key term - Xy = k

Definition

The equation xy = k, where k is a constant, represents a relationship between two variables x and y where their product is a fixed value. This type of relationship is known as an inverse variation or a hyperbolic function, and it is commonly encountered in the context of modeling using variation.

5 Must Know Facts For Your Next Test

1. The equation xy = k represents an inverse variation relationship, where the product of the two variables is a constant.
2. In an inverse variation, as one variable increases, the other variable decreases, and their product remains the same.
3. The graph of an inverse variation function is a hyperbola, which is a curved shape that opens to the left and right.
4. Inverse variation relationships are commonly found in real-world situations, such as the relationship between speed and time in a fixed distance, or the relationship between the volume and pressure of a gas.
5. Understanding the properties of inverse variation and the equation xy = k is crucial for modeling and analyzing various phenomena in the context of 5.8 Modeling Using Variation.

Review Questions

• Explain the relationship between the variables x and y in the equation xy = k.
  • In the equation xy = k, where k is a constant, the variables x and y have an inverse variation relationship. This means that as one variable increases, the other variable decreases, and their product remains a fixed value. For example, if x increases, y must decrease in order to maintain the constant product k. This inverse relationship is often represented by a hyperbolic function graph, where the curve opens to the left and right.
• Describe how the equation xy = k can be used to model real-world situations in the context of 5.8 Modeling Using Variation.
  • The equation xy = k can be used to model various real-world phenomena that exhibit an inverse variation relationship. For instance, it can be used to model the relationship between the volume and pressure of a gas (Boyle's law), the relationship between speed and time for a fixed distance, or the relationship between the cost and quantity of a product. By understanding the properties of the inverse variation relationship represented by xy = k, you can analyze and make predictions about these types of situations in the context of 5.8 Modeling Using Variation.
• Analyze how the graph of the equation xy = k would change if the constant k were to increase or decrease.
  • $$\text{If the constant } k \text{ in the equation xy = k \text{ increases, the graph of the hyperbolic function will become narrower and taller.}}\\\text{Conversely, if the constant } k \text{ decreases, the graph of the hyperbolic function will become wider and shorter.}\\\text{This is because the product of the variables x and y must equal the constant k, so as k changes, the shape of the hyperbola adjusts to maintain this relationship.}$$

Related terms

Inverse Variation:

A relationship between two variables where one variable is inversely proportional to the other, meaning as one variable increases, the other decreases, and their product is a constant.

Hyperbolic Function: A function that has the form y = k/x, where k is a constant, and it represents an inverse variation relationship between the variables x and y.

Proportionality: A relationship between two variables where one variable is directly proportional to the other, meaning as one variable increases, the other increases proportionally.