In this activity, students are introduced to the idea that some functions can be defined by a rule, and the rule can be described in words or with expressions and equations. Students examine some simple rules and make connections between their verbal and algebraic representations. Doing so prompts them to look for and make use of structure (MP7).

The algebraic statements are written in function notation, so the work also reinforces students’ understanding of the notation and expands their capacity to use it to describe functions.

Invite students to briefly share how they matched the equations and verbal descriptions in the first question. Discuss questions such as:

• “The expressions for functions and both involve multiplying by 3 and subtracting 7. How are they different?” (The order in which the operations happen is different. Function first multiplies the input by 3, and then 7 is subtracted from the result. Function first subtracts 7, then multiplies the result by 3.)
• “The expressions for and both involve subtracting 7 and dividing by 3. How did you decide which one corresponds to description A and which one corresponds to D?” (By looking at what is done to first. In , is divided by 3 before 7 is subtracted, so it must correspond to D.)

Next, ask students how they determined which function has  as an input-output pair and which function has the greatest output when is 0 and when is 10. Highlight explanations that mention evaluating each function at those input values and seeing which one generates -3 for the output or gives the greatest output.

In this activity, students analyze functions that relate two quantities in a situation and work to define the relationship between the quantities with a rule. They do so by creating a table of values and generalizing the process of finding one quantity given the other. Students also plot the values in each table to see the graphical representation of the functions.

The mathematical reasoning here is not new. Students have done similar work earlier in the course, when investigating expressions and equations. What is new is seeing these relationships as functions and using function notation to describe them.

Students are likely to graph the functions by plotting the values in the tables and then connecting the points with a curve. As students work on the second set of questions about a perimeter function, which is linear, look for those who relate to a linear equation, namely, , and then graph a line with a vertical intercept of and a slope of 2. Invite them to share their thinking during the whole-class discussion.

If students struggle to graph the functions, suggest that they use the coordinate pairs in the tables to help them.

Select students to share the rule they wrote for the perimeter function (from the second set of questions) and how they determined the rule. Students may have written expressions of different forms for :

Record and display the variations for all to see, and discuss whether they all give the value of . Ask students to explain how they know these expressions are equivalent and define the same function.

Next, discuss how students sketched the graph of the function. If no students made a connection between the slope and vertical intercept of the graph of to the parameters in their equation, ask them about it. For example, display the graph of and ask students to use it to write an equation for the line.