# 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.

A:

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

B:

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

C:

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

D:

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

### 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.

A:

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.

B:

The tuition cost increased 7.5% each year.

C:

The tuition cost increased about 7.2% each year.

D:

The tuition cost roughly doubles in 10 years.

E:

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
1. By what factor does the population grow every two years? Use this information to fill out the table for 4 years and 6 years.
2. By what factor does the population grow every year? Explain how you know, and use this information to complete the table.
(From Unit 4, Lesson 3.)

### Problem 5

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

(From Unit 4, Lesson 4.)

### 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.

(From Unit 4, Lesson 5.)