Complete The Coordinate Table For The Given Equation Xy 12

Completing the Coordinate Table for the Equation xy = 12: A Comprehensive Guide

The equation xy = 12 represents a rectangular hyperbola. Understanding its characteristics and plotting its graph requires a systematic approach, particularly when constructing a coordinate table. This article will guide you through the process of completing a coordinate table for this equation, exploring various methods, analyzing its properties, and demonstrating its practical applications. We'll delve into the intricacies of hyperbolas, offering a detailed, step-by-step approach accessible to all levels of mathematical understanding.

Understanding the Equation xy = 12

Before we start filling out the coordinate table, let's establish a clear understanding of the equation itself. xy = 12 is a type of inverse relationship. This means that as the value of 'x' increases, the value of 'y' decreases proportionally, and vice versa. The product of x and y always remains constant at 12. This constant product defines the hyperbola's shape and properties. Unlike linear equations, there are no x or y intercepts; the graph never crosses the x or y axes.

Constructing the Coordinate Table: A Systematic Approach

To create a comprehensive coordinate table, we'll select various values for 'x' and calculate the corresponding 'y' values using the equation xy = 12. Remember, we can choose any value for x (except 0, as this would make y undefined), and the corresponding y value is found by solving the equation for y: y = 12/x. Let's begin with a table structured as follows:

Let's populate this table with several data points. We'll choose both positive and negative values for x to illustrate the symmetry of the hyperbola.

• Positive x values:

Let's start with simple integer values:

• If x = 1, then y = 12/1 = 12. This gives us the point (1, 12).

• If x = 2, then y = 12/2 = 6. This gives us the point (2, 6).

• If x = 3, then y = 12/3 = 4. This gives us the point (3, 4).

• If x = 4, then y = 12/4 = 3. This gives us the point (4, 3).

• If x = 6, then y = 12/6 = 2. This gives us the point (6, 2).

• If x = 12, then y = 12/12 = 1. This gives us the point (12, 1).

• Negative x values:

Now let's include negative x values. Remember that a negative x multiplied by a negative y will still yield a positive 12.

• If x = -1, then y = 12/(-1) = -12. This gives us the point (-1, -12).

• If x = -2, then y = 12/(-2) = -6. This gives us the point (-2, -6).

• If x = -3, then y = 12/(-3) = -4. This gives us the point (-3, -4).

• If x = -4, then y = 12/(-4) = -3. This gives us the point (-4, -3).

• If x = -6, then y = 12/(-6) = -2. This gives us the point (-6, -2).

• If x = -12, then y = 12/(-12) = -1. This gives us the point (-12, -1).

• Fractional and Decimal Values:

To get a more complete picture of the curve, let's add some points with fractional or decimal x values.

• If x = 0.5, then y = 12/0.5 = 24. This gives us the point (0.5, 24).
• If x = 1.5, then y = 12/1.5 = 8. This gives us the point (1.5, 8).
• If x = -0.5, then y = 12/(-0.5) = -24. This gives us the point (-0.5, -24).
• If x = -1.5, then y = 12/(-1.5) = -8. This gives us the point (-1.5, -8).

Now, let's update our coordinate table with all these points:

Graphing the Hyperbola

With our completed coordinate table, we can now plot these points on a Cartesian coordinate system (x-y plane). You'll observe that the points form a curve known as a rectangular hyperbola. This hyperbola has two branches, one in the first quadrant (where both x and y are positive) and one in the third quadrant (where both x and y are negative). The branches approach but never touch the x and y axes. The axes themselves are asymptotes of the hyperbola – lines that the curve approaches indefinitely but never intersects.

Properties of the Hyperbola xy = 12

Let's summarize the key properties of the hyperbola represented by the equation xy = 12:

• Asymptotes: The x and y axes (x = 0 and y = 0) are the asymptotes.
• Branches: The hyperbola has two branches, symmetrically located across the origin.
• Symmetry: The hyperbola is symmetric with respect to the origin. If (x, y) is a point on the hyperbola, then (-x, -y) is also a point on the hyperbola.
• Inverse Relationship: The relationship between x and y is inversely proportional. As one variable increases, the other decreases.
• No Intercepts: The hyperbola does not intersect either the x-axis or the y-axis.

Applications of Rectangular Hyperbolas

Rectangular hyperbolas, while seemingly abstract, have several practical applications in various fields:

• Physics: They appear in descriptions of certain physical phenomena, such as the relationship between pressure and volume of an ideal gas under isothermal conditions (Boyle's Law).
• Economics: In economics, they can model inverse relationships between price and quantity demanded in certain market situations.
• Engineering: Hyperbolic curves find use in the design of certain structures and engineering components.
• Mathematics: They serve as important examples in the study of conic sections and advanced mathematical concepts.

Conclusion: Mastering the Rectangular Hyperbola

Constructing a coordinate table for the equation xy = 12 and subsequently graphing the resulting hyperbola provides valuable insights into the properties of this important mathematical function. By understanding its behavior and applications, you gain a deeper appreciation for the interconnectedness of mathematics and the real world. Remember to choose a diverse range of x-values – positive, negative, integer, and fractional – to obtain a clear and accurate representation of the hyperbola. This comprehensive approach ensures a thorough understanding of the inverse relationship inherent in the equation and the unique characteristics of the resulting curve. The systematic approach detailed here makes constructing similar coordinate tables for other inverse relationships relatively straightforward. Practice makes perfect, so keep exploring and expanding your understanding of mathematical functions!