What is a proportional relationship?

A proportional relationship, also known as a directly proportional relationship, exists between two variables when their ratio is constant. This means that as one variable increases, the other increases proportionally, and as one decreases, the other decreases proportionally. The constant ratio is called the constant of proportionality (often represented by k).

Here's a breakdown of key characteristics:

• Constant Ratio: The defining feature. If you take the ratio of one variable to the other (y/x, for example), you always get the same value, k, regardless of the specific values of x and y.

• Equation: Proportional relationships can be expressed by the equation y = kx, where:

  • y is the dependent variable
  • x is the independent variable
  • k is the constant of proportionality

• Graph: When plotted on a coordinate plane, a proportional relationship always forms a straight line that passes through the origin (0,0). This is because when x=0, y=0.

• Table of Values: A table of values representing a proportional relationship will show a consistent ratio between corresponding x and y values.

• Examples: Many real-world scenarios exhibit proportional relationships, such as:

  • The relationship between the number of hours worked and the amount of money earned at a constant hourly rate.
  • The relationship between the number of items purchased and the total cost at a fixed price per item.
  • The relationship between the distance traveled and the time taken at a constant speed.

Non-Examples: Relationships where adding a constant to one variable or squaring/cubing a variable, etc. are not proportional. For example, y = x + 2 or y = x² are not proportional relationships. They might be linear or quadratic, but not proportional.