Problem 51 - 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. $$ ((0,5),(-3,4),(-2,-5),(7,-3),(0,0)) $$

Step-by-Step Solution

Answer

Domain: {-3, -2, 0, 7}, Range: {-5, -3, 0, 4, 5}. Max x: 7, Min x: -3, Max y: 5, Min y: -5. Plot includes these points.

Step 1: Identify the Domain

The domain of a relation is the set of all first coordinates (or x-values) in the ordered pairs. From the given relation, the x-values are: 0, -3, -2, 7, and 0. The domain, therefore, is the set of these values, removing duplicates: {-3, -2, 0, 7}.

Step 2: Identify the Range

The range of a relation is the set of all second coordinates (or y-values) in the ordered pairs. From the given relation, the y-values are 5, 4, -5, -3, and 0. The range is the set of these values: {-5, -3, 0, 4, 5}.

Step 3: Determine Maximum and Minimum of x-values

To find the maximum and minimum of the x-values from the domain, we see the list of values: {-3, -2, 0, 7}. The minimum x-value is -3, and the maximum x-value is 7.

Step 4: Determine Maximum and Minimum of y-values

To find the maximum and minimum of the y-values from the range, we look at: {-5, -3, 0, 4, 5}. The minimum y-value is -5, and the maximum y-value is 5.

Step 5: Label Appropriate Scales on the Axes

For the x-axis, label from -3 to 7 based on the minimum and maximum x-values. For the y-axis, label from -5 to 5 for the minimum and maximum y-values.

Step 6: Plot the Points

Plot each point from the given relation: (0,5), (-3,4), (-2,-5), (7,-3), and (0,0) on the labeled axes. Make sure each point is marked clearly on the graph.

Key Concepts

Ordered Pairs

Ordered pairs are like little addresses that tell you exactly where points are located on a coordinate plane. An ordered pair is written like this: \((x, y)\)where the first number inside the parentheses represents the position on the x-axis, and the second number represents the position on the y-axis. The x-value is called the "abscissa," and the y-value is known as the "ordinate." To better understand, think of it as a pair of directions you need to follow to get to a spot on a map.Here’s what makes ordered pairs so special:

• They are always written in the order of x first, then y.
• If you swap the numbers or order, you end up at a completely different location.
• Even if the x or y-value is zero, the ordered pair guides you to a precise location on the graph.

A cool tip is remembering that ordered pairs are a key component in defining relations and functions, which connect two sets of data. In our exercise example, pairs like (0, 5) and (-3, 4) were important starting points for graphing the relation.

Coordinate Plane

The coordinate plane is a two-dimensional space where each point is determined by an ordered pair of numbers: \((x, y)\)This plane is divided into four sections known as quadrants, each helping to precisely locate points. The horizontal line is called the x-axis, and the vertical line is the y-axis. These axes intersect at a point called the origin, identified by the ordered pair (0, 0).Here's more on the coordinate plane:

• Quadrant I, both x and y are positive.
• Quadrant II, x is negative and y is positive.
• Quadrant III, both x and y are negative.
• Quadrant IV, x is positive and y is negative.

When using a coordinate plane for graphing the relation, label the axes based on the range and domain provided. It helps in setting clear boundaries, so in the exercise, we labeled the x-axis from -3 to 7, and the y-axis from -5 to 5. This setup ensures every point on your list of ordered pairs fits snugly within the graph.

Relation Graphing

Relation graphing involves plotting a collection of ordered pairs on the coordinate plane. By connecting these dots, you explore the connection or 'relation' between differing sets of x and y-values. This process involves:

• Identifying ordered pairs.
• Labeling the correct scales on the axes based on the ordered pairs.
• Plotting each point precisely.
• Checking for connections or patterns among the plotted points.

In our exercise, plotting points such as (0, 5) and (-3, 4) helps visualize how the x and y values relate. It's like piecing together a puzzle to uncover the full picture. Once all points are on the graph, you can easily identify maximums, minimums, and determine any patterns or trends. Essentially, relation graphing provides a visual way to understand data and relationships between variables, enhancing comprehension beyond numerical data alone.