In a previous lesson, students used equations defining functions to reason about graphs. Here they work in the opposite direction, analyzing graphs and creating corresponding equations. In the first question, the coordinates given are not of two consecutive values of , so students need to reason about the decay factor indirectly. They may:

• Try different factors that, when multiplied by  twice, give .
• Use the graph to estimate , and then use that estimate to find a reasonable decay factor.
• Find the factor that takes to (or to ), recognize that number as , and then find .

The second question has no context and is thus more abstract. This prepares students to interpret graphs in the next activity, where minimal information is given.

If students have trouble getting started, ask them to consider the graph and say what they know is true about the situation that the graph represents. Most likely, they will answer some parts of the first question by doing this, and will be in a better position to figure out any questions they didn't answer.

Select students to share their responses to the first question. Focus the discussion on how students found the decay factor for the computer. Make sure students understand that the key is to notice that every 2 years, the computer's value is multiplied by . This means that the annual decay factor is , because  .

For the last graph, discuss questions such as:

• “What information is needed to describe an exponential function?” (the factor of growth or decay and the initial value)
• “Where on the graph do we see the initial value?” (the -intercept of the graph)
• “What points on the graph can we use to help us find the growth factor?” (the points and , or and )
• “Why might and be a more strategic choice than and for finding the factor of growth?” (The coordinates of first two points differ by 1, so the quotient of the coordinates of those points can be used to find the factor of decay. It is a more direct way to find the factor of decay, which is the factor by which the output of changes when the input, , changes by 1. In the second pair, the coordinate changes by 2.)

This activity presents students with two graphs without numerical values assigned to any points on the graphs. The only labeled point is where the two graphs intersect. Students will need to use their understanding that tripling each day is a greater growth factor than doubling each day. This is their only means to decide which function represents which situation. Students then interpret the graphs in terms of the situations that they represent.

Some students may think that the dashed graph represents the mold that is tripling because it has a greater vertical intercept. Ask these students to think carefully about the second question.

Select students to share their responses and explanations. Focus on how they deciphered the meanings of the graphs. Discuss questions such as:

• The solid graph started off lower than the dashed graph. What does that mean in this context? (The mold population represented by the solid graph is initially smaller or covers a smaller area on the wall than the one represented by the dashed graph.)
• After some point, the solid graph passes the dashed one. What does that mean in this context? (The mold population that started out covering a greater area on the wall is surpassed by the other population, so it is growing at a slower rate.)
• What does the intersection of the two graphs mean? What do and represent? (The two mold populations are the same at time with population covering an area of square inches.)