key term - Y = k/x

Definition

The equation y = k/x represents an inverse variation, where the dependent variable y is inversely proportional to the independent variable x. This means that as x increases, y decreases, and vice versa. The constant k represents the rate of change or the proportionality constant between the two variables.

5 Must Know Facts For Your Next Test

1. The graph of y = k/x is a hyperbola, which opens either left or right depending on the sign of the constant k.
2. The constant k determines the steepness of the hyperbolic curve, with larger values of k resulting in a steeper curve.
3. Inverse variation relationships are commonly seen in real-world situations, such as the relationship between an object's speed and the time it takes to travel a fixed distance.
4. The equation y = k/x can be rearranged to x = k/y, which is useful for solving for one variable given the other and the constant k.
5. Inverse variation relationships are often used in modeling and optimization problems, where the goal is to minimize or maximize one variable while the other is constrained.

Review Questions

• Explain how the constant k in the equation y = k/x affects the shape and behavior of the hyperbolic curve.
  • The constant k in the equation y = k/x represents the proportionality constant between the variables x and y. The value of k determines the steepness and orientation of the hyperbolic curve. A larger positive value of k will result in a steeper, narrower hyperbola that opens left or right, depending on the sign of k. Conversely, a smaller positive value of k will produce a flatter, wider hyperbolic curve. The sign of k also affects the orientation of the curve, with a positive k resulting in a curve that opens left or right, and a negative k producing a curve that opens up or down.
• Describe how the inverse variation relationship represented by y = k/x can be used in real-world modeling and optimization problems.
  • The inverse variation relationship expressed by the equation y = k/x is commonly used in various modeling and optimization problems. For example, in the context of physics, the equation can model the relationship between an object's speed and the time it takes to travel a fixed distance, where the constant k represents the distance traveled. In economics, the equation can model the relationship between the price and quantity of a good, where the constant k represents the total revenue. In these types of problems, the goal is often to minimize or maximize one variable, such as time or cost, while the other variable is constrained. By understanding the inverse variation relationship and the role of the constant k, researchers and analysts can develop models and make informed decisions to optimize the desired outcome.
• Analyze how the equation y = k/x can be rearranged and applied to solve for one variable given the other and the constant k.
  • The equation y = k/x can be easily rearranged to solve for one variable given the other and the constant k. By isolating the variable you wish to solve for, the equation can be transformed into the form x = k/y or y = k/x. This rearrangement is useful in various applications, such as when you need to find the value of one variable based on the known values of the other variable and the constant k. For example, in a physics problem involving an object's speed and the time it takes to travel a fixed distance, you can use the rearranged equation to solve for the object's speed given the time and the distance traveled. Similarly, in an economic problem involving price and quantity, you can use the rearranged equation to solve for the price given the quantity and the total revenue. By understanding how to manipulate the inverse variation equation, you can effectively solve for the unknown variable in a wide range of real-world scenarios.

Related terms

Inverse Variation:

A relationship between two variables where one variable is inversely proportional to the other, meaning as one increases, the other decreases proportionally.

Proportionality Constant: A fixed value, represented by the variable k, that determines the rate of change between the two variables in an inverse variation relationship.

Hyperbolic Function: A function that has the form y = k/x, which results in a hyperbolic curve when graphed.