Lesson 17

In this warm-up, students make sense of a linear function with a negative rate of change, set in a familiar context. Students have seen similar relationships between the amount of water in a tank and time in an earlier unit (on solving and graphing equations). Here, they think about the quantities in terms of input and output, interpret statements in function notation, and sketch a graph of the function.

The work here prepares students to find and interpret the inverse of functions written in function notation, and to use inverse functions to solve problems.

Arrange students in groups of 2. Provide access to scientific or four-function calculators, if requested.

Select students to share their responses, including their graph.

Make sure students recall that in an equation such as \(w(t)=80-2.5t\), \(w(t)\) is the output when the input is \(t\), that they can interpret the equation in terms of the situation, and see why the graph is a downward-slanting line.

In this activity, students revisit the same function they saw in the warm-up, which was given in function notation. They work to find and represent the inverse function and think about what it tells us in this situation.

Point out to students that the equations we have seen in the past few lessons use variables for the input and output. Sometimes, however, an equation that defines a function is written using function notation, so the output has the form \(w(t)\), or \(\text {function name (input variable)}\).

Tell students that they will now think about how to represent the inverse of a function defined using function notation.

Invite students to share their equation and interpretation of the inverse function of \(w\) and their prediction of what its graph would look like.

Discuss questions such as:

• "In the inverse of function \(w\), what is the input and what is the output?" (\(w(t)\) is now the input and \(t\) is the output.)
• "The equation \(t = \frac {80 - w(t)}{2.5}\) represents the inverse function of \(w\). What does it tell us?" (It tells us the time in minutes as a function of the amount of water in liters. It allows us to find the time that has passed if we know how much water is left in the tank.)
• "Suppose that, instead of using the function name and writing \(w(t)\), we designate another variable, say \(r\), to represent it, so \(r = w(t)\). How would that affect the equation for the inverse function?" It would be written as \(t = \frac {80 - r}{2.5}\).)
• (Display the graphs of both function \(w\) and its inverse.) "On the graph of \(w\), which point represents the time the tank had 80 liters of water?" (The vertical intercept, \((0,80)\).)
• "On the graph of the inverse function, which point represents that same time and same amount of water?" (The horizontal intercept, \((80,0)\).)

  

  

  

  

In this activity, students revisit another familiar situation: a function that gives the percentage of homes with only cell phones as a function of time since year 2004. Here, the focus is on finding a linear function that models the given data and using the model to answer questions about the situation.

Previously, students saw questions about percentages, such as: “What percentage of homes had only cell phones in 2010?” Here, they see questions such as: “In what year did 50% of all homes use only cell phones?” or "Solve \(P(t)=30\) for \(t\)." Students are not explicitly asked to find the inverse of their function, but the questions about time can be more efficiently answered by first finding the inverse of the function, motivating students to do so.

To write an equation for a possible linear model, students may find a line of best fit (and then determine its slope and vertical intercept, by hand or using technology), or they may find an average rate of change between two meaningful points on the data. Students engage in aspects of modeling (MP4) as they make such decisions.

Making graphing technology available gives students an opportunity to choose appropriate tools strategically (MP5).

Arrange students in groups of 2–3. Provide access to graphing technology, in case requested.

Students may not recall how to write an equation to model a set of data, even if they see that the points on the scatter plot appear to be linear. Ask students how they might find the rate of change in the situation, or the slope of a line that could represent the trend in the data. Prompt them with questions such as, "How quickly does the percentage of homes with only cell phones grow?" or "By how many percent, roughly, does it grow each year? How can we find out?" Once they see how to estimate a rate of change or to calculate the slope of a line that fits the data, ask what other information they might need to write a linear equation.

Invite students to share how they wrote an equation to model the relationship in the data. Then, focus the discussion on how they answered questions about time, such as how they solved \(P(t)=30\) for \(t\), or found the years in which the percentages of cell-phone-only homes reached 50%, 75%, and so on.

When writing an equation for the last question, students may have solved for \(t\) but not realize that what they have done is writing an inverse of their original function. If so, highlight this connection and discuss how the inverse function could help us answer the questions about time.