How To Determine Increasing And Decreasing Intervals On A

Julian Assange

2 min read

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03-01-2025

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Determining the intervals where a function is increasing or decreasing is a fundamental concept in calculus. It allows us to understand the behavior of a function and its rate of change. This guide will walk you through the process, focusing on graphical analysis.

Understanding Increasing and Decreasing Functions

A function is considered increasing on an interval if its y-values consistently increase as its x-values increase. Conversely, a function is decreasing on an interval if its y-values consistently decrease as its x-values increase. Visually, this means:

• Increasing: The graph rises from left to right.
• Decreasing: The graph falls from left to right.

Identifying Intervals Graphically

The most straightforward method for determining increasing and decreasing intervals is by visually inspecting the graph of the function. Here's a step-by-step approach:

1. Examine the Graph: Carefully look at the graph of the function. Pay attention to the overall trend of the curve.

2. Identify Turning Points: Locate the points where the function changes from increasing to decreasing, or vice versa. These points are called critical points or turning points. They often represent local maxima (peaks) or local minima (valleys).

3. Determine Intervals: Based on the turning points, divide the x-axis into intervals. Each interval will either be increasing or decreasing.

4. Specify Intervals: Express the increasing and decreasing intervals using interval notation. Remember to use parentheses ( and ) for open intervals (where the endpoints are not included) and brackets [ and ] for closed intervals (where the endpoints are included). Infinity symbols, ∞ and -∞, are used to represent unbounded intervals.

Example:

Let's say a function's graph increases from x = -∞ to x = 2, then decreases from x = 2 to x = 5, and finally increases from x = 5 to x = ∞. The intervals would be described as:

• Increasing: (-∞, 2) ∪ (5, ∞)
• Decreasing: (2, 5)

Important Note: If the function is constant over an interval (a horizontal line segment), it is neither increasing nor decreasing on that interval.

Beyond Graphical Analysis

While graphical analysis is intuitive, it's important to note that for precise determinations, especially with complex functions, using calculus (specifically, the first derivative test) is necessary. The first derivative test identifies critical points and determines the function's behavior around those points. This method provides a more rigorous and accurate assessment of increasing and decreasing intervals, especially when dealing with functions that are not easily visualized graphically.