The mathematical work in this activity centers on interpreting graphical and symbolic representations of a piecewise-defined function.

First, students interpret a graph of a piecewise function and make sense of the rules in terms of a situation. In particular, students consider the meaning of the open and closed circles on the graph and the input values that appear to "break" the graph.

Next, students interpret a couple of equations in cases notation that could define the function, making connections between the symbols and numbers in the notation and features of the graph.

Invite students to share their analyses of Mai's and Kiran's work and identify the errors each person made. Highlight explanations that point out that:

• In Kiran's rules, a letter that weighs 1, 2, or 3 ounces each has two possible rates. For example, for 1 ounce, it could be $0.50 or $0.71. For 2 ounces, it could be $0.71 or $0.92.
• Mai's rules, on the other hand, excludes letters that weigh exactly 1, 2, or 3 ounces each. No rates are specified for those weights.
• In both cases, the problem is with the inequality symbols that are used.

Discuss what the rules should be, making sure to connect the notation with the features on the graph. Ask students:

• "How do we know whether to use the  symbol or the  symbol by looking at the graph?" (A point with an open circle means the coordinate values of that point are not included, but values that are greater or less—depending on the graph—are included. So an open circle corresponds to the  symbol or the  symbol. A point with a closed (or solid) circle means the coordinate values of that point are included, so it corresponds to the  symbol or the  symbol.)
• "By looking at the graph, how can you tell how many cases should be included in the equation?" (Each piece of the graph is a case.)

If time permits, ask students about the domain and range for this function. Assuming that a mail item heavier than 3.5 ounces no longer qualifies as a letter, the domain would include weights that are greater than 0 but no more than 3.5 ounces (\(0). The corresponding range includes 0.50, 0.71, 0.92, and 1.13, in dollars.

Display the table in the Task Statement, and ask students what values should go in the column for cost. If there are disagreements about the cost for a certain number of minutes of rental, ask students who disagree to share their reasoning and discuss until they reach an agreement.

Then, select a student to display their graph (or, if there are variations in students' graphs, select a few students to share, and ask the class to compare and contrast the graphs). Discuss whether all parts of each graph accurately represent the different cases or intervals of rental time. Ask questions such as:

• "In the rules, the number 30 shows up in the first two cases. How can we tell if the cost for 30 minutes of rental is $2.50 or $5.00?" (The rule says if is less than or equal to 30, then the cost is $2.50.)
• "How do we mark the point for on the graph?" (We use a closed circle for and and open circle for .)
• "The rules show the cost for renting up to 720 minutes, but the horizontal scale of the graph goes to only 240. How do we graph the last case?" (There is an open circle on the left end, at ), followed by a horizontal line all the way to , without special markings at the end. An open circle is needed only if we're excluding that point, which is not the case here.)

Ask students about the domain and range of the function. Assuming that 720 minutes is the maximum length of rental, the domain would include all values of greater than 0 and not over 720 (or ). Point out that the range of the function is not all numbers between $2.50 and $15.00 because the cost always increases at an increment of $2.50. It is not possible for a rental to cost $11.00, for example.

If time permits, ask student:

• “Is a function of ? How do you know?” (Yes. The price is determined by the rental time. Every time somebody rents a bike for the length of time, they will pay the same amount.)
• “Is a function of ? How do you know?” (No. Knowing that somebody paid $2.50 to rent the bike doesn't tell us how long they rented the bike. It could be anywhere from 1 second to 30 minutes.)

This activity is optional. It gives students an opportunity to reason about and graph the rules of piecewise functions without a context.

Students are given the equations that define two piecewise functions, along with strips of paper, each containing a part of a graph and a portion of the horizontal axis (no scale is shown). Their job is to arrange the strips, apply a scale on each axis, and add open and closed circles to the graph to accurately represent the function values at each interval of input.

Unlike the piecewise functions students saw earlier in the lesson, each function here contains intervals defined by non-constant linear expressions. Support students as needed in reasoning about those intervals. If desired, this activity could be done over two class periods. (Students could piece together the graph of the first function in one class period and do the same for the second function at another time.)

To create an accurate graph, students need to make sense of the values, expressions, and inequality symbols in each equation and persevere in discerning their connections to the graph (MP1).

Cut each graph in the blackline master along the dashed lines and the vertical axis before giving it to students.

Invite students to display their completed graphs and explain their reasoning. In particular, discuss how students determined the appearance of the intervals defined by non-constant linear expressions, such as , , or .

If different groups created different graphs for the same function, ask them to analyze one another's work and try to reach an agreement.

If time permits, ask students to identify the domain and range of each function.