## Task

An epidemic of influenza spreads through a city.
The figure below
is the graph of $I=f(w)$, where $I$ is the number of individuals
(in thousands) infected $w$ weeks after the epidemic begins.

- Estimate $f(2)$ and explain its meaning in terms of the epidemic.
- Approximately how many people were infected at the height of the
epidemic? When did that occur? Write your answer in the form
$f(a)=b$.
- For approximately which $
w$ is $f(w)=4.5$; explain what the estimates mean in terms of
the epidemic.
- An equation for the function used to plot the image above is $f(w)=6w(1.3)^{-w}$. Use
the graph to estimate the solution of the inequality $6w(1.3)^{-w}\geq
6$. Explain what the solution means in terms of the epidemic.
(Task from Functions Modeling Change: A Preparation for Calculus, Connally et al., Wiley 2010.)

## IM Commentary

The principal purpose of this task is to probe students' ability to correlate symbolic statements about a function using function notation with a graph of the function, and to interpret their answers in terms of the quantities between which the function describes a relationship. It can be used in assessment, or in instruction to bring out some common frailties of student understanding, such as not really understanding what it means for a point to lie on the graph of a function, and, in part (d), not being comfortable with interchanging a function value expressed in function notation and an expression for the function. As it is impossible to read exact numerical data from the graph, students will have to approximate coordinates of data points on the graph, providing a good opportunity for an instructor to address MP6 (attend to precision).