Some students may choose Option B because of the words of what is left in the prompt. Ask them what fraction of the quantity is left if of it is spent. If students are still not convinced, ask them how much of the 180 is spent in week 1? How much is left? Use your equation to verify that 120 is left. 

The goal of this lesson is to examine how the numbers  and  influence the graph representing a function, , defined by an equation of the form .

A money context is used in the first question to facilitate access. A bank account with more money is generally desirable, so students will be looking for the function that grows most quickly. The growth of the accounts is not realistic (unless more money is being added on a regular basis), and has been chosen to motivate a close analysis of the growth factors.

As students work and discuss in groups, notice those who can articulate how and in an exponential function affect a graph that represents it. Ask them to share later.

This activity works best when each student has access to devices that can run the Desmos applet because students will benefit from seeing the relationship in a dynamic way.

Once students have used graphing technology to graph four functions on the same set of axes, they may need help choosing a graphing window so that they can see salient features of all four graphs. They may also need help understanding how to determine which graph represents which function. (Some tools make it easier to distinguish than others.)

Invite previously selected students or groups to share their responses. Discuss:

• “On the graph, where do you see the of each function? How does the size of affect the graph?” (The is the -intercept. The greater the , the higher it is on the vertical axis.)
• “On the graph, where do you see the of each function? How does the size of affect the graph?” (The appears in the steepness of the curve. When the function is characterized by exponential growth, the larger the , the steeper the curve, or the more quickly the quantity is growing.)
• “How is the graph of different from other graphs?” (It represents a situation characterized by exponential decay: decreases as increases.)
• “How is the value for function different from those of the other functions?” (The value of is less than 1.)

This activity complements the previous one. For a situation characterized by exponential growth, a larger base value, , means faster growth. For a situation characterized by exponential decay, however, a smaller value of (between 0 and 1) corresponds to faster decay. Understanding how the parameter influences the graph of an exponential function will prepare students to make sense of and model situations involving repeated percentage change later in the unit.

Focus the discussion on the connection between the numbers in the equation (especially the ) and the features of the graph. Discuss questions such as:

• “In the first question, what does the point that the graphs share on the -axis say about the situation?” (All of the accounts started with $200.)
• “How does Graph A compare to Graph D?” (Graph A appears more horizontal, so the function is decaying at a slower rate.)
• “What does the largest factor () tell us? To which graph does it correspond?” (The function loses only of its value when increases by 1, so it is decaying the most slowly. Graph A must be the graph of   because it shows the slowest decay.)
• “What might a graph representing a function , given by  , look like?” (It would look very close to a horizontal line because nearly all of its value remains as increases by 1, so it is decaying very slowly.)
• “What might a graph representing a function , given by  , look like?” (It will go toward 0 extremely quickly because it loses of its value each time increases by 1, so it is decaying very quickly.)

Emphasize that, in situations characterized by exponential growth (when ), a larger value of means a curve that is more vertical. In situations characterized by exponential decay, where is between 0 and 1, the closer is to 1, the more the graph approaches a horizontal line. Conversely, the smaller the value of , the more swiftly it heads toward 0 (the more vertical the curve is) before it flattens out and approaches a horizontal line.