Unit 6, Lesson 4
Representing Exponential Decay
Algebra 1
4.1
Warm-up
Use the patterns you notice to complete the tables. Show your reasoning.
Table A
| 0 | 1 | 2 | 3 | 4 | 25 | |
| 2.5 | 10 | 17.5 | 25 |
Table B
| 0 | 1 | 2 | 3 | 4 | 25 | |
| 2.5 | 10 | 40 | 160 |
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In order 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, is given by the equation . Time, , is measured in weeks.
Create a graph to represent when 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.
Once a glow stick begins to glow, it can glow for hours. The graph shows the luminescence, in lumens, of a glow stick over time, in hours.
| glowing time (hours) | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| luminescence (lumens) | 9 | 6.3 | 4.4 | 3.1 |
Here is a graph showing the luminescence of a glow-in-the-dark paint, measured in lumens, over a period of time, measured in hours. The luminescence of this glow-in-the-dark paint can be modeled by an exponential function.
Notice that the amounts are decreasing over time. The graph includes the point . This means that when the glow-in-the-dark paint started glowing, its glow measured 12 lumens. The point tells us the glow measured 6 lumens 1 hour later. Between 3 and 4 hours after the glow-in-the-dark paint began to glow, the luminescence fell below 1 lumen.
We can use the graph to find out what fraction of luminescence stays each hour. Notice that and . As each hour passes, the luminescence that stays is multiplied by a factor of .
If is the luminescence, in lumens, and is time, in hours, then this situation is modeled by the equation:
We can confirm that the data is changing exponentially because it is multiplied by the same value each time. When the growth factor is between 0 and 1, the quantity being multiplied decreases, the situation is sometimes called “exponential decay,” and the growth factor may be called a “decay factor.”