Problem 26 - Precalculus Mathematics for Calculus

Question

Sketch the region given by the set. $$\\{(x, y) | x=-1\\}$$

Step-by-Step Solution

Answer

The set represents a vertical line at \(x = -1\).

Step 1: Understand the Set Notation

The set \[\{(x, y) | x = -1\}\] represents all ordered pairs \((x, y)\) where the value of \(x\) is constantly \(-1\). This suggests that \(y\) can be any real number, but \(x\) is fixed at \(-1\). This defines a vertical line on the Cartesian plane.

Step 2: Identify the Characteristics of the Set

Since \(x = -1\) for all elements in the set, all points \((x, y)\) are such that the \(x\) coordinate is \(-1\). The \(y\) coordinate can take any value, implying that the set describes a vertical line at \(x = -1\) that extends infinitely in both the positive and negative \(y\) directions.

Step 3: Sketch the Region

To sketch the region, draw the Cartesian coordinate plane with the \(x\) and \(y\) axes. Plot a vertical line at \(x = -1\). This line should be parallel to the \(y\)-axis and should intersect the \(x\)-axis at the point \((-1, 0)\). Since \(y\) is not restricted, this line extends infinitely up and down.

Step 4: Highlight the Region

Clearly mark the vertical line at \(x = -1\) to indicate it's the object of interest. You may optionally shade or highlight the line to distinguish it from any other graphical elements on the plane.

Key Concepts

Vertical Line

A vertical line on the Cartesian plane is a straight line that moves up and down but never left or right. In mathematical terms, a vertical line is defined by the equation \(x = a\), where \(a\) is a constant. This means that no matter the value of \(y\), the \(x\)-coordinate will always remain the same. This is why vertical lines appear as columns of ordered pairs that are stacked directly above or below each other.

To visualize, let's take \(x = -1\). This vertical line crosses the \(x\)-axis at the point (-1, 0) and extends infinitely upwards and downwards. The line is parallel to the \(y\)-axis and never intersects it. It's a crucial aspect in understanding the delineation of regions in graphing, particularly when dealing with restrictions in set notation.

Ordered Pairs

Ordered pairs are a fundamental concept in graphing and coordinate geometry. An ordered pair, usually written as \((x, y)\), represents a specific point on the Cartesian plane. The first component, \(x\), refers to the horizontal position, while the second component, \(y\), indicates the vertical position.

When a set is defined for ordered pairs, as seen with \(\{(x, y) | x = -1\}\), it specifies a selection of points based on conditions given to either \(x\) or \(y\). Here, the condition restricts \(x\) to always be -1, but leaves \(y\) unrestricted. This creates a pattern of points aligned in a vertical line at the \(x\)-coordinate of -1. Understanding how ordered pairs function helps in plotting precise and accurate locations on graphs.

Graphing Equations

Graphing equations involves plotting points on a Cartesian plane to visualize the relationship described by an equation. Take the example of \(x = -1\) from our problem. This equation is quite simple, as it defines a straight, vertical line.

Here's how you graph this equation:

• Begin with identifying the constant in the equation, which is \(x = -1\).
• Draw a straight line that runs parallel to the \(y\)-axis, crossing the \(x\)-axis at \(x = -1\).
• Note that this line represents every point where \(x\) is -1, forming a visually continuous line through these points.

Understanding graphing is vital for reading and interpreting equations, as well as identifying sets like those described using set notation.

Set Notation

Set notation is a powerful language used in mathematics to denote a collection of objects or numbers that meet certain criteria. In our example, the set \(\{(x, y) | x = -1\}\) captures all ordered pairs where \(x\) remains a constant \(-1\).

In detailed terms:

• Curly braces \(\{ \}\) signify a set, containing all elements that fit the description.
• The vertical bar \(|\) is read as "such that", introducing conditions those elements must meet.
• This notation precisely communicates the criteria without ambiguity, allowing anyone familiar with the syntax to immediately visualize the vertical line described.

This efficient means of summarizing relationships is widely used across various branches of mathematics, including algebra, calculus, and more.