3.1 Exponential Growth and Decay

Introduction

Exponential functions of form

Determining formulas for exponential functions

Suppose that at age $20$ you have ＄$20000$ and you can choose between one of two ways to use the money: you can invest it in a mutual fund that will, on average, earn $8$% interest annually, or you can purchase a new automobile that will, on average, depreciate $12$% annually. Let's explore how the ＄$20000$ changes over time.

Let $I(t)$ denote the value of the ＄$20000$ after $t$ years if it is invested in the mutual fund, and let $V(t)$ denote the value of the automobile $t$ years after it is purchased.

Determine $I(0)$, $I(1)$, $I(2)$, and $I(3)$.

What is the domain of $g(t) = ab^t$?

What is the range of $g(t) = ab^t$?

What is the $y$-intercept of $g(t) = ab^t$?

How does changing the value of $b$ affect the shape and behavior of the graph of $g(t) = ab^t$? Write several sentences to explain.

For what values of the growth factor $b$ is the corresponding growth rate positive? For which $b$-values is the growth rate negative?

Consider the graphs of the exponential functions $p$ and $q$ provided in the following figure. If $p(t) = ab^t$ and $q(t) = cd^t$, what can you say about the values $a$, $b$, $c$, and $d$ (beyond the fact that all are positive and $b \ne 1$ and $d \ne 1$)? For instance, can you say a certain value is larger than another? Or that one of the values is less than $1$?

Suppose that $p$ is an exponential function and we know that $p(2) = 11$ and $p(5) = 18$. Determine the exact values of $a$ and $b$ for which $p(t) = ab^t$.

Suppose the car's value $t$ years after its purchase is given by the function $V(t)$ and that $V$ is exponential with form $V(t) = ab^t$, what are the values of $a$ and $b$? Find $a$ and $b$ both exactly and approximately.

Using the exponential model determined in (a), determine the purchase value of the car and then use Desmos to estimate when the car will be worth less than \ 1000$.

Suppose instead that the car's value is modeled by a linear function $L$ and satisfies the values stated at the outset of this activity. Find a formula for $L(t)$ and determine both the purchase value of the car and when the car will be worth \ 1000$.

Which model do you think is more realistic? Why?