Lesson 9

Lin and Diego are discussing two expressions: \(x^2\) and \(2^x\).

• Lin says, “I think the two expressions are equivalent.”
• Diego says, “I think the two expressions are only equal for some values of \(x\).”

Do you agree with either of them? Explain or show your reasoning.

1. What is the statement \(f(9) \approx 6\) saying about this situation?
2. What is \(f(4)\)? What about \(f(3.5)\)? What do these values represent in this context?
3. When \(f(t)=45\), what is \(t\)? What does that value of \(t\) represent in this context?
4. By what factor did the cost of solar cells change each year? (If you get stuck, consider creating a table.)

1. The thickness \(t\) in millimeters of a folded sheet of paper after it is folded \(n\) times is given by the equation \(t = (0.05) \boldcdot 2^n\).
   1. What does the number \(0.05\) represent in the equation?
   2. Use graphing technology to graph the equation \(t = (0.05) \boldcdot 2^n\).
   3. How many folds does it take before the folded sheet of paper is more than 1 mm thick? How many folds before it is more than 1 cm thick? Explain how you know.
2. The area of a sheet of paper is 93.5 square inches.
   1. Find the remaining, visible area of the sheet of paper after it is folded in half once, twice, and three times.
   2. Write an equation expressing the visible area \(a\) of the sheet of paper in terms of the number of times it has been folded \(n\).
   3. Use graphing technology to graph the equation.
   4. In this context, can \(n\) take negative values? Explain your reasoning.
   5. Can \(a\) take negative values? Explain your reasoning.

Your teacher will give you either a problem card or a data card. Do not show or read your card to your partner.

Pause here so your teacher can review your work. Ask your teacher for a new set of cards and repeat the activity, trading roles with your partner.

Earlier, we used equations to represent situations characterized by exponential change. For example, to describe the amount of caffeine \(c\) in a person’s body \(t\) hours after an initial measurement of 100 mg, we used the equation \(c = 100\boldcdot \left(\frac{9}{10}\right)^t\).

Notice that the amount of caffeine is a function of time, so another way to express this relationship is \(c = f(t)\) where \(f\) is the function given by  \(f(t)=100 \boldcdot \left( \frac{9}{10} \right)^t\)

We can use this function to analyze the amount of caffeine. For example, when \(t\) is 3, the amount of caffeine in the body is \(100 \boldcdot\left(\frac{9}{10}\right)^3\)or \(100 \boldcdot \frac{729}{1,000}\), which is 72.9. The statement \(f(3) = 72.9\) means that 72.9 mg of caffeine are present 3 hours after the initial measurement.

We can also graph the function \(f\) to better understand what is happening. The point labeled \(P\), for example, has coordinates approximately \((10,35)\) so it takes about 10 hours after the initial measurement for the caffeine level to decrease to 35 mg.

A graph can also help us think about the values in the domain and range of a function. Because the body breaks down caffeine continuously over time, the domain of the function—the time in hours—can include non-whole numbers (for example, we can find the caffeine level when \(t\) is 3.5). In this situation, negative values for the domain would represent the time before the initial measurement. For example \(f(\text-1)\) would represent the amount of caffeine in the person's body 1 hour before the initial measurement. The range of this function would not include negative values, as a negative amount of caffeine does not make sense in this situation.