19.1: Graph of Which Function? (5 minutes)
At the beginning of this unit, students compared linear and exponential growth. They return to this comparison in this lesson. This warm-up aims to show that, visually, it could be very difficult to distinguish linear and exponential growth for some domain of the function. While any exponential function eventually grows very quickly, it also can look remarkably linear over a portion of the domain.
Here is a graph.
- Which equation do you think the graph represents? Use the graph to support your reasoning.
- \(y=120 + (3.7)\boldcdot x\)
- \(y = 120 \boldcdot (1.03)^x\)
- What information might help you decide more easily whether the graph represents a linear or an exponential function?
Invite students to share the rationales for their decision and their ideas for improving the clarity of the graph.
Help students understand that the graph of a linear function always looks like a line regardless of the domain where it is plotted. An exponential function, however, can look linear depending on the domain and range. Graphs are very helpful for seeing the general behavior of a function but not always for determining what kind of function is being graphed.
Consider showing students this image of the graph showing a larger domain and range as well as the original window (the red rectangle in the lower left).
Or show a dynamic graph of \(y=120\boldcdot(1.03)^x\) starting with a small window where the graph looks linear and then zooming out until the curve is visible
19.2: Simple and Compound Interests (15 minutes)
In this activity, students compare linear and exponential growth in a context involving simple and compound interest. The initial balances are chosen so that the two options stay pretty close for small values of time. The intersection of their graphs is far enough from 0 that it is not easily noticed with only a few calculations. The graph for simple interest is linear. The graph for compound interest is exponential, but it is relatively flat for small values of time. As the domain values increase, students may notice that the values of the two options get closer and closer, or they may notice from the graph that the gap between the two graphs gets smaller.
Look for students who make different choices for the investment option, as the better option would depend on the length of investment, which is unspecified. As in the case of the genies' offers in the opening lesson, simple interest is better if the investment term is short, but compound interest becomes increasingly more favorable as time passes. Some students may not recognize this until they graph the two functions in the last question. Identify students who decide to change their earlier choice and can defend the change. Ask them to share later.
Arrange students in groups of 2. Encourage students to think quietly about the questions before conferring with their partner. Provide access to graphing technology.
Design Principle(s): Support sense-making
Supports accessibility for: Conceptual processing; Memory
A family has $1,000 to invest and is considering two options: investing in government bonds that offer 2% simple interest, or investing in a savings account at a bank, which charges a $20 fee to open an account and pays 2% compound interest. Both options pay interest annually.
Here are two tables showing what they would earn in the first couple of years if they do not invest additional amounts or withdraw any money.
|years of investment||amount in dollars|
|years of investment||amount in dollars|
- Bonds: How does the investment grow with simple interest?
- Savings account: How are the amounts $999.60 and $1,019.59 calculated?
- For each option, write an equation to represent the relationship between the amount of money and the number of years of investment.
- Which investment option should the family choose? Use your equations or calculations to support your answer.
- Use graphing technology to graph the two investment options and show how the money grows in each.
Students may have trouble choosing an option without additional information. Ask them to think about what might make sense for their own family or to give circumstances that would justify a choice.
Consider surveying the class to see the choices students made. Select students who made different choices to explain their responses, starting with those who opted for the bonds. Display their reasoning and graphs for all to see. If no students consider a time frame beyond 8 or 9 years, ask students which option would be better if the money could be left alone for 20 years. Discuss questions such as:
- How does the fee for the savings account affect the function? (It reduces the account balance, at the beginning, by $20. In the equation, there is a subtraction of 20.)
- Where can we see the fees on the graph? (The vertical intercept of the graph for the savings account is at 980, which is 20 less than 1,000.)
- When might the bonds be the better investment, if ever? (When the investment is shorter than a decade, as that is when the savings account surpasses it.)
- When might the savings account be the better investment, if ever? (When it is left for more than 10 years, after the compounding effects of interest make up for the initial fees.)
Point out that the exponential function here (the balance of the savings account) has a relatively slow rate of growth. It takes it a relatively long time before it overtakes the linear function, but it eventually does.
19.3: Reaching 2,000 (15 minutes)
This activity prompts students to again compare a linear function with an exponential one, but this time without a context, and the exponential function grows much more slowly over a long period of time. Even if students predict that the exponential function will grow more quickly because it is exponential, they still need to decide which one reaches 2,000 first. To do that, students may try:
- continuing the table of values with increasingly larger values
- using graphing technology
- finding when \(f(x) = 2,\!000\) and substituting this value (\(x=1,\!000\)) into \(g(x)\)
Making graphing and spreadsheet technology available gives students an opportunity to choose appropriate tools strategically (MP5).
