key term - Decreasing Function

Definition

A decreasing function is a type of function where, as the input values (or x-values) increase, the output values (or y-values) decrease. This means that for any two points on the function, if the first point has a larger x-value than the second, then the y-value of the first point will be less than the y-value of the second. Recognizing decreasing functions is crucial when analyzing graphs and understanding how transformations, like shifts or reflections, affect their behavior.

5 Must Know Facts For Your Next Test

1. In a decreasing function, if you pick any two points on the graph, moving from left to right shows that the graph goes downwards.
2. The slope of a decreasing linear function is negative, which means that the rise over run will yield a negative value.
3. Transformations such as vertical shifts or reflections can change a previously increasing function into a decreasing one.
4. A continuous decreasing function can have intervals where it is still decreasing without any breaks or jumps in its graph.
5. Graphically, a decreasing function can be identified as having a downward slope in its graphical representation.

Review Questions

• How can you identify whether a function is decreasing based on its graph?
  • To identify if a function is decreasing from its graph, look for sections where moving from left to right results in lower y-values. If you see that as x increases, y decreases consistently across an interval, then that part of the graph represents a decreasing function. It’s also helpful to observe if there are any horizontal segments or increases in between; they indicate that not every part is decreasing.
• What effects do transformations have on a decreasing function and how might they change its characteristics?
  • Transformations can significantly alter the nature of a decreasing function. For example, applying a vertical shift downward may maintain its decreasing nature but lower its overall position on the graph. Conversely, reflecting it across the x-axis can convert it into an increasing function. Understanding how these transformations impact a function's slope and behavior helps in predicting how graphs will look after adjustments.
• Evaluate how recognizing decreasing functions aids in understanding real-world scenarios like economics or physics.
  • Recognizing decreasing functions is crucial in fields like economics and physics because they often represent relationships where one variable decreases as another increases. For instance, in economics, demand often decreases as price increases, modeled by a decreasing function. Analyzing these functions helps predict outcomes and make informed decisions based on trends, highlighting their significance beyond mere mathematics.

Related terms

Increasing Function:

An increasing function is a type of function where, as the input values increase, the output values also increase.

Constant Function: A constant function is a type of function that always produces the same output value regardless of the input value.

Transformation: A transformation refers to any operation that alters the graph of a function, including shifts, reflections, and stretches.