Introduction

Linear functions have constant average rate of change and model many important phenomena. In other settings, it is natural for a quantity to change at a rate that is proportional to the amount of the quantity present. For instance, whether you put $ or $ or any other amount in a mutual fund, the investment’s value changes at a rate proportional the amount present. We often measure that rate in terms of the annual percentage rate of return.

Suppose that a certain mutual fund has a % annual return. If we invest $, after year we still have the original $, plus we gain % of $, so

If we instead invested $, after year we again have the original $, but now we gain % of $, and thus We therefore see that regardless of the amount of money originally invested, say , the amount of money we have after year is .

If we repeat our computations for the second year, we observe that

The ideas are identical with the larger dollar value, so and we see that if we invest dollars, in years our investment will grow to .

Of course, in years at %, the original investment will have grown to . Here we see a new kind of pattern developing: annual growth of % is leading to powers of the base , where the power to which we raise corresponds to the number of years the investment has grown. We often call this phenomenon exponential growth.

Suppose that at age you have $ 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 % interest annually, or you can purchase a new automobile that will, on average, depreciate % annually. Let’s explore how the changes over time.

Let denote the value of the $ after years if it is invested in the mutual fund, and let denote the value of the automobile years after it is purchased.

a.
Determine , , , and .
b.
Note that if a quantity depreciates % annually, after a given year, % of the quantity remains. Compute , , , and .
c.
Based on the patterns in your computations in (a) and (b), determine formulas for and .
d.
Use Desmos to define and . Plot each function on the interval and record your results on the axes below, being sure to label the scale on the axes. What trends do you observe in the graphs? How do and compare?

Exponential functions of form

In the exploration above, we encountered the functions and that had the same basic structure. Each can be written in the form where and are positive constants and . Based on our earlier work with transformations, we know that the constant is a vertical scaling factor, and thus the main behavior of the function comes from , which we call an “exponential function”.

For an exponential function , we note that , so an exponential function of this form always passes through . In addition, because a positive number raised to any power is always positive (for instance, and ), the output of an exponential function is also always positive.

Because we will be frequently interested in functions such as and with the form , we will also refer to functions of this form as “exponential”, understanding that technically these are vertical stretches of exponential functions according to our definition of exponential function. In the exploration above, we found that and . It is natural to call the “growth factor” of and similarly the growth factor of . In addition, we note that these values stem from the actual growth rates: for and for , the latter being negative because value is depreciating.

Suppose that at age you have $ 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 % interest annually, or you can purchase a new automobile that will, on average, depreciate % annually. Let’s explore how the changes over time.

Let denote the value of the $ after years if it is invested in the mutual fund, and let denote the value of the automobile years after it is purchased.

a.
What is the domain of ?
b.
What is the range of ?
c.
What is the -intercept of ?
d.
How does changing the value of affect the shape and behavior of the graph of ? Write several sentences to explain.
e.
For what values of the growth factor is the corresponding growth rate positive? For which -values is the growth rate negative?
f.
Consider the graphs of the exponential functions and provided in the figure below. If and , what can you say about the values , , , and (beyond the fact that all are positive and and )? For instance, can you say a certain value is larger than another? Or that one of the values is less than ?

Determining formulas for exponential functions

To better understand the roles that and play in an exponential function, let’s compare exponential and linear functions. In the tables below, we see output for two different functions and that correspond to equally spaced input.

In the leftside table for , we see a function that exhibits constant average rate of change since the change in output is always for any change in input of . Said differently, is a linear function with slope . Since its -intercept is , the function’s formula is .

In contrast, the function given by rightside table for does not exhibit constant average rate of change. Instead, another pattern is present. Observe that if we consider the ratios of consecutive outputs in the table, we see that

So, where the differences in the outputs in the table for are constant, the ratios in the outputs in the table for are constant. The latter is a hallmark of exponential functions and may be used to help us determine the formula of a function for which we have certain information.

If we know that a certain function is linear, it suffices to know two points that lie on the line to determine the function’s formula. It turns out that exponential functions are similar: knowing two points on the graph of a function known to be exponential is enough information to determine the function’s formula. In the following example, we show how knowing two values of an exponential function enables us to find both and exactly.

The value of an automobile is depreciating. When the car is years old, its value is $; when the car is years old, its value is $.
a.
Suppose the car’s value years after its purchase is given by the function and that is exponential with form . What are the exact values of and ?
b.
Use the exponential model determined in (a), determine the purchase value of the car and estimate when the car will be worth less than $1000.
c.
Suppose instead that the car’s value is modeled by a linear function and satisfies the values stated at the outset of this activity. Find a formula for and determine both the purchase value of the car and when the car will be worth $.
d.
Which model do you think is more realistic? Why?

Recall that a function is increasing on an interval if its value always increasing as we move from left to right. Similarly, a function is decreasing on an interval provided that its value always decreases as we move from left to right.

If we consider an exponential function with a growth factor , such as the function pictured in the left-hand graph above, then the function is always increasing because higher powers of are greater than lesser powers (for example, ). On the other hand, if , then the exponential function will be decreasing because higher powers of positive numbers less than get smaller (e.g., ), as seen for the exponential function in the right-hand graph above.

An additional trend is apparent in the graphs in above. Each graph bends upward and is therefore concave up. We can better understand why this is so by considering the average rate of change of both and on consecutive intervals of the same width. We choose adjacent intervals of length and note particularly that as we compute the average rate of change of each function on such intervals,

Thus, these average rates of change are also measuring the total change in the function across an interval that is -unit wide. We now assume that and and compute the rate of change of each function on several consecutive intervals.

The average rate of change of

The average rate of change of

From the data in the first table about we see that the average rate of change is increasing as we increase the value of . We naturally say that appears to be “increasing at an increasing rate”. For the function , we first notice that its average rate of change is always negative, but also that the average rate of change gets less negative as we increase the value of . Said differently, the average rate of change of is also increasing as we increase the value of . Since is always decreasing but its average rate of change is increasing, we say that appears to be “decreasing at an increasing rate”. These trends hold for exponential functions generally according to the conditions given below. It takes calculus to justify this claim fully and rigorously.

Observe how a function’s average rate of change helps us classify the function’s behavior on an interval: whether the average rate of change is always positive or always negative on the interval enables us to say if the function is always increasing or always decreasing, and then how the average rate of change itself changes enables us to potentially say how the function is increasing or decreasing through phrases such as “decreasing at an increasing rate”.

For each of the following prompts, give an example of a function that satisfies the stated characteristics by both providing a formula and sketching a graph.

a.
A function that is always decreasing and decreases at a constant rate.
b.
A function that is always increasing and increases at an increasing rate.
c.
A function that is always increasing for , always decreasing for , and is always changing at a decreasing rate.
d.
A function that is always increasing and increases at a decreasing rate. (Hint: to find a formula, think about how you might use a transformation of a familiar function.)
e.
A function that is always decreasing and decreases at a decreasing rate.