# Lesson 6

Writing Equations for Exponential Functions

- Let’s decide what information we need to write an equation for an exponential function.

### Problem 1

A population of 1,500 insects grows exponentially by a factor of 3 every week. Select** all** equations that represent or approximate the population, \(p\), as a function of time in days, \(t\), since the population was 1,500.

\(p(t) = 1,\!500 \boldcdot 3^t\)

\(p(t) = 1,\!500 \boldcdot 3^{\frac{t}{7}}\)

\(p(t) = 1,\!500 \boldcdot 3^7t\)

\(p(t) = 1,\!500 \boldcdot \left(3^\frac17\right)^t\)

### Problem 2

The tuition at a public university was $21,000 in 2008. Between 2008 and 2010, the tuition had increased by 15%. Since then, it has continued to grow exponentially.

Select **all** statements that describe the growth in tuition cost.

The tuition cost can be defined by the function \(f(y) = 21,\!000 \boldcdot (1.15)^\frac{y}{2}\), where \(y\) represents years since 2008.

The tuition cost increased 7.5% each year.

The tuition cost increased about 7.2% each year.

The tuition cost roughly doubles in 10 years.

The tuition cost can be approximated by the function \(f(d) = 21,\!000 \boldcdot 2^d\), where \(d\) represents decades since 2008.

### Problem 3

Here is a graph that represents \(g(x) = a \boldcdot b^x\). Find the values of \(a\) and \(b\). Show your reasoning.

### Problem 4

The number of fish in a lake is growing exponentially. The table shows the values, in thousands, after different numbers of years since the population was first measured.

years | population |
---|---|

0 | 10 |

1 | |

2 | 40 |

3 | |

4 | |

5 | |

6 |

- By what factor does the population grow every two years? Use this information to fill out the table for 4 years and 6 years.
- By what factor does the population grow every year? Explain how you know, and use this information to complete the table.

### Problem 5

The value of a home increases by 7% each year. Explain why the value of the home doubles approximately once each decade.

### Problem 6

Here is the graph of an exponential function \(f\).

The coordinates of \(A\) are \(\left(\frac{1}{4},3\right)\). The coordinates of \(B\) are \(\left(\frac{1}{2},4.5\right)\). If the \(x\)-coordinate of \(C\) is \(\frac{7}{4}\), what is its \(y\)-coordinate? Explain how you know.