In this activity, students work with rules of functions that arise from situations and continue to make connections between different representations of functions. They use rules to evaluate functions and interpret the input and output values in context. For instance, students see that represents the cost of using the service 1 month beyond the trial period, and find its value by computing .

Students also use rules of functions to solve for an unknown input value. For example, given the rule and the statement , they work to find a value of that makes true. Monitor for students who use these strategies:

• Guessing and checking
• Analyzing the graph of and identifying the -value when is 280
• Reasoning that a budget of $280 means that only $30 is available for additional months, and at $10 per month, it means 3 additional months of service.
• Solving algebraically

Plan to have students present in this order to support moving students toward more precise algebraic thinking.

Students may question if  is a function at all because unlike or other function rules they have seen so far,  is defined with a constant instead of an expression containing the dependent variable. Or they may wonder why  has the same value no matter what the input value is. Ask students to recall the definition of function and to consider whether each input value gives only one output value. Because it does, even though it is always the same output value, is still a function.

The purpose of this discussion is to explore linear functions that represent situations and examine how to solve functions when the input or output value is known.

Select students to share their interpretations of the two options. Make sure students see that:

• The equation tells us that, regardless of the number of months the service is used, , the cost, is always 600.
• The in the rule of tells us that each extra month of service after the trial period costs $10, and that there is a $250 price for the console.

Explain to students that the two functions here are linear functions because the output of each function changes at a constant rate relative to the input. Option B involves a rate of change of $10 per additional month after the trial period, while Option A has a rate of change of $0 per additional month after the trial period.

Then, ask students how they went about graphing the functions. Students are likely to have plotted some input-output pairs of each function. If no students mention identifying the slope and vertical intercept of each graph, ask them about it.

Invite previously selected groups to share how they solved the last question. Sequence the discussion of the strategies by the order listed in the Activity Narrative. If possible, record and display their work for all to see.

Connect the different responses to the learning goals by asking questions, such as:

• “How did the console price show up in your work of finding the number of months Elena could get for $280?” (It is the constant 250 in the function. I subtracted that from the budgeted $280 to see that there is $30 left for additional months.)
• “How would these methods change if Elena had budgeted $300?” (I would replace the 280 with 300 in my equation and solve the new equation for .)
• “Why is the graph of a horizontal line?” ( is the output of function and is represented by vertical values on a coordinate plane. The vertical values are typically labeled with the variable , so we can write and graph to represent function .)
• “How can we see the slope and intercept of the graph of Option B in the function?” (  is the output of function and is represented by vertical values on a plane. We can write or to see that the slope is 10 and the -intercept is 250.)

If time permits, invite students to share which option they believe Elena should choose and why.

In this optional activity, students learn to use graphing technology to graph an equation in function notation, evaluate the function at a specific input value, create a table of inputs and outputs, and identify the coordinates of the points along a graph. They also revisit how to set an appropriate graphing window.

In the digital version of the activity, students use an applet to find precise function values. The applet allows students to click on the graph of a function to find precise coordinate pairs. If available, use the digital version to increase precision and allow students to focus on the notation rather than estimation.

Invite students to share any insights they had while using the graphing tool and techniques to evaluate expressions and solve equations. In what ways might the tool and techniques be handy? When might they be limited?

Also discuss any issues, technical or otherwise, that students encountered while completing the task.

If desired, consider showing another way to obtain input-output pairs of a function in Desmos.

Let’s assign a new input variable, say , to function . If we enter in the expression list, activate a slider for , and enable the option to label points, the graph will show the coordinate pair for any value of .