Lesson 9

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.