Key Features of Functions – Mathberry Lane

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!

Interval notation is a way to describe a set of numbers. In elementary school, students are introduced to number lines and inequalities. Number lines are a wonderful visual tool for students to make sense of numbers and to process whether a value is greater than or less than another value. Eventually, these ideas morph into a solution set when they reach pre-algebra. For example, x is greater than 3 has infinitely many solutions. Its solution set could be displayed on a number line with an open ended ray.

Students first learn to write this analytically as an inequality, x > 3. We can write this same interval of values with interval notation such as (3,∞). In algebra, we start to study intervals of numbers such as the domain and range. Instead of using an inequality to represent an interval of real numbers, interval notation is often used instead. When a set of numbers does not include the endpoint, as shown above, a parenthesis is used to indicate that the interval approaches that number, but does not include it in the interval.

Suppose we would like to include the endpoint of an interval. Let’s look at the interval when x is less than or equal to 2.

As an inequality we would write this as x ≤ 2. However as an interval of values, we would use a square bracket to show that the endpoint 2 is included in the interval such as (-∞, 2]. Note that one can never actually reach infinity, so infinity will always have a parenthesis and not a square bracket.

Sometimes an interval of numbers has a starting and ending point on the number line. This is sometimes referred to as an “and” compound inequality.

As an inequality we would write -3<x<1. And then for interval notation it would look like (-3,1).

We can make a union of intervals when the rays go in opposite directions. This would be an “or” compound inequality.

The set of all real numbers can also be written with interval notation.

A common mistake I students make when using interval notation is to write the larger value first and then the smaller value. So, make sure that when you introduce interval notation to your students that you remind them it looks like (lower bound, upper bound).

You can grab my free interval notation graphic organizer here. I also have a full version Interval Notation Lesson available to purchase in my TPT store that is part of my Key Features Unit.

Category: Key Features of Functions

Teaching Increasing and Decreasing Functions

Teaching Interval Notation