Lynn Marecek; MaryAnne Anthony-Smith; Andrea Honeycutt Mathis; Jay Abramson; Sharon North; Amy Baldwin; Alyssa Howell

Activity

To control an algae bloom in a lake, scientists introduce some treatment products.

Once the treatment begins, the area covered by algae in square yards, $A$ , is given by the equation $A = 240 \cdot (\frac{1}{3})^{t}$ . Time, $t$ , is measured in weeks.

1. In the equation, what does 240 tell us about the algae?

Compare your answer:

The algae initially covers 240 square yards.

2. What does the $\frac{1}{3}$ tell us?

Compare your answer:

Every week, the area covered is $\frac{1}{3}$ of the area covered the previous week.

3. Create a graph to represent $A = 240 \cdot (\frac{1}{3})^{t}$ when $t$ is 0, 1, 2, 3, and 4. Think carefully about how you choose the scale for the axes. If you get stuck, consider creating a table of values. Use the graphing tool or technology outside the course. Graph the equation that represents this scenario using the Desmos tool below.

Compare your answer:

A scatter plot showing a decrease in area (square yards) over time (weeks), with points at (0, 250), (1, 80), (2, 25), (3, 10), and (4, 5). Both axes are labeled and gridlines are visible.

$t$ (Weeks)	$A$ (Square Yards)
0	240
1	80
2	27 (rounded)
3	9
4	3

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4. Approximately how many square yards will the algae cover after 2.5 weeks? Be prepared to show your reasoning.

Your answers may vary, but here is an example:

The algae will cover fewer than 27 square yards and more than 9 square yards because the area covered by the algae is decreasing. A number between 15 and 20 would be a good guess.

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Extending Your Thinking

The scientists estimate that to keep the algae bloom from spreading after the treatment concludes, they will need to get the area covered under one square foot. How many weeks should they run the treatment to achieve this?

Compare your answer:

7 weeks. There are 9 square feet in every square yard, so 1 square foot is $\frac{1}{9}$ square yard (or about 0.111 square yard). At 7 weeks is the first time the area covered by algae is less than $\frac{1}{9}$ because $240 \cdot (\frac{1}{3})^{6} \approx 0.329$ , and $240 \cdot (\frac{1}{3})^{7} \approx 0.110$ .

Self Check

Which of the following is the graph of

y = 12(\frac{1}{2})^x

?

Additional Resources

Connecting Tables and Graphs in Exponential Decay Functions

A population $p$ of migrating butterflies satisfies the equation $p = 100, 000 \cdot (\frac{4}{5})^{w}$ , where $w$ is the number of weeks since they began their migration.

Answer the following questions that would complete the table and create a graph to show the population $p$ of migrating butterflies at the number of weeks since they began their migration.

$w$	0	1	2	3	4
$p$

The number of butterflies when they began their migration.
The number of butterflies 1 week after they began their migration.
The number of butterflies 3 weeks after they began their migration.
Create a graph that best represents the butterfly population.
What is the vertical intercept, or y-intercept, of the graph? What does it tell you about the butterfly population?
When does the butterfly population reach approximately 50,000?
What is the asymptote of the graph? What does it tell you about the butterfly population over time?

When you have the equation that represents the situation, you can substitute values in and use the order of operations to find the values for the table.

Part 1 is asking for the initial value. Since the equation $p = 100, 000 \cdot (\frac{4}{5})^{w}$ is in the form $y = a b^{x}$ , you know the initial value is $a$ or 100,000.
Part 2 is 1 week after migration, so replace the $w$ in the equation with 1 to get 80,000.
In part 3, the $w$ can be replaced with 3 to get 51,200.
The rest of the values of the table can be found the same way, and once you have the completed table, you can plot the points. Once you have plotted the points, you can connect them with a smooth curve.

$w$	0	1	2	3	4
$p$	100,000	80,000	64,000	51,200	40,960

Graph showing the relationship between number of weeks and population of butterflies, plotted points include (0, 100000), (1, 80000), (2, 64000), (3, 51200), and (4, 40960).

Part 5 asks about the vertical intercept and what it means. The vertical intercept is 100,000, which is the initial amount of the butterfly population. This means that when the butterflies start their migration (at $w = 0$ weeks), the population is initially 100,000.
The $y$ -intercept is the value of $p$ when $w = 0$ , which corresponds to the initial condition or starting point of the migration. Substituting $w = 0$ into the equation, we have $p = 100000 \cdot (\frac{4}{5})^{0} = 100000 \cdot 1 = 100000$ .
For part 6, you could continue the table or use the graph to see where it is 50,000, which is around 3 weeks.

