Problem 56 - College Algebra with Modeling and Visualization

Question

Complete the following. (a) Find the domain and range of the relation. (b) Determine the maximum and minimum of the \(x\) -values and then of the y-values. (c) Label appropriate scales on the xy-axes. (d) Plot the relation. $$ ((-1.2,1.5),(1.0,0.5),(-0.3,1.1),(-0.8,-1.3)] $$

Step-by-Step Solution

Answer

Domain: \(\{-1.2, 1.0, -0.3, -0.8\}\); Range: \(\{1.5, 0.5, 1.1, -1.3\}\); Max x: 1.0, Min x: -1.2; Max y: 1.5, Min y: -1.3.

Step 1: Determine the Domain

The domain of a relation is the set of all the first elements (x-values) from the ordered pairs. In the given relation \(((-1.2,1.5),(1.0,0.5),(-0.3,1.1),(-0.8,-1.3))\), the x-values are \(-1.2, 1.0, -0.3, \) and \(-0.8\). Thus, the domain is \(\{-1.2, 1.0, -0.3, -0.8\}\).

Step 2: Determine the Range

The range of a relation is the set of all the second elements (y-values) from the ordered pairs. Looking at the given relation \(((-1.2,1.5),(1.0,0.5),(-0.3,1.1),(-0.8,-1.3))\), the y-values are \(1.5, 0.5, 1.1,\) and \(-1.3\). Therefore, the range is \(\{1.5, 0.5, 1.1, -1.3\}\).

Step 3: Find Maximum and Minimum of X-values

To find the maximum and minimum of the x-values, we compare all the x-values from the ordered pairs: \(-1.2, 1.0, -0.3,\) and \(-0.8\). The maximum of the x-values is \(1.0\) and the minimum is \(-1.2\).

Step 4: Find Maximum and Minimum of Y-values

Similarly, find the maximum and minimum of the y-values: \(1.5, 0.5, 1.1,\) and \(-1.3\). The maximum y-value is \(1.5\) and the minimum y-value is \(-1.3\).

Step 5: Label Scales on XY-Axes

To label appropriate scales, consider the ranges of the x-values and y-values. The x-axis should adequately cover from \(-1.2\) to \(1.0\), and the y-axis should cover from \(-1.3\) to \(1.5\). A good choice might be increments of 0.2 on the x-axis and increments of 0.5 on the y-axis.

Step 6: Plot the Relation

Plot the given points \((-1.2, 1.5)\), \((1.0, 0.5)\), \((-0.3, 1.1)\), and \((-0.8, -1.3)\) on the coordinate plane using the scales identified. Ensure each point is placed at the correct coordinate.

Key Concepts

Relations in Algebra

In algebra, a relation is a set of ordered pairs, where each pair consists of two elements. These elements are typically represented as \(x,y\) coordinates. The first element denotes the x-value, and the second element denotes the y-value. Understanding relations helps us understand how these elements (or numbers) can be associated or linked together. This is foundational in mathematics as it allows for the representation of real-world problems in a mathematical format.

• Domain: This is the set of all possible x-values in the relation. It tells us which values are considered as inputs for the relation.
• Range: This is the set of all possible y-values that correspond to the domain. It contains all the outputs that come from substituting domain values into the relation.

Determining the domain and range of a relation is crucial, as it helps set the boundaries within which we analyze and interpret data.

Coordinate Plane

The coordinate plane, sometimes referred to as the Cartesian plane, is a two-dimensional plane formed by the intersection of two number lines: the x-axis and the y-axis. This plane provides a space where we can graphically represent relations and functions. It makes it possible to visualize mathematical concepts and their relationships.

• The x-axis is the horizontal line that runs left to right, with negative values to the left and positive values to the right.
• The y-axis is the vertical line, with positive values going upwards and negative values going downwards.

At the point where both axes meet is the origin, denoted as (0,0). Using this plane, we can plot any ordered pair (x, y) by moving x units horizontally from the origin and y units vertically. Labeling axes appropriately with scales is vital for accurate representation. The scale should reflect the range of data being analyzed, allowing for clarity and precision in graphing.

Graphing Relations

Graphing relations allows for the visualization of the connection between certain values on the coordinate plane. When graphing a set of ordered pairs, it is essential to plot each pair precisely to maintain the integrity of the data's representation.
To begin graphing:

• First, ensure that the scales on the x-axis and y-axis adequately cover the range of data points. This setup ensures that every point can be plotted correctly.
• Next, graph each point by locating its position according to its x and y coordinates.

The advantage of graphing is that it provides a clear visual depiction of how variables may interact. For example, you can quickly see how the y-values change as the x-values increase or decrease. This allows for immediate analysis of trends and patterns, helping to interpret the data presented in a meaningful way. Furthermore, graphing can reveal outliers or data trends that may be less obvious with numerical only data.