# 13 Section 3.1: Exponential Growth and Decay

Example 1

Determine if each of the following functions are exponential. If they are, write them in the form [latex]f(t) = ab^t[/latex].

- [latex]f(t) = 3^{t+1}[/latex]
- [latex]p(t) = 2^t+ 6^t[/latex]
- [latex]g(t) = \frac{2^{3t}}{7}[/latex]
- [latex]k(t) = 27^{\frac{t}{3}}[/latex]
- [latex]h(t) = 4^t \cdot 2^{2t}[/latex]

## Show Solution

a. Yes, exponential. [latex]f(t) = 3^{t+1} = 3^t \cdot 3 = 3\cdot 3^t[/latex]

b. Not exponential.

c. Yes, exponential. [latex]g(t) = \frac{2^{3t}}{7} = \frac{(2^{3)^t}}{7} = \frac{1}{7} \cdot 8^t[/latex].

d. Yes, exponential. [latex]k(t) = 27^{\frac{t}{3}} = (27^{1/3})^t = 3^t.[/latex]

e. Yes, exponential. [latex]h(t) = 4^t \cdot 2^{2t} = 4^t \cdot (2^2)^t = 4^t \cdot 4^t = 4^{t+t} = 4^{2t} = (4^2)^t = 16^t.[/latex]

Example 2

Supposes we have two cities growing at different rates. The population in millions at time t in years for city A is modeled by [latex]P(t) = 120(1.036)^t[/latex] and the population in millions at time t in years for city B is modeled by [latex]Q(t) = 99(1.039)^t[/latex].

a. Which city is growing at a faster rate? What is the annual growth rate of that town?

b. Which city has the largest initial population?

c. After 20 years which town will have more people?

## Show Solution

a. City B is growing at a faster rate. The annual growth rate is 3.9%.

b. City A has the largest initial population of 120 million.

c. Plugging in t = 20 into both models we get [latex]P(20) = 120(1.036)^{20} \approx 243.43[/latex] and [latex]Q(20) = 99(1.039)^{20} \approx 212.79[/latex]. So, city A will have more people after 20 years.

Example 3

Find a formula for an exponential function that contains the points [latex](2,48)[/latex] and [latex](4,768)[/latex].

## Show Solution

First plug in both points into the equation [latex]f(t) = ab^t[/latex] to get [latex]768 = ab^4[/latex] and [latex]48 = ab^2[/latex]. Then, take the ratio of the two equations to get [latex]\frac{768}{48} = \frac{ab^4}{ab^2}[/latex]. Then simplifying we get [latex]16 = b^2[/latex], so [latex]b = 4[/latex]. Then plugging [latex]b = 4[/latex] back into equation 2 we get [latex]48 = a(4)^2[/latex]. Solving for a we get [latex]a = 3[/latex]. So, [latex]f(t) = 3 \cdot 4^t[/latex].

Example 4

Suppose a bank account is opened initially with $[latex]3500[/latex] and earns [latex]4.2%[/latex] annually each year. Write a model [latex]P(t)[/latex] that models the money in the bank account after t years.

## Show Solution

[latex]P(t) = 3500(1.042)^t[/latex].