Graph of a decreasing exponential function that represents the population of butterflies as a function of number of weeks labeled points include (0, 100000), (1, 80000), (2, 64000), (3, 51200), and (4, 40960).

For part 7, an asymptote is a line or curve that a function approaches but does not intersect or touch. In other words, as the input values of a function approach a certain value or tend towards infinity or negative infinity, the function values get arbitrarily close to the asymptote without actually crossing it. Asymptotes can provide valuable information about the behavior of a function and can help in understanding its limits and range.

To determine the asymptote, we examine the behavior of the function as 𝑤 approaches infinity. In this case, as 𝑤 becomes larger and larger, the term $(\frac{4}{5})^{𝑤}$ will grow exponentially smaller. Consequently, the population 𝑝 will also grow infinitely smaller as it gets closer and closer to zero. Thus, the horizontal asymptote in this equation is the $x$ -axis. The graph of the function will continue to decrease towards the $x$ -axis as 𝑤 increases, approaching $y = 0$ . The butterfly population will decrease over time due to the migration.

Try it

Try It: Connecting Tables and Graphs in Exponential Decay Functions

The number of copies of a book sold the year it was released was 600,000. Each year after that, the number of copies sold decreased by $\frac{1}{2}$ .

1. Complete the table showing the number of copies of the book sold each year.

Years Since Published	Number of Copies Sold
0
1
2
3
$y$

Compare your answer:

Here is how to turn an exponential decay situation into a table and a graph:

To complete the table, you enter 600,000 in the first line of the table, since it is the initial value or the starting point. For each entry after the first, you multiply each term by $\frac{1}{2}$ . The table below has an extra row with the process worked out.

Years Since Published	Process	Number of Copies Sold
0	$600, 000 (\frac{1}{2})^{0}$	$600, 000$
1	$600, 000 (\frac{1}{2})^{1}$	$300, 000$
2	$600, 000 (\frac{1}{2})^{2}$	$150, 000$
3	$600, 000 (\frac{1}{2})^{3}$	$75, 000$
$y$	$600, 000 (\frac{1}{2})^{y}$	$600, 000 (\frac{1}{2})^{y}$

2. Write an equation representing the number of copies, $c$ , sold $y$ years after the book was released.

Compare your answer:

The last row of the table is also the equation for the function, $c = 600, 000 (\frac{1}{2})^{y}$ .

3. Use the graphing tool or technology outside the course. Graph the equation that represents the situation, using the Desmos tool below.

Compare your answer:

Use the points in the table to create a graph.

Graph of a decreasing exponential function that represents the number of copies sold as a function of the number of years since published. The graph has y-intercept of 600,000 and passes through the points (1, 300000) and (2, 150000).

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4. Use your equation or graph to find $c$ when $y = 6$ . What does this mean in terms of the book sales?

Compare your answer:

If you use the equation you would substitute 6 in for $y$ , or $c = 600, 000 (\frac{1}{2})^{6}$ , and evaluate to find the solution.

The answer 9,375 means that six years after the book was published, 9,375 copies of the book were sold.

5. What is the vertical intercept, or y-intercept of the graph? What does it tell you about the number of copies sold?

Compare your answer:

The $y$ -intercept represents the value of the function when $t = 0$ , which corresponds to the year the book was released. Substituting $t = 0$ into the function, we have $f (0) = 600, 000 \cdot (21 )^{0} = 600, 000 \cdot 1 = 600, 000$ .
The $y$ -intercept is 600,000. This means that, in the year the book was released (at $t = 0$ ), 600,000 copies were sold.

6. What is the asymptote of the graph? What does it tell you about the number of copies sold over time?

Compare your answer:

To determine the asymptote, we need to consider the behavior of the function as $t$ approaches infinity. As $t$ increases, the exponent $(\frac{1}{2}) t$ becomes smaller and smaller. Eventually, as $t$ approaches infinity, the exponent approaches 0.
The function $f (t)$ approaches the value 0 as $t$ tends towards infinity. In other words, the horizontal asymptote of the function is $y = 0$ .
The asymptote is $y = 0$ , indicating that as the number of years increases indefinitely, the number of copies sold approaches zero.

5.5.3 Using Graphs to Represent Exponential Decay

Activity

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Extending Your Thinking

Additional Resources

Connecting Tables and Graphs in Exponential Decay Functions

Try It: Connecting Tables and Graphs in Exponential Decay Functions