Lesson 8

The goal of this warm-up is to review the meaning of a function presented graphically. While students do not need to use function notation here, interpreting the graph in terms of the context will prepare them for their work with functions in the rest of the unit.

Tell students to close their books or devices. Display the graph for all to see. Ask students to observe the graph and be prepared to share one thing they notice and one thing they wonder. Select students to briefly share one thing they noticed or wondered to help ensure all students understand the information conveyed in the graph. Ask students to open their books or devices and answer the questions about the graph. Follow with a whole-class discussion.

If students struggle to see from the graph how the accumulated rainfall is a function of time but time is not a function of accumulated rainfall, consider displaying the data in a table. Shown here is the data for the first 20 days of 2017. Help students see that for every value of \(t\), the time in days, there is one value of \(r\), the accumulated rainfall in inches, but this is not true the other way around.

Make sure students understand why the accumulated rain is a function of time but not the other way around.

Then, recall the notation for writing, using function notation, accumulated rain fall as a function of time. If \(r\) represents the amount of rainfall in inches and \(t\) is time in days, then \(r(t)\) is the amount of rain that has fallen in the first \(t\) days of 2017. For example, \(r(2) = 0\) tells us that the accumulated rainfall after in the first two days of the year was 0 inches. \(r(48)=1\) means that there was 1 inch of accumulated rain in the first 48 days of the year. Ask students to write and explain the meaning of a few other statements using function notation.

In this activity, students represent a situation using a table of values, a graph, and an equation. From the exponential equation, it is a short step to thinking of the relationship between the quantities as a function.

Note that it is possible and acceptable to think of time as a function of area, but expressing this using an equation is out of the scope of this course. Students could, however, represent such a function with a graph, table, or description.

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

Students may have trouble understanding how to account for time in the first question. They may benefit from writing the area after 1 day has passed, 2 days have passed, etc. A table is a convenient way to gather this information.

Discuss why the area covered by mold is a function of the number of days that have passed. Attend explicitly to language students learned in the prior unit on functions: The area of the mold \(A\) is a function of the number of days \(d\) since the mold was spotted, \(A = f(d)\). The function \(f\) expressing the mold relationship can be written as \(f(d) = 1 \boldcdot 2^d\), where \(d\) measures days since the mold was spotted and \(f(d)\) gives the area covered by the mold in square millimeters.

Discuss whether a discrete graph or a curve is more appropriate and what domain would be suitable in this context. Ask questions such as:

• “Can the independent variable be something like 1.5, a number that is not a whole number? Is there an area that is associated with 1.5 days?” (Yes, some area of the bread is covered by mold at any point in time. The mold doesn't disappear after being spotted and then reappear at exactly 1 full day, 2 full days, etc.)
• “What would be the meaning of a point on the graph where the value of \(d\) is, for instance, between 2 and 3?” (It would mean the area covered by mold at some point longer than 2 days but less than 3 days after mold was spotted.)
• “What domain would be appropriate for this function? Can the mold grow indefinitely?” (Since the area of the bread (the range) is limited, the exponential growth cannot continue indefinitely. By the end of one week, more than 1 square cm will be covered and, by the end of the second week, the values of the function will be close to or will exceed the total area of the bread.)

Students using paper and pencil may decide that it makes sense to connect the points on the graph but they will not yet know how to do so. Consider stating that they are connected (in a very specific way) and their properties will be studied later.

Students using the digital version (or graphing technology along with the paper and pencil version) will see the continuous graph. If desired, you may want to demonstrate how using function notation to write the equation like \(f(d) = 1 \boldcdot 2^d\) can be put to use. Try typing \(A(2)\) or \(A(\text-1)\).

Students have described and analyzed situations involving exponential change, using graphs, tables, and equations. Now they revisit several of these contexts, viewing them as functions and expressing them using function language and notation.

Each situation involves two quantities and students will need to choose one of these to be the independent variable and one to be the dependent variable. For all of these relationships, it is possible to choose either variable as the independent but one choice gives a logarithmic function (which is out of the scope of this course), while the other gives an exponential function. In each case, however, students have previously worked with the context.

Look for students who explicitly state the meaning of their variables, including units, and invite them to share during the discussion. For example, in the second situation, if \(t\) represents the number of years and \(v\) represents the value of the car, in dollars, after \(t\) years then \(v\) can be viewed as a function \(f\) of \(t\) where \(f(t) = 18,\!000 \boldcdot \left(\frac{2}{3}\right)^t\). The meaning and units for both \(t\) and \(f(t)\) (or \(v\)) are vital elements to answering the question fully.

Tell students that they will now revisit some previously seen situations. Ask students to read the three situations in the task, then solicit a few ideas on why all of them can be seen as functions.

Students may confuse the terms "independent" and "dependent." Help them to think about which variable depends on the other in context.

Invite selected students to present their equations, making sure to indicate what each variable represents, as well as the units of the variable.

For the third question, point out that it is a short step from an equation for the area covered by the algae \(A =240 \boldcdot \left(\frac{1}{3}\right)^t\) to a similar equation with function notation. The equation gives the area covered by the algae at each time \(t\), so a function \(f\) can be defined using the same expression: \(f(t) =240 \boldcdot \left(\frac{1}{3}\right)^t\) and \(A = f(t)\).

This optional activity further addresses the skill of choosing an appropriate graphing window when using graphing technology. In an earlier activity, students looked at how adjusting the graphing window affects the usefulness of the graph. Here they gauge the reasonableness of a graphing window given an equation and a description of a function.

Because exponential functions eventually grow very quickly, the graph tends to quickly go off the screen if the domain is too large. The graphing window can be adjusted to display large values for the vertical axis, but in doing so, the output values for most of the domain will all look like they are essentially 0.

To decide on the reasonableness of the given graphing boundaries, students may evaluate the function at the endpoints of the domain for the graphing window. Look for students who think carefully about the domain, observing that, based on the context, the equation probably does not model even modest negative values of the input variable. This activity represents scaffolded practice for an important aspect of mathematical modeling (MP4).

Provide access to graphing technology.

Select students to share their graphs, the one created using the given horizontal and vertical boundaries, as well as the improved versions. Or display the graphs in the sample responses for all to see. Discuss questions such as:

• “Why were the given boundaries not good?” (The domain and range are both too wide. The range is so large that almost all the points look like they're about 0. It is even hard to tell what the starting value is.)
• “How can the context help us choose an appropriate graphing window?” (Negative values of time probably are not relevant because the amount of medicine in the body before the injection is likely insignificant or is not modeled by the equation. Large values of time (say \(t=100\)) are not likely important because the effects of the medicine after 100 hours are likely to be minimal.)

Graphing exponential functions can be challenging because they can increase or decrease very quickly. Emphasize that an appropriate graphing window can often be selected using the context. Once the relevant domain of a function and the horizontal boundaries of a graph are chosen, the vertical boundaries can be selected based on the output values for that domain so that any meaningful trends (for example, exponential decay) are visible.