In this lesson, students deepen their understanding of functions by comparing representations of several functions relating the same pair of quantities. They analyze two or more graphs simultaneously, interpreting their relative features and their average rates of change in context.
Students also study comparative statements in function notation, such as \(A(x) = B(x)\) or \(B(10) > A(10)\), and explain them in terms of changes in population, changes in the trends of phone ownership, and the popularity of different television shows.
Students pay close attention to the intersection of two graphs in this lesson. Earlier in their study, they learned that a solution to a system of linear equations in two variables is a point where the graphs of the equations in the system intersect. Here, students recognize equations of the form \(f(x)=g(x)\) to mean functions \(f\) and \(g\) having the same output value at the same input value. They see that a solution to such an equation is the \(x\)coordinate of a point where the graphs of \(f\) and \(g\) intersect.
Making comparisons involves looking beyond individual pieces of information. To accurately relate the information from multiple representations requires careful and precise use of mathematical language and notation (MP6). Students continue reasoning abstractly and quantitatively (MP2) as they use their analyses of representations of functions to draw conclusions about the quantities in situations.
Lesson overview
 9.1 Warmup: Population Growth (5 minutes)

9.2 Activity: Wired or Wireless? (20 minutes)
 Includes "Are you Ready for More?" extension problem
 9.3 Optional Activity: Audience of TV Shows (15 minutes)
 9.4 Activity: Functions \(f\) and \(g\) (10 minutes)
 Lesson Synthesis
 9.5 Cooldown: A Toy Rocket and a Drone Again (5 minutes)
Learning goals:
 Compare key features of graphs of functions and interpret them in context.
 Interpret equations of the form \(f(x)=g(x)\) in context and recognize that the solutions to such an equation are the \(x\)coordinates of the points where the graphs of \(f\) and \(g\) intersect.
 Interpret statements about two or more functions written in function notation.
Learning goals (student facing):

Let’s compare graphs of functions to learn about the situations they represent.
Learning targets (student facing):
 I can compare the features of graphs of functions and explain what they mean in the situations represented.
 I can make sense of an equation of the form \(f(x)=g(x)\) in terms of a situation and a graph, and know how to find the solutions.
 I can make sense of statements about two or more functions when they are written in function notation.
Standards:
 This lesson builds towards the standard:CCSS.HSAREI.D.11MS.AREI.11MO.A1.REI.C.6
IM Algebra 1, Geometry, Algebra 2 is copyright 2019 Illustrative Mathematics and licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).
The Illustrative Mathematics name and logo are not subject to the Creative Commons license and may not be used without the prior and express written consent of Illustrative Mathematics.