Lesson 9
Increasing and Decreasing Functions
These materials, when encountered before Algebra 1, Unit 4, Lesson 9 support success in that lesson.
9.1: Comparing Values (10 minutes)
Warmup
In this warmup, students compare values to write true equations or inequalities. In the associated Algebra 1 lesson, students will compare functions. Students are supported by considering individual values in this warmup.
Student Facing
For each pair of numbers, write \(=,<\), or \(>\) in the blank to make a true equation or inequality. Be prepared to share your reasoning.
 6 \(\underline{\hspace{.5in}}\) 9
 \(\frac{7}{3}\ \underline{\hspace{.5in}}\ \frac{13}{6}\)
 5.2 \(\underline{\hspace{.5in}}\ \frac{53}{11}\)
 \(5 (3  6)\ \underline{\hspace{.5in}}\ 15  6\)
 Let \(f(x) = 5  2x\).
 \(f(3)\ \underline{\hspace{.5in}}\ f(5)\)
 \(f(\text{}3)\ \underline{\hspace{.5in}}\ f(\text{}4)\)
 \(f(\text{}1)\ \underline{\hspace{.5in}}\ f(1)\)
Student Response
For access, consult one of our IM Certified Partners.
Activity Synthesis
The purpose of the discussion is to recognize methods students used to compare the values. Select students to share their solutions and reasoning. After each solution shared, ask if there are other methods students used to determine the solution. In particular, listen for students who:
 manipulate fractions to get common denominators
 rewrite fractions in decimal form
 solve for the values of a function or expression and compare
 recognize that the function is decreasing, so greater inputs will result in lesser outputs
If no students share the method using the decreasing function, point it out.
9.2: What Could It Be? (15 minutes)
Activity
In this activity, students invent situations that could be represented by given graphs. It is not essential that students describe situations that would be perfectly linear. Particular attention should be given to discussion about how the graphs increase and decrease. In the associated Algebra 1 lesson, students look at more complex functions and the average rate of change over certain intervals is essential to the comparison. Students are supported in this activity by gaining a better understanding of real situations when the functions increase or decrease. These situations can be referenced when discussing the concept with students in future lessons and activities.
Student Facing
Describe \(f(x)\) and \(g(x)\) with a situation that could fit the given graphs. Explain your reasoning.
Student Response
For access, consult one of our IM Certified Partners.
Activity Synthesis
The purpose of the discussion is to provide some concrete examples for students to reference when thinking about increasing and decreasing functions. Select students to share solutions. As they share, ask students if there are any interesting points in the graph and what they would mean in the situation described. After each solution shared, ask if there are additional descriptions from other students.
9.3: Cities, Towns, and Villages (15 minutes)
Activity
In this activity, students draw graphs that represent given situations. Particular focus should be given to describing the functions using the terms “increasing” and “decreasing” as well as recognizing regions when one function is greater than the other or when they are equal.
Student Facing
Draw an example of a graph that shows two functions as they are described. Make sure to label the functions.

The population of 2 cities as functions of time so that city A always has more people than city B.

The population of 2 towns as functions of time so that town A is larger to start, but then town B gets larger.

The population of 2 villages as functions of time so that village A has a steady population and village B has a population that is initially large, but decreases.
Student Response
For access, consult one of our IM Certified Partners.
Activity Synthesis
The purpose of the discussion is to compare features of functions on a graph that can be the same and different based on basic descriptions. There are many possible ways to interpret the descriptions to create correct graphs. There are some aspects that must be the same and some that could be different. For each situation, select at least 3 graphs to display for the class. Ask students to compare and contrast the graphs for the situation to find things that are the same and different.
Some things that may be different:
 the actual populations of the regions
 how quickly the populations change
 the way in which the populations change (for example, linear or curved graphs)
Some things that may be the same:
 The graph of the cities must not intersect.
 The graph of the towns must have a point where the two curves intersect.
 Village B must have a graph that is decreasing overall.
 Village A must have a graph that is relatively constant.
 The graph of town A must start higher than town B, but after the intersection, the graph for town B is higher.
Here are some additional questions about the graphs:
 “What is happening at the point where the graphs of the populations intersect?” (The two regions have the same population at the same time.)
 “Does the graph for city A or city B have to increase?” (Not necessarily. They could both decrease or A could increase and B could decrease.)
 “Does the graph for town B have to increase?” (Not necessarily. It could decrease at a slower rate than town A or remain steady while town A’s population decreases.)