To determine the open intervals on which a function is increasing by examining its graph, follow these steps:
- Understand the Graph: Start by looking at the graph of the function. An increasing function is characterized by a slope that moves upward as you move from left to right. This means that for any two points on the curve, if you move from the left point to the right one, the y-value should rise.
- Identify Critical Points: Look for critical points where the function changes its behavior; these are typically the points where the derivative is zero or undefined. At these points, the function might switch from increasing to decreasing or vice versa. Mark these points on your graph.
- Analyze Intervals: Divide the x-axis into intervals based on the critical points identified. Each interval will represent a portion of the graph between two critical points or the endpoints of the graph.
- Examine the Slope: For each interval, determine whether the function is increasing or decreasing. You can do this by selecting a test point from each interval and observing the y-value. If the y-value increases as x increases, the function is increasing on that interval.
- Write the Open Intervals: For all intervals where the function is increasing, express these as open intervals in interval notation. For example, if the function is increasing from x = a to x = b, you would write this as (a, b).
By following these steps, you will be able to effectively identify the open intervals where the function is increasing, giving you insights into its behavior.