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Section3.1Introduction to Exponential Functions

You may have heard, “The world is constantly changing”. But the big question is “How?” Is there a pattern? In math, patterns have names.

Linear is the name of a pattern. A linear equation may look like this \begin{equation*} y=mx+b \end{equation*} and its graph is a straight line.

If you look at a table of values from a linear pattern, you will notice the data have a constant rate of change; in other words, a constant slope.

In this section we study a pattern in which something remains constant, but not the rate of change; it is the percent change. An exponential pattern has the characteristic of having constant percent change.

Subsection3.1.1Percent Change, Multiplication and the Growth Factor

Before we explore exponential functions, we should have a very good understanding of percent, what it means and how it is calculated.

Example3.1.1

Quickly test yourself:

  1. Can you increase the number \(745\) by \(12 \%\text{?}\)

  2. Can you decrease the number \(38.2\) by \(25 \%\text{?}\)

  3. Which of the following expressions will find \(60 \%\) of \(1450\text{?}\)

    • \(1450(60)\)

    • \(1450(0.60)\)

    • \(1450 + 0.60\)

    • \(1450 - 1450(0.40)\)

  4. What is the decimal equivalent of \(0.14 \%\text{?}\)

You may want a more in-depth review on the meanings of the terms “decimal equivalent” and “growth factor”. Follow this link to an interactive review of that material: review 3.3

In short, to change a number by a percent, we multiply.

For instance if we start with the number \(8\text{,}\) then \(8(1.25)\) will increase the number \(8\) by one and a quarter of itself. You get all of the \(8\) and \(25 \%\) more.

So, the equation \(8(1.25) = 10\) means that \(10\) is \(25 \%\) greater than \(8\)

On the other hand, if we again start with \(8\text{,}\) then \(8(0.75)\) is a fraction of 8; not all of it, only a portion. In fact, it means \(75 \%\) of \(8\text{.}\)

Notice that keeping \(75 \%\) of 8 automatically means a loss of \(25 \%\text{.}\) So, the equation \(8(0.75) = 6\) says that \(6\) is \(75 \%\) of \(8\text{,}\) which also means that \(6\) is \(25 \%\) less than \(8\text{.}\)

The number we multiply by to get the percent change is called the growth factor — even though it may actually result in a percent decrease. In this book we will refer to the growth factor with the variable b.

Example3.1.2

Given an initial value of \(165\text{,}\) identify the growth factor \(b\) in each expression and describe the percent change.

  • \(165(1.12)\)

  • \(165(0.08)\)

  • \(165(2.69)\)

Solution
Exercise3.1.3

Use the growth factor to determine the percent change on the given initial value. The growth factor is shown in parentheses.

We can use a growth factor to determine the overall percent change between any two numbers.

Exercise3.1.4
Percent Increase:

A population increases from \(2{,}000\) to \(10{,}000\text{.}\) Find the overall percent increase in the population.

Exercise3.1.5
Percent Decrease:

A population decreases from \(10{,}000\) to \(2{,}000\text{.}\) What is the overall percent decrease in the population?

Subsection3.1.2Linear vs. Exponential Patterns

Looking at data, lists of numbers, is part of almost everyone's profession. We search for patterns in the data from past or current events to try to predict future events. We have questions, like

  • How long will a mechanical part last?

  • When can we administer the next dose of medication?

  • When will we get to our destination?

Two common patterns are linear and exponential. The news is full of them.

Example3.1.6

Exponential gains in computing power, along with innovations in software, analytical techniques and the rise of Big Data, mean that many white-collar occupations are due for disruption by machines too. According to a 2013 study by two Oxford University professors, almost half of all jobs in the United States are susceptible to “computerization” over the coming decade or two.

—“How Efficiency Is Wiping Out the Middle Class”, Another View by David Diegel, New York Times January 25, 2017

In writing, “Exponential gains in computing power” this article claims computing power is increasing by a constant percent. It must be assumed that the increases happen over equal time intervals, like every year or every \(10\) years for the claim to be valid.

Example3.1.7

In 2012 the average rent was $\(2,225\text{.}\) Every year it has risen roughly a hundred dollars. But in late 2016, the rental market suddenly became flat.

“If I were to bet, I would bet that prices are on the way down. I would bet the tenants and buyers are going to have the upper hand probably for the next two or three years,” said real estate broker Peter Zalewski.

—“After Years Of Price Hikes, South Florida's Rental Market Entering New Era Of Decline”, David Sutta, CBS Miami, March 29, 2017

This article is reporting a pattern of constant rate of change (monthly rent). The real estate broker quoted in the article is using a linear model to predict the rental market for the next two or three years.

Distiguishing between linear and exponential patterns is as easy as knowing the difference between addition and multipication. The patterns allow us to find formulas for linear or exponential situations.

A linear pattern of constant rate of change is essentially repeated addition. A linear formula counts how many times you add the same number.

An exponential pattern of constant percent change is essesntially repeated multiplication. An exponential formula counts how many times you multiply by the same number.

Knowing how a pattern works, we can then generate our own data in the form of a table.

