Graph Equations to Solve Problems

5.9.3 Graph Equations to Solve Problems

5.9.3 • Graph Equations to Solve Problems

Activity

1. The thickness $t$ in millimeters of a folded sheet of paper after it is folded $n$ times is given by the equation $t = (0.05) \cdot 2^{n}$ .

a. What does the number 0.05 represent in the equation?

Compare your answer:

The sheet of paper is 0.05 mm thick.

b. What does the number 2 represent in the equation?

Compare your answer:

The number 2 indicates that the thickness doubles with each increment of $n$ , the number of folds.

c. Use graphing technology to graph the equation $t = (0.05) \cdot 2^{n}$ .

Compare your answer:

Here is a table of values for the thickness of the paper after different numbers of folds:

$n$	$t$
0	0.05
1	0.1
2	0.2
3	0.4
4	0.8
5	1.6

Scatter plot showing the number of folds on the x-axis (0 to 8) and thickness in millimeters on the y-axis (0 to 4). The points rise sharply, indicating increasing thickness with more folds.

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d. 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.

Compare your answer:

It takes 5 folds before the paper is more than 1 mm thick. After 4 folds, the paper is 0.8 mm thick, and after 5 folds, it is 1.6 mm thick. It takes 8 folds before the paper is more than 1 cm or 10 mm thick (1.28 cm to be precise).

2. The area of a sheet of paper is 93.5 square inches.

a. Find the remaining visible area of the sheet of paper after it is folded in half once, twice, and three times.

Compare your answer:

46.75, 23.375, 11.6875 square inches

b. Write an equation expressing the visible area $a$ of the sheet of paper in terms of the number of times it has been folded, $n$ .

Compare your answer:

$a = 93.5 \cdot (\frac{1}{2})^{n}$

c. Use graphing technology to graph the equation.

Compare your answer:

A scatter plot showing area in square inches decreasing as the number of folds increases from 0 to 10. The data points illustrate an inverse relationship between number of folds and area.

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d. In this context, can $n$ take negative values? Be prepared to show your reasoning.

Compare your answer:

The variable $n$ cannot take negative values (unless the original sheet of paper can be unfolded).

e. Can $a$ take negative values? Be prepared to show your reasoning.

Compare your answer:

No. The area of the sheet of paper can never be negative, no matter how many times it is folded.

f. Identify the domain of the graph using an inequality.

Compare your answer:

$n \geq 0$

g. Identify the range of the graph using an inequality.

Compare your answer:

$0 < a \leq 93.5$ . Note that the area cannot be negative, or even 0, because the paper will not disappear.

h. What equation represents the horizontal asymptote?

Compare your answer:

$y = 0$

3. Use the equation to make a prediction of the thickness with 7 folds.

Compare your answer:

$t = (0.05) \cdot 2^{7} = (0.05) (128) = 6.4$

4. The $y$ -intercept represents the initial thickness of the paper $t$ before folding (when $n$ is zero). Calculate the value of the $y$ -intercept.

Compare your answer:

$t = (0.05) \cdot 2^{0} = (0.05) (1) = 0.05$

5. The asymptote represents the possible limit of thickness of the paper as it is folded. What does this mean?

Compare your answer:

In this equation, the value of thickness of the paper $t$ becomes larger as the number of folds $n$ increases (approaches infinity). This means that, as the paper is folded an increasing number of times, the thickness has the potential to be infinite, but in reality, it can never quite reach it.

Self Check

The graph below represents the function that gives the value of a computer in dollars, $y$ , as a function of time, $x$ , measured in years since the time of purchase.

What is the range for this graph and situation?

Graph of a decreasing exponential function, x y plane, origin O.

$y \geq 0$
$0<y 400$<br="" \leq=""></y>
$x \geq 0$
$y \leq 400$

Additional Resources

Domain and Range of Exponential Graphs

The graph below describes the amount of caffeine, $c$ , in a person’s body $t$ hours after an initial measurement of 100 mg. The equation of this curve is $y = 100 (1 - 0.1)^{t}$ .

A graph showing caffeine (mg) decreasing over time (hours); caffeine starts at 100 mg and gradually declines as time increases from 0 to 14 hours. A point P is marked on the curve.

What is the smallest value $c$ can reach?

Since $c$ is the amount of caffeine, it can get close to 0.

Can $t$ ever take on negative values?

No, time will not be negative in this relationship because time is moving forward.

What are the lowest and highest values for $t$ ?

At the beginning, $t = 0$ , and $t$ can go as long as it takes for $c$ to get so close to 0 it is insignificant.

The domain (or values of $t$ ) written as an inequality is $t \geq 0$ .

The range (or values of $c$ ) written as an inequality is $0 < c \leq 100$ .

Try it

Try It: Domain and Range of Exponential Graphs

The dollar value of a car is a function, $f$ , of the number of years, $t$ , since the car was purchased. The car was purchased at $12,000. Tell the domain and range. The equation of this curve is $y = 12, 000 (1 - 0.25)^{t}$ .

Line graph showing a downward curve; x-axis labeled Years Since Purchase (0 to 6), y-axis labeled Value in Dollars (0 to 14,000). Value starts near $12,000 and declines rapidly over time.

Compare your answer:

Here is how to determine the domain and range of an exponential function:

To determine the domain, look at the $t$ values. The car is bought when $t = 0$ , so the domain is $t \geq 0$ .

To determine the range, look at the $f$ values, or the possible dollar values of the car. The car is bought at $12,000 and will lose value until the value is close to $0, but it will never be worth negative dollars.

The range is $0 < f \leq 12000$ .