Lesson 12

The goal of this warm-up is to practice modeling a situation characterized by exponential decay with an equation.

One strategy students may use to choose the right equation is by evaluating each one for some value of \(t\). For instance, they may choose 1 for the value of \(t\), evaluate the expression, and see if the result makes sense for the amount of money after 1 week. Though this is a valid approach, encourage students to think about what is happening in the situation and see if it is correctly reflected in the way each expression is written.

For instance, in Option A, the term \(\frac13 t\) in the expression would mean “a third of the time in weeks” (rather than a third of the gift money), suggesting that this expression doesn’t accurately reflect the situation. Similarly, in Option C, \( \frac13 \boldcdot 180^t\) would mean \(\frac13\) being multiplied by 180 repeatedly (\(t\) times), which also does not represent the situation.

Some students may choose Option B because of the words \(\frac{1}{3}\) of what is left in the prompt. Ask them what fraction of the quantity is left if \(\frac13\) 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. 

Ask students for the correct response, and prompt them to explain why the other ones are not correct.

Make sure students understand that if a third of the balance is spent every week, two-thirds of the balance is what remains every week, so the value of \(b\) is \(\frac{2}{3}\), and the amount available after \(t\) weeks would be \(180 \boldcdot (\frac23)^t\).

The goal of this lesson is to examine how the numbers \(a\) and \(b\) influence the graph representing a function \(f\) defined by an equation of the form \(f(x) = a \boldcdot b^x\).

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 which 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 \(a\) and \(b\) 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.

Arrange students in groups of 2 and provide access to graphing technology.

Tell students to read the first question and ask them: "If the account were yours, which function would you choose: \(f\), \(g\)\(, \) \(h\), or \(j\) ?" Give students a moment to discuss their choice and rationale with their partner.  Then, tell students to read the second question and discuss with their partner which functions—\(p\), \(q\), or \(r\)—they would choose.

Ask one partner to graph the functions in the first question and the other partner to graph the functions in the second question and then analyze their graphs together.

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

This activity complements the previous one. For a situation characterized by exponential growth, a larger base value \(b\) means faster growth. For a situation characterized by exponential decay, however, a smaller value of \(b\) (between 0 and 1) corresponds to faster decay. Understanding how the parameter \(b\) 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.

Present the following scenario: “Suppose you are presented with four functions \(m\), \(n\), \(p\), and \(q\) that describe the amount of money, in dollars, in a bank account as a function of time \(x\), in years. If the account is yours (and more money is better), which function would you choose? Why?” Here are equations defining the functions.

\(\displaystyle m(x) = 200 \boldcdot \left(\frac14 \right)^x\)

\(\displaystyle n(x) = 200 \boldcdot \left(\frac12 \right)^x\)

\(\displaystyle p(x) = 200 \boldcdot \left(\frac34 \right)^x\)

\(\displaystyle q(x) = 200 \boldcdot \left(\frac78 \right)^x\)

Give students a minute of quiet think time and ask students to share their responses. Ask a student who chose the first option and one who chose the last option to share their reasoning. Tell students they will now consider the options graphically before confirming their choice.

Arrange students in groups of 2. Provide access to graphing technology.

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

• “In the first question, what does the point that the graphs share on the \(y\)-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 (\(\frac78\)) tell us? To which graph does it correspond?” (The function loses only \(\frac18\) of its value when \(x\) increases by 1, so it is decaying the most slowly. Graph A must be the graph of \(q(x) = 200 \boldcdot \left(\frac78 \right)^x\) since it shows the slowest decay.)
• “What might a graph representing a function \(v\) given by \(v(x) = 200 \boldcdot \left(\frac{99}{100}\right)^x\) look like?” (It would look very close to a horizontal line because nearly all of its value remains as \(x\) increases by 1, so it is decaying very slowly.)
• “What might a graph representing a function \(w\) given by \(w(x) = 200 \boldcdot \left(\frac{1}{100}\right)^x\) look like?” (It will go toward 0 extremely quickly because it loses \(\frac {99}{100}\) of its value each time \(x\) increases by 1, so it is decaying very quickly.)

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