Graph The Function Xy 16 0

Graphing the Function xy = 16: A Comprehensive Guide

The equation xy = 16 represents a rectangular hyperbola. Understanding its graph involves exploring its key features, including asymptotes, intercepts, and its behavior in different quadrants. This comprehensive guide will delve into these aspects, providing a thorough understanding of how to graph this function and analyze its properties.

Understanding the Equation xy = 16

The equation xy = 16 defines a relationship between two variables, x and y, where their product is always 16. This immediately tells us that neither x nor y can be zero. This constraint significantly influences the shape and behavior of the graph. We can also express this equation as y = 16/x, which highlights the inverse relationship between x and y. As x increases, y decreases, and vice versa.

Key Features of the Hyperbola

Several key features help us accurately and efficiently graph the rectangular hyperbola xy = 16:

1. Asymptotes

Asymptotes are lines that the graph approaches but never actually touches. For the function xy = 16, the asymptotes are the x-axis (y = 0) and the y-axis (x = 0). To visualize this, consider what happens as x approaches infinity or negative infinity. In either case, y approaches zero. Similarly, as y approaches infinity or negative infinity, x approaches zero. These are the lines our graph gets infinitely close to, but will never cross.

2. Intercepts

Intercepts are points where the graph crosses the x-axis or the y-axis. Since neither x nor y can be zero in the equation xy = 16, the graph has no x-intercepts and no y-intercepts. This reinforces the concept that the graph does not cross either axis.

3. Symmetry

The equation xy = 16 exhibits symmetry about the line y = x. This means that if (a, b) is a point on the graph, then (b, a) is also a point on the graph. This symmetry is easily demonstrable: if we substitute x = b and y = a into the equation, we still get ab = 16, satisfying the original condition.

4. Branches

The graph of xy = 16 is composed of two distinct branches. One branch resides in the first quadrant (x > 0, y > 0), where both x and y are positive. The other branch lies in the third quadrant (x < 0, y < 0), where both x and y are negative. The product of two negative numbers is positive, maintaining the condition xy = 16.

Step-by-Step Graphing Process

Let's outline a systematic approach to graphing the function xy = 16:

1. Identify the asymptotes: Draw the x-axis (y=0) and the y-axis (x=0) as dashed lines on your graph. These are the asymptotes that the hyperbola will approach.

2. Plot key points: Choose several values for x, and calculate the corresponding y values using the equation y = 16/x. For instance:

   • If x = 1, y = 16
   • If x = 2, y = 8
   • If x = 4, y = 4
   • If x = 8, y = 2
   • If x = 16, y = 1

   You can also choose negative values for x:

   • If x = -1, y = -16
   • If x = -2, y = -8
   • If x = -4, y = -4
   • If x = -8, y = -2
   • If x = -16, y = -1

   Plot these points on your graph.

3. Sketch the branches: Connect the points in the first quadrant to form one branch of the hyperbola. Similarly, connect the points in the third quadrant to form the second branch. Remember, the branches should approach the asymptotes but never touch them.

4. Apply symmetry: Verify the symmetry about the line y=x. If a point (a,b) is on the graph, the point (b,a) should also be on the graph. This helps ensure accuracy in your sketch.

Analyzing the Behavior of the Function

The equation xy = 16 displays several interesting behaviors:

• Inverse Relationship: The inverse relationship between x and y is clearly evident. As one variable increases, the other decreases proportionally.

• Asymptotic Behavior: The approach to the asymptotes highlights the limits of the function. The graph gets arbitrarily close to the x and y axes but never actually intersects them.

• Domain and Range: The domain and range of the function are both all real numbers except 0. This is because neither x nor y can be equal to zero. We can represent this using interval notation: (-∞, 0) U (0, ∞) for both the domain and range.

• Continuity: The function is continuous within each of its branches. However, there is a discontinuity at x = 0, resulting in two separate branches.

Extensions and Applications

The concept of a rectangular hyperbola, as represented by xy = 16, has various applications in different fields:

• Physics: In physics, inverse relationships are common. For example, the relationship between pressure and volume of an ideal gas (Boyle's Law) at a constant temperature can be modeled using a similar hyperbola.

• Economics: Certain economic models might use hyperbolas to represent relationships between variables such as supply and demand under specific conditions.

• Mathematics: Hyperbolas are essential in conic sections and coordinate geometry, and understanding their properties is crucial for advanced mathematical studies.

Conclusion

Graphing the function xy = 16 provides a valuable exercise in understanding the properties of rectangular hyperbolas. By systematically analyzing the asymptotes, intercepts, symmetry, and branches, we can accurately sketch the graph and comprehend the inverse relationship between x and y. The application of this knowledge extends beyond mathematics, offering insight into various scientific and economic models that demonstrate similar inverse proportional relationships. Remember, practice is key to mastering the graphing of such functions, so try experimenting with different values and observing how the graph changes. This detailed analysis should equip you with a comprehensive understanding of this fundamental mathematical concept.