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, where the product of the two variables is constant.
2. The graph of the equation xy = k is a hyperbolic function, which has a characteristic U-shaped curve.
3. In an inverse variation, as one variable increases, the other variable decreases, and vice versa, maintaining a constant product.
4. The constant k in the equation xy = k represents the fixed product of the two variables, and it determines the shape and behavior of the hyperbolic function.
5. Inverse variation relationships are commonly observed in various scientific and real-world applications, such as Hooke's law, Boyle's law, and the relationship between speed and time in motion problems.

Review Questions

• Explain the relationship between the variables x and y in the equation xy = k.
  • In the equation xy = k, the variables x and y have an inverse variation relationship. This means that as one variable increases, the other variable decreases, and vice versa, while their product remains constant at the value of k. For example, if x doubles, y will be halved, maintaining the fixed product of k. This inverse relationship is characteristic of hyperbolic functions, which exhibit a U-shaped curve on a graph.
• Describe how the constant k in the equation xy = k affects the behavior of the inverse variation relationship.
  • The constant k in the equation xy = k represents the fixed product of the two variables, x and y. This constant determines the specific characteristics of the inverse variation relationship and the shape of the resulting hyperbolic function. A larger value of k will result in a wider, more open hyperbolic curve, while a smaller value of k will produce a narrower, more compressed curve. The value of k also affects the rate at which the variables x and y change in relation to each other, with a larger k leading to a slower rate of change compared to a smaller k.
• Analyze how the equation xy = k can be used to model real-world situations involving inverse variation.
  • The equation xy = k can be used to model a wide range of real-world situations that exhibit inverse variation relationships. Examples include Hooke's law, which describes the relationship between the force applied to a spring and its resulting displacement, Boyle's law, which describes the inverse relationship between the pressure and volume of a gas, and the relationship between speed and time in motion problems, where speed is inversely proportional to the time taken to cover a fixed distance. By understanding the properties of the inverse variation relationship represented by the equation xy = k, students can apply this knowledge to analyze and solve problems in various scientific and engineering contexts.

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 vice versa.

Hyperbolic Function: A function that has the form $f(x) = \frac{k}{x}$, where k is a constant, and it exhibits a characteristic hyperbolic shape.

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