no boundaries on how far left or right the x-values can extend.

Real numbers are crucial because:

• They allow lines, such as \( y = -2 \), to extend infinitely.
• They include both positive and negative numbers as well as zero, covering the entire x-axis.
• They ensure continuity, meaning there are no gaps in the line on the graph.

Real numbers provide the flexibility needed in graphing equations without limitations on how far a line can stretch across the plane.

Basic Techniques in Graph Sketching

Graph sketching is a fundamental skill in coordinate geometry. To sketch a line like \( y = -2 \), first identify its nature (in this case, a horizontal line). Begin by locating the y-coordinate on the y-axis. For \( y = -2 \), this means finding \( y = -2 \) on the axis and drawing a straight, horizontal line through it.

Here's how to make sure your sketch is accurate:

• Ensure the line is parallel to the x-axis. Use a ruler if necessary to maintain straightness.
• Verify the line crosses the y-axis at the correct point, \( y = -2 \).
• Check a sample of points on the line to confirm they all have the same y-coordinate (-2 here).

By following these steps, you will create a precise graph representation of the equation \( y = -2 \).