In this Warm-up, students are prompted to compare function values. To do so, they need to interpret statements in function notation and connect their interpretations to the graph of the function.

Previously, students recognized that if a point with coordinates is on the graph of a function, the 2 is an input and the 20 is a corresponding output. Here, they begin to see the second value in a coordinate pair more abstractly, as when the input is 2 or as when the input is 4.

Invite students to share how they compared each pair of outputs. If not mentioned in students’ explanations, point out that for each pair, what we are comparing are the vertical values of the points on the graph at different horizontal values. Even though the statements don’t tell us the values of, say,  and , and the vertical axis shows no scale, we can tell from the graph that the function has a greater value when is 3 than when is 7.

Tell students that the coordinates represent the starting point of the drone. Ask students, “What are the coordinates of the points when the drone starts leveling off? When it starts to descend? When it lands?” (, and , respectively)

Highlight that the coordinates of each point on a graph of a function are .

Discuss with students how they compared and (in the last question). Make sure students see that the could be less than, equal to, or greater than , depending on the value of .

• If is 2, then is 3. From the graph, we can see that is equal to because the graph has the same vertical value when is 2 and 3.
• If is 5, then is 6. From the graph, we can see that is greater than because the graph has a greater vertical value when  is 5 than when is 6.

In this activity, students continue to interpret statements in function notation in terms of a situation. Several things are new here, all of which provide opportunities to attend to precision (MP6) and to reason quantitatively and abstractly (MP2).

• The output of the function is measured in millions, so students need to attend carefully to the units or otherwise may misinterpret the situation (for example, saying that about 2 thousand people rather than about 2 billion people owned smartphones in 2017).
• The input of the function is measured in “years after 2000,” so a positive input value needs to be interpreted as year . For example, the year 2010 should be associated with .
• One of the input values is negative, so students will need to reason about what this means in this situation.

Students also consolidate various pieces of information about the function, given in descriptions and function notation, in order to sketch the graph of a function. It is possible for students to sketch many different graphs through the given points, but the context suggests that, over time, the function values are increasing roughly exponentially (though students do not need to know that term at this point).

Previously, students learned that each point on the graph of a function is of the form for input and corresponding output . They analyzed and plotted input-output pairs in which both values were known.

In this activity, students reason about unknown output values by relating them to values that are known (by interpreting inequalities such as ), using a graph and a context to support their reasoning. The work here prompts students to reason quantitatively and abstractly (MP2).

Students take turns explaining their interpretation of statements in function notation and making sense of their partner’s interpretation, discussing their differences if they disagree. In so doing, students practice constructing logical arguments and critiquing the reasoning of others (MP3).

Identify students who can correctly interpret statements such as in terms of the situation and in relation to the graph of the function. Also, look for partners whose graphs are very different but are both correct. Invite them to share their responses during the whole-class discussion.