For some reason students tend to have difficulty with this concept. At first glance, it seems rather straightforward. If a graph is going up, then the function is increasing. If the graph is going down, then the function is decreasing. If the graph is a horizontal line, then the function is constant. This is the level of this key feature of functions taught in my algebra 2 class.
In calculus, we take it a step further and look at intervals where the first derivative is positive to see where the function is increasing. And likewise, intervals where the first derivative is negative tells us where the function is decreasing. If the first derivative equals zero, then the function is constant.
Next, are the intervals for increasing and decreasing open or closed? Different textbooks will have different notations. Some will use open intervals while others will use closed. I used to be in the open interval camp, but have switched over to closed. I discuss this with my students about how textbook publishers and even math teachers cannot agree on this topic. With that said, if a function is increasing as it goes to an asymptote or infinity, then an open interval would be used.
Let’s look at the definition for a function increasing or decreasing on an interval.
I start this topic with a what do you notice prompt for the graph shown above. It is an entry point into the lesson where each student can share what they see. We then get out some crayons or colored pencils to shade in different portions of the graph.
Next I ask students to try sketching their own examples of graphs that are increasing, decreasing or constant, followed by a graph that exhibits more than one of these behaviors.
Before we start creating intervals for increasing and decreasing, we first review interval notation. Click HERE if you are interested in my free graphic organizer for interval notation.
Finally, we analyze graphs to identify intervals of increasing, decreasing or constant. After students feel comfortable identifying these intervals, I give them intervals with specified criteria and ask them to create the graph that would meet these criteria.
If you are interested in my lesson, it is available in my TPT shop as part of my Key Features of Functions Unit. I have included 2 versions: closed interval notation and open interval notation. Cheers!
