Problem 10 - Precalculus

Question

Graph the given relation. $$ \\{(2, y) \mid y \leq 5\\} $$

Step-by-Step Solution

Answer

A vertical line at x=2, extending downwards from y=5, including all y-values ≤ 5.

Step 1: Understand the Relation

The relation \( \{(2, y) \mid y \leq 5\} \) describes a set of points where all x-values are fixed at 2, and y-values are all possible values less than or equal to 5.

Step 2: Identify the Range of y-values

Since the condition given is \(y \leq 5\), the y-values range from negative infinity to 5, inclusive. This means any y-value less than or equal to 5 is part of the relation.

Step 3: Plot the Fixed x-value

On a coordinate plane, locate the line where x is constant at 2. This is a vertical line parallel to the y-axis passing through the point (2,0).

Step 4: Indicate the Region of y-values

From the point (2,5), draw a vertical line downward. Shade or mark the portion of the line where the y-values are less than or equal to 5, indicating all possible y-values until reaching up to (2,-∞). The point (2,5) will be marked as a solid dot to indicate that y=5 is included in the set.

Key Concepts

The Coordinate Plane

The coordinate plane is a two-dimensional surface where we can plot points, lines, and curves. It's like a piece of graph paper and is essential for understanding relationships between two variables. The plane is defined by two perpendicular axes:

• The x-axis, which runs horizontally.
• The y-axis, which runs vertically.

These axes intersect at the origin, labeled as (0,0). Each point on the coordinate plane is defined by a pair of numbers (x, y).
These numbers indicate the position of the point relative to the origin. For example, in the point (2, y), 2 is the x-coordinate, and y is the y-coordinate. Understanding how to navigate the coordinate plane helps in plotting points and graphing relations effectively.

Understanding Fixed x-value

In our relation \(\{(2, y) \,|\, y \leq 5\}\), the x-value is fixed at 2. This means that no matter what the value of y is, x will always be 2. Let's break this down further:

• A fixed x-value creates a vertical line on the coordinate plane.
• This line is parallel to the y-axis.
• Every point on this line has the same x-coordinate, which is 2 in this case.

This concept is crucial because it means when you're plotting this relation, you're only looking at variations in the y-values while x remains constant. Knowing this helps simplify the graphing process since you only focus on one variable changing.

Exploring the Range of y-values

The range of y-values for the relation \(\{(2, y) \,|\, y \leq 5\}\) includes all numbers less than or equal to 5. Here's how it works:

• Y-values can include positive numbers, negative numbers, and zero, as long as they don't exceed 5.
• The notation \(y \leq 5\) signifies that y can be equal to 5, but not greater.
• This extends the range downwards infinitely, as indicated by \( -\infty \).

When graphing, we use a solid dot at the point (2,5) to show that 5 is included in the range. The line continues downwards to represent all smaller y-values. Understanding this concept helps in shading or marking the correct portion of the graph.

Visualizing Vertical Lines

A vertical line on a coordinate plane is a line where all points have the same x-coordinate. In this exercise, we are dealing with the vertical line at \(x = 2\). Here are some key points about vertical lines:

• Vertical lines run parallel to the y-axis.
• They have undefined slopes, which means they don't tilt left or right.
• Every point on this line has x-coordinates equal to 2, regardless of their y-coordinate.

To represent the set of \(\{(2, y) \,|\, y \leq 5\}\), you would draw a straight vertical line at \(x = 2\) and extend it from \(y = 5\) downwards to include all y-values less than or equal to 5. Understanding and plotting vertical lines are essential skills for graphing relations where the x-value is fixed.