Problem 53 - 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. $$ \\{(2,2),(-3,1),(-4,-1),(-1,3),(0,-2)\\} $$

Step-by-Step Solution

Answer

Domain: \{-4, -3, -1, 0, 2\}; Range: \{-2, -1, 1, 2, 3\}; Max x: 2, Min x: -4, Max y: 3, Min y: -2.

Step 1: Determine the Domain and Range

The domain of a relation is the set of all possible \( x \)-values, and the range is the set of all possible \( y \)-values from each ordered pair. For the relation \( \{(2,2),(-3,1),(-4,-1),(-1,3),(0,-2)\} \), the domain is \( \{-4, -3, -1, 0, 2\} \). The range is \( \{-2, -1, 1, 2, 3\} \).

Step 2: Find Maximum and Minimum of x- and y-values

To find the maximum and minimum \( x \)-values, identify the smallest and largest values from the domain. Here, the minimum \( x \)-value is \( -4 \), and the maximum \( x \)-value is \( 2 \). For \( y \)-values, find the smallest and largest values from the range. The minimum \( y \)-value is \( -2 \), and the maximum \( y \)-value is \( 3 \).

Step 3: Label Appropriate Scales on Axes

To plot the points on a graph, choose scales for the \( x \)-axis and \( y \)-axis based on minimum and maximum values. For the \( x \)-axis, label from \( -5 \) to \( 3 \). For the \( y \)-axis, label from \( -3 \) to \( 4 \). The scale should ideally be 1 unit per tick for clarity and spread the points comfortably on the graph.

Step 4: Plot the Points on the Graph

Now, plot each point from the relation on the graph using the scales determined: \((2,2)\), \((-3,1)\), \((-4,-1)\), \((-1,3)\), and \((0,-2)\). Ensure that each coordinate is correctly placed according to the axes' scales.

Key Concepts

Domain and Range in Coordinate Geometry

In coordinate geometry, the domain and range are vital concepts used to define a set of values for a relation or a function. Let's break these down:

• Domain: This is the collection of all possible input values or the set of all x-values that a relation can have. For instance, in the relation \( \{(2,2),(-3,1),(-4,-1),(-1,3),(0,-2)\} \), the domain includes all x-values extracted from each coordinate, which are \( \{-4, -3, -1, 0, 2\} \).
• Range: Conversely, the range is the set of all possible output values or all y-values from the relation. From the same set of pairs, we can identify the range as \( \{-2, -1, 1, 2, 3\} \).

Understanding domain and range helps in predicting the behavior of a graph and solving real-world problems where only certain input/output values make sense. Also, recognizing these sets can be the first step in graphing a relation or a function.

Graphing Ordered Pairs

Graphing is a key technique in coordinate geometry that visually represents relations or functions on a plane. This involves plotting points that represent data step-by-step.

• Labeling the Axes: Before plotting, choose appropriate scales for the axes considering the maximum and minimum values of the data. For our relation, the x-values range from -4 to 2 and y-values range from -2 to 3, so you might label the x-axis from -5 to 3 and the y-axis from -3 to 4.
• Plotting Points: Once the scales are set, mark each pair of x and y values on the graph. For instance, the point (2, 2) corresponds to x = 2 and y = 2. All points like (2,2), (-3,1), (-4,-1), (-1,3), and (0,-2) are plotted accurately to reflect the relation visibly.

This graphing process reveals the overall structure and behavior of the relation, which could help in identifying patterns or comparing with other data or relations.

Identifying Maximum and Minimum Values

The concepts of maximum and minimum values are crucial in understanding the extent or limits of a relation. Here's how they apply:

• For x-values: Look at the domain to find the smallest and largest values. Using the domain \( \{-4, -3, -1, 0, 2\} \), the minimum x-value is -4, and the maximum is 2.
• For y-values: Do the same with the range to identify the smallest and largest outputs. The range \( \{-2, -1, 1, 2, 3\} \) gives us a minimum y-value of -2 and a maximum of 3.

These values are not just numbers; they help in visualizing the spread and distribution of the data set on a graph. Additionally, maximums and minimums are useful when solving optimization problems, determining potential limits, or understanding constraints in real-life situations.