In this activity students make sense of a linear function in context. They interpret the rate of change and function values, given inputs that are in and out of the domain that makes sense for the situation. As students interpret values in the context of the situation, they reason abstractly and quantitatively (MP2). In the associated Algebra 1 lesson students use inverse functions to discover information about situations. This activity supports students by asking them to think about a situation based on the function that represents it.

The purpose of the discussion is to recall the connection between functions and the situations they describe. Select students to share their solutions. Here are some questions for discussion:

• “What is a reasonable domain for the function in this situation? Explain your reasoning.” ( because it only makes sense after the trip has started and the car trip probably wouldn’t go longer than 10 hours at a time.)
• “Is a linear function realistic for this situation? Explain your reasoning.” (It is not very realistic because the car will likely slow down or stop due to things like traffic or stop lights or similar things rather than maintain a constant 34 mph for the entire trip.)
• “Even though the linear function may be unrealistic to completely describe the car trip, is there any use to using a linear function to model a car trip? Explain your reasoning.” (There is some use. If the average speed of the car is 34 mph, a linear function like this may give an approximate reading on the odometer, which could be useful to know when the car may be running low on gas or need maintenance.)

In this activity students use a description of a situation to write a function representing the situation and then describe the domain and range of the function and evaluate what input is associated with a given output. Students reason abstractly and quantitatively when connecting the situations with equations and interpreting values (MP2). In the associated Algebra 1 lesson, students use functions and their inverses to answer questions about situations. Students are supported in this lesson by recalling the connection of functions to situations and using some inverse thinking to get a desired input for a given output.

The purpose of this discussion is to draw additional connections between a function and the situation it describes. Select students to share their solutions and reasonings. Here are some questions for discussion:

• “How did you think about writing the function to represent the situation?” (First I recognized what was already there when the time started and used that as the constant term. Then I found the amount that was added after each unit of time and used that rate of change as the coefficient of .)
• “How did you find the value of given the function value? For the question here, is it easier to use the inverse to find the value of or use the original function?” (To find , I substituted the value I was given for the function in my function equation and solved for . Because there was only 1 question, it was easier for me to not use the inverse. If I needed to find the value for for several function values, it might make sense to use the inverse like in the Warm-up.)