Exponential Situations as Functions

5 min

Narrative

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.

Launch

Ask students to recall the meaning of the term "function." (A relationship between two variables for which each input corresponds to only 1 output value.)

Student Task

Here is a graph of the accumulated rainfall in Las Vegas, Nevada, in the first 60 days of 2017.

<p>Graph, accumulated rainfall.</p>
Line graph titled, Accumulated Rainfall, Las Vegas, Nevada. Horizontal axis, day of the year 2017, from 0 to 60, by 10's. (There are 4 equally spaced hashmarks/tickmarks between each labeled number.) Vertical axis, rainfall in inches, from 0 to 1 point 6 by 0 point 2's. The graph begins at 0 comma 0 and moves horizontally and to the right until it reaches 11 comma 0. It then slants upward and to the right until it reaches 13 comma 0 point 1. The graph then moves horizontally and to the right until reaching 19 comma 0 point 1. It then slants upwards and to the right until it reaches 20 comma 0 point 4. The graph moves horizontally and to the right until reaching 21 comma 0 point 4. It then slants upwards and to the right until it reaches 22 comma 0 point 9. It slants upwards and to the right again until it reaches 23 comma 0 point 95. The graph then moves horizontally and to the right until reaching 41 comma 0 point 95. It slants upwards and to the right until 43 comma 1. It moves to horizontally and to the right until 47 comma 1. It slants upwards and to the right until 48 comma 1 point 02, and slants upwards and to the right again until 49 comma 1 point 45. The graph moves horizontally and to the right until 60 comma 1 point 45.  

Use the graph to support your answers to the questions.

  1. Is the accumulated amount of rainfall a function of time?
  2. Is time a function of accumulated rainfall?

Sample Response

  1. Yes. Sample explanation: For each number of days dd between 0 and 60, a certain amount of rain fell up to that time, so the there is one particular value for the accumulated rain at any point in time.
  2. No. Sample explanation: There was no rainfall in the first 10 days. This means that for the amount of rainfall 0 inches or for the input value of 0, there are lots of times dd that could be the output.
Activity Synthesis (Teacher Notes)

Make sure that 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 rainfall as a function of time. If rr represents the amount of rainfall in inches and tt is time in days, then r(t)r(t) is the amount of rain that has fallen in the first tt days of 2017. For example, r(2)=0r(2) = 0 tells us that the accumulated rainfall after in the first two days of the year was 0 inches. r(48)=1r(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.

Anticipated Misconceptions

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 tt, the time in days, there is one value of rr, the accumulated rainfall in inches, but this is not true the other way around.

tt (days) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
rr (inches) 0 0 0 0 0 0 0 0 0 0 0 0.03 0.1 0.1 0.1 0.1 0.1 0.1 0.11 0.38
Standards
Building On
  • HSF-IF.A·Understand the concept of a function and use function notation.
Addressing
  • F-IF.B·Interpret functions that arise in applications in terms of the context
  • HSF-IF.B·Interpret functions that arise in applications in terms of the context.

15 min

15 min