How To Determine Increasing And Decreasing Intervals On A Graph

Waren Burn

2 min read

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24-11-2024

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Determining the intervals where a function is increasing or decreasing is a fundamental concept in calculus. Understanding this allows you to analyze the behavior of a function and its rate of change. This guide provides a clear, step-by-step method for identifying these intervals directly from a graph.

Understanding Increasing and Decreasing Functions

Before we delve into the graphical analysis, let's establish the definitions:

• Increasing Function: A function is increasing on an interval if, for any two points x₁ and x₂ in that interval, where x₁ < x₂, f(x₁) < f(x₂). Visually, this means the graph is rising as you move from left to right.

• Decreasing Function: A function is decreasing on an interval if, for any two points x₁ and x₂ in that interval, where x₁ < x₂, f(x₁) > f(x₂). Visually, this means the graph is falling as you move from left to right.

• Constant Function: A function is constant on an interval if, for any two points x₁ and x₂ in that interval, f(x₁) = f(x₂). Visually, this means the graph is a horizontal line.

Identifying Intervals from a Graph

Analyzing a graph to find increasing and decreasing intervals is straightforward. Follow these steps:

1. Examine the Graph: Carefully observe the overall trend of the graph. Look for sections where the graph consistently rises or falls.

2. Identify Turning Points: Locate any points where the graph changes from increasing to decreasing (local maximum) or from decreasing to increasing (local minimum). These points are crucial for defining the intervals. Note that these points themselves are neither increasing nor decreasing.

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

4. Specify Intervals: Using interval notation (e.g., (a, b), [a, b), (a, b], [a, b]), describe the intervals where the function is increasing and decreasing. Remember to use parentheses for open intervals (where the endpoints are not included) and brackets for closed intervals (where the endpoints are included). Unless otherwise specified, assume open intervals.

Example

Let's consider a graph with a local maximum at x = 2 and a local minimum at x = 5. Assume the graph extends infinitely to the left and right.

• Increasing Interval: The function is increasing on the interval (-∞, 2) and (5, ∞).

• Decreasing Interval: The function is decreasing on the interval (2, 5).

Important Considerations

• Vertical Asymptotes: Vertical asymptotes represent breaks in the function's domain and should be considered as boundaries when identifying intervals. The function is neither increasing nor decreasing at the asymptote.

• Horizontal Asymptotes: Horizontal asymptotes indicate the behavior of the function as x approaches positive or negative infinity. They do not affect the determination of increasing/decreasing intervals within the function's domain.

• Accuracy: When reading intervals from a graph, aim for the best possible approximation. The accuracy depends on the scale and clarity of the graph.

By following these steps, you can accurately and efficiently determine the increasing and decreasing intervals of a function directly from its graph, providing valuable insight into its behavior.