# Lesson 7

Using Negative Exponents

Let’s look more closely at exponential graphs and equations.

### Problem 1

A forest fire has been burning for several days. The burned area, in acres, is given by the equation \(y = (4,\!800) \cdot 2^d\), where \(d\) is the number of days since the area of the fire was first measured.

- Complete the table.
- Look at the value of \(y = 4,\!800 \cdot 2^d\) when \(d= \text-1\). What does it tell you about the area burned in the fire? What about when \(d= \text-3\)?
- How much area had the fire burned a week before it measured 4,800 acres? Explain your reasoning.

\(d\), days since first measurement |
\(y\), acres burned since fire started |
---|---|

0 | |

-1 | |

-2 | |

-3 | |

-5 |

### Problem 2

The value of a home in 2015 was $400,000. Its value has been doubling each decade.

- If \(v\) is the value of the home, in dollars, write an equation for \(v\) in terms of \(d\), the number of decades since 2015.
- What is \(v\) when \(d = \text-1\)? What does this value mean?
- What is \(v\) when \(d = \text-3\)? What does this value mean?

### Problem 3

A fish population, \(p\), can be represented by the equation \(p= 800 \boldcdot \left(\frac{1}{2}\right)^{t}\) where \(t\) is time in years since the beginning of 2015.

What was the fish population at the beginning of 2012?

100

800

2,400

6,400

### Problem 4

The area, \(A\), of a forest, in acres, is modeled by the equation \(A = 5,\!000 \boldcdot \left(\frac{5}{4}\right)^d\) where \(d\) is the number of decades since the beginning of the year 1950.

- Is the area of the forest increasing or decreasing with time? Explain how you know.
- What was the area of the forest in 1950?
- What was the area of the forest in 1940?
- Was the area of the forest less than 1,000 acres in 1900? Explain how you know.

### Problem 5

A population of mosquitos \(p\) is modeled by the equation \(p = 1,\!000 \boldcdot 2^w\) where \(w\) is the number of weeks after the population was first measured.

- Find and plot the mosquito population for \(w = 0, 1, 2, 3, 4\).
- Where on the graph do you see the 1,000 from the equation for \(p\)?
- Where on the graph can you see the 2 from the equation?

### Problem 6

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 \(\frac{1}{2}\).

- Complete the table showing the number of copies of the book sold each year.
- Write an equation representing the number of copies, \(c\), sold \(y\) years after the book was released.
- Use your equation to find \(c\) when \(y = 6\). What does this mean in terms of the book?

years since published | number of copies sold |
---|---|

0 | |

1 | |

2 | |

3 | |

\(y\) |

### Problem 7

The graph shows a population of butterflies, \(t\) weeks since their migration began.

- How many butterflies were in the population when they started the migration? Explain how you know.
- How many butterflies were in the population after 1 week? What about after 2 weeks?
- Write an equation for the population, \(q\), after \(t\) weeks.