Decreasing Function

Definition

A decreasing function is a function where the output values decrease as the input values increase. In other words, as the independent variable increases, the dependent variable decreases. This term is particularly relevant in the context of understanding rates of change and the behavior of graphs, as well as analyzing the properties of linear functions.

5 Must Know Facts For Your Next Test

1. A decreasing function has a negative rate of change, meaning the function value decreases as the input variable increases.
2. The graph of a decreasing function will slope downward from left to right, with the dependent variable values getting smaller as the independent variable values get larger.
3. Decreasing functions can be linear, quadratic, exponential, or any other type of function as long as the output values decrease as the input values increase.
4. The rate of change, or slope, of a decreasing linear function will be a negative number, indicating the inverse relationship between the independent and dependent variables.
5. Decreasing functions are often used to model real-world situations where a quantity, such as cost or time, decreases as another quantity, such as quantity or distance, increases.

Review Questions

• Explain how the graph of a decreasing function differs from the graph of an increasing function.
  • The graph of a decreasing function will slope downward from left to right, with the dependent variable values getting smaller as the independent variable values get larger. In contrast, the graph of an increasing function will slope upward from left to right, with the dependent variable values getting larger as the independent variable values get larger. The key distinction is the direction of the slope, which indicates the relationship between the independent and dependent variables.
• Describe the relationship between the rate of change and the behavior of a decreasing function.
  • The rate of change, or slope, of a decreasing function will be a negative number, indicating an inverse relationship between the independent and dependent variables. As the independent variable increases, the dependent variable decreases. This negative rate of change is a defining characteristic of decreasing functions and is crucial for understanding their behavior and how they differ from increasing functions, which have a positive rate of change.
• Analyze how the properties of a decreasing linear function can be used to model real-world situations.
  • Decreasing linear functions are often used to model situations where a quantity, such as cost or time, decreases as another quantity, such as quantity or distance, increases. For example, the cost per unit of a product may decrease as the quantity purchased increases, or the time it takes to travel a distance may decrease as the speed increases. By understanding the properties of decreasing linear functions, including their negative rate of change and downward-sloping graphs, we can effectively model and analyze these types of real-world relationships.

Related terms

Increasing Function: A function where the output values increase as the input values increase. The dependent variable increases as the independent variable increases.

Constant Function: A function where the output value remains the same regardless of the input value. The dependent variable does not change as the independent variable changes.

Linear Function: A function where the relationship between the independent and dependent variables is a straight line. Linear functions can be either increasing or decreasing.