Present the two equations that define two functions \(f\) and \(g\): \(f(x) = 2x\) and \(g(x) = (1.01)^x\). Give students a moment to share what they notice and wonder about the functions. Ask questions such as:
- What is happening in the first function? What about in the second function?
- Which function do you think will reach a value of 2,000 first?
Design Principle(s): Cultivate conversation; Maximize meta-awareness
Supports accessibility for: Organization; Conceptual processing; Attention
- Complete the table of values for the functions \(f\) and \(g\).
\(x\) \(f(x)\) \(g(x)\) 1 10 50 100 500
- Based on the table of values, which function do you think grows faster? Explain your reasoning.
- Which function do you think will reach a value of 2,000 first? Show your reasoning. If you get stuck, consider increasing \(x\) by 100 a few times and record the function values in the table.
Are you ready for more?
Consider the functions \(g(x)=x^5\) and \(f(x)=5^x\). While it is true that \(f(7)>g(7)\), for example, it is hard to check this using mental math. Find a value of \(x\) for which properties of exponents allow you to conclude that \(f(x)>g(x)\) without a calculator.
Select students who used contrasting approaches to share. The exponential function \(g\) grows sufficiently slowly that it looks like it will not catch up with or ever overtake the linear function \(f\). This is true for a table of values as well as using a graph. Invite students who used contrasting methods to share, in the order shown in the Activity Narrative.
Emphasize that the table needs to be continued for a long time to identify when the values of \(g\) become greater than those of \(f\). Similarly, the window of the graph needs to be chosen carefully. A very efficient method is to identify that \(f(1,\!000)=2,\!000\) and then evaluate \(g(1,\!000)\). Since \(g(1,\!000)\) is much greater than 2,000, \(g\) reaches 2,000 first.
If not already illustrated by students who used graphing, show this dynamic sketch and zoom out:
Though for quite a while it doesn't seem like the values of \(g\) would ever catch up with those of \(f\), if the growth is allowed to take place long enough, exponential eventually surpasses linear.
In this course, we have looked at many examples of exponential functions and linear functions. We have studied their graphs, equations, and tables of values. Help students review the features and behaviors of the two kinds of functions by presenting these pairs of functions (all of which represent growth):
- two linear functions \(f\) and \(g\), for example: \(f(x)=\frac12 x+100\) and \(g(x) = 2x\)
- two exponential functions \(p\) and \(q\), for example: \(p(x)= 100 \boldcdot (1.5)^x\) and \(q(x) = 5 \boldcdot 2^x\)
- a linear function \(m\) and an exponential function \(n\), for example: \(m(x) = 300x+ 1\!,000\) and \(n(x) = (0.4)\boldcdot 2^x\)
For each pair for functions, ask students these questions:
- How can we tell which one grows more quickly?
- Will the one that started out with the lower initial value catch up with the function that starts with a higher initial value?
- What are some ways to check if one function will overtake another?
Reiterate that even though an exponential function might seem to be growing too slowly to catch up to a linear function with large numbers for its \(y\)-intercept and slope, at some value of \(x\) down the line, the exponential function will catch up with the linear. Remind students how this was the case in the genie activity early in the unit. (The starting value for the linear function was 100,000 times that of the exponential function, and the slope of the linear function was 2,000 while the growth factor of the exponential function was 2. But within 3 weeks, the exponential function surpassed the linear.)
19.4: Cool-down - Which One Gets There First? (5 minutes)
Student Lesson Summary
Suppose that you won the top prize from a game show and are given two options. The first option is a cash gift of \$10,000 and \$1,000 per day for the next 7 days. The second option is a cash gift of 1 cent (or \$0.01) that grows tenfold each day for 7 days. Which option would you choose?
In the first option, the amount of money increases by the same amount (\$1,000) each day, so we can represent it with a linear function. In the second option, the money grows by multiples of 10, so we can represent it with an exponential function. Let \(f\) represent the amount of money \(x\) days after winning with the first option and let \(g\) represent the amount of money \(x\) days after winning with the second option.
Option 1: \(f(x) = 10,\!000 + 1,\!000x\)
. . .
Option 2: \(g(x) = (0.01) \boldcdot 10^x\)
. . .
For the first few days, the second option trails far behind the first. Because of the repeated multiplication by 10, however, after 7 days it surges past the amount in the first option.
What if the factor of growth is much smaller than 10? Suppose we have a third option, represented by a function \(h\). The starting amount is still \$0.01 and it grows by a factor of 1.5 times each day. If we graph the function \(h(x)=(0.01) \boldcdot (1.5)^x\), we see that it takes many, many more days before we see a rapid growth. But given time to continue growing, the amount in this exponential option will eventually also outpace that in the linear option.