Problem 11 - Precalculus

Question

Graph the given relation. $$ \\{(-2, y) \mid-3

Step-by-Step Solution

Answer

Graph a vertical line at \(x = -2\) from \(y = -3\) (not included) to \(y = 4\) (included).

Step 1: Identify the Domain and Range

The given relation is \((-2, y)\) where \(-3 < y \leq 4\). The domain consists of a single x-value, which is \(x = -2\). The range comprises the y-values such that \(y\) is greater than -3 and less than or equal to 4. Thus, the range is \(-3 < y \leq 4\).

Step 2: Understand the Graph Type

Given that \(x = -2\) for all points in the relation, the result will be a vertical line at \(x = -2\) on the coordinate plane. This line will extend from \(y = -3\) to \(y = 4\), specifically without including \(y = -3\) and including \(y = 4\).

Step 3: Plot the Graph

To graph the relation, draw a vertical line at \(x = -2\). This line will start slightly above the point \((-2, -3)\) (represented with an open circle to denote exclusion) and extend to the point \((-2, 4)\) (filled circle to indicate inclusion). This indicates all integer and decimal values for y between -3 and 4, inclusive of 4.

Key Concepts

Understanding Domain and Range

The domain and range are critical concepts in graphing relations. The domain refers to all possible x-values a relation can have. In our case, the domain is simple: just a single value,

• \(x = -2\)

This tells us that, no matter what the value of \(y\) is, \(x\) will always be -2.
The range, on the other hand, is the set of possible y-values. In our example, the range is all \(y\) such that:

• \(-3 < y \leq 4\)

This means that \(y\) can be any value greater than -3 and up to, and including, 4.

Concept of the Vertical Line in Graphs

A vertical line in the coordinate plane appears quite differently than other types of lines. For any given x-value in a vertical line, the y-values can vary while x remains constant. In simple terms:
The graph of our relation at \(x = -2\) forms a vertical line since x is constant regardless of how the y-value changes. Unlike horizontal lines which have a slope of zero, vertical lines break the rule by having an undefined slope. This is because:

• Slope = \(\frac{{\text{Change in } y}}{\text{Change in } x} = \frac{\Delta y}{0}\)
• This results in division by zero, which is undefined.

Exploring the Coordinate Plane

The coordinate plane is essential for graphing and consists of two axes:

• The x-axis (horizontal)
• The y-axis (vertical)

These axes intersect at the origin,

• Point (0,0)

The coordinate plane enables us to navigate and place points using ordered pairs written as (x, y). Each point corresponds to a specific location where:
x gives the horizontal position and y gives the vertical position. In our relation,\((-2, y)\),since \(x = -2\) is fixed, the vertical arrangement of y-values is depicted along a single vertical line at this x-coordinate.

Plotting Points to Visualize Relations

Plotting points involves marking positions on a graph based on their coordinates. For our relation, this means marking all positions with the fixed x-value of -2. Here's how you plot the given relation:

• Start at x = -2 on the coordinate plane.
• Draw a vertical line at this position.
• Represent the beginning of the y-range slightly above (-2, -3) with an open circle since -3 is not included.
• End the line at (-2, 4) with a filled circle to show inclusion of y = 4.

Through this process, we map every potential y-value that exceeds -3 up to and includes 4, creating a complete visual representation of the given relation.