Exercise3.1.10

A patient's hourly dosage rate is reduced by \(5\) mg/hour.

Exercise3.1.11

A patient's hourly dosage rate is reduced by \(2 \%\) every hour.

Subsection3.1.3A General Formula for Exponential Growth/Decay: Discrete Model

You will find the word compounded in the exercises that follow.

Compounding is the process of repeatedly increasing or decreasing by a percent. It refers to the fact that each succesive percent change is based on a number already greater/lesser than the original.

  • Compounded monthly means a percent is calculated each month.

  • Compounded annually means a percent is calculated each year.

  • In the expression \(500\cdot 1.15\cdot 1.15\cdot 1.15\) the value \(500\) has been compounded three times by \(15 \%\text{.}\)

A common form for a linear equation is \(y = mx + b\text{,}\) although there are others. Exponential equations have their own form too. In this part of the activity we use a table of values to find a pattern in constant percent growth or decay then find a form for exponential functions.

To find a pattern within numerical data, it is best to record how we get our results instead of the results themselves.

As a way to convince you, here is an experiment.

Exercise3.1.13

Given a sequence of values, predict (or guess) what value comes next in the sequence.

Hopefully, you're convinced that finding a pattern in a sequence is much easier if you write out how you get each result instead of the actual result.

Exercise3.1.14Exponential Equation: General Form

A population of bacteria increases by a constant percent each hour. Find a formula to model the size of the popultion \(P(t)\) as a function of time, \(t\) in hours.

Now we have general forms for both linear \begin{equation*} y=mx+b \end{equation*} and exponential \begin{equation*} y=ab^{x} \end{equation*} equations.

In the linear equation \begin{equation*} y=mx+b \end{equation*} the variable \(b\) is sometimes referred to as the initial value. On a graph it represents the vertical intercept while \(m\) represents the rate of change (slope). The rate of change tells us the rate of increase (or decrease) per unit input.

The variable \(x\) counts how many times the rate of change is added to the initial value.

In the exponential equation \begin{equation*} y = ab^{x} \end{equation*} the initial value is represented by \(a\text{,}\) and \(b\) is the growth factor. This equation, or model, is called the discrete exponential model.

The growth factor tells us the percent increase (or decrease) per unit input while the variable \(x\) in this case counts how many times the growth factor is multiplied.

Finding the growth factor per unit in an exponential formula can be tricky. We have been thinking in terms of linear (constant rate of change)for so long, we often think everything is linear.

We need a way to convert a statement like “\(60 \%\) over \(3\) years” into one about the percent growth per year. A good start is to remind ourselves how repeated addition is different than repeated multiplication.

Exercise3.1.15

Is one decrease of \(60\) students the same as three decreases of \(20\) students?

Exercise3.1.16

Is one decrease by \(60 \%\) of students the same as three decreases by \(20 \%\) of students?

As we found in the last exercise, given a change of \(60 \%\) over three years, we cannot simply divide \(60 \%\) by \(3\) to find the annual percent change. Because of compounding, three \(20 \%\) increases is actually more than one \(60 \%\) increase.

To derive an exponential formula of the form \(y=ab^{x}\text{,}\) the goal is to find the values \(a\) and \(b\) for the specific problem, then you're done!

Practice finding exponential formulas for three different examples.

In each example we will use time, \(t\text{,}\) as the input. But the strategy will be the same regardless of what variable we use:

  • Assume an exponential with \(t\) representing time as in \(f(t)=ab^{t}\text{.}\)

  • Find the overall growth factor between to output values.

  • Use the overall growth factor to calculate the growth factor per unit, \(b\text{.}\)

  • Use the newly found growth factor \(b\) to determine the initial value \(a\) if it is not already known.

Exercise3.1.17

Derive a formula for a population \(P(t)\) of bacteria that increases by a constant percent every \(3\) hours. Let \(t\) be time in hours.

Strategy: Find the overall growth factor for the given span of time and use to it to find the hourly growth factor (per \(1\) hour).

Exercise3.1.18

Derive an exponential formula \(f(t)\) for the temperature of a pie, in degrees Fahrenheit, as a function of time, \(t\) in hours. The initial temperature and one other ordered pair are known.

Strategy: Find the overall growth factor for the given span of time and use to it to find the hourly growth factor (per \(1\) hour).

What if the initial value is not known? Can we still find an exponential formula?

Exercise3.1.19

Derive an exponential formula \(f(t)\) for the temperature of a pie, in degrees Fahrenheit, as a function of time, \(t\) in hours. Two ordered pairs are known, not including the intitial temperature.

Strategy: Find the overall growth factor, and use it to find the hourly growth factor.

The initial value is found using any known ordered pair.

Subsection3.1.4Graphs of Exponential Functions

To explore the graphs of exponential functions it may help to compare them to the graphs of linear functions since we already have some experience with them.

In the following examples we track the position of an ant over time using both a linear model and an exponential model.

Exercise3.1.20

An ant is initially \(150\) inches from a wall, and each minute it walks \(25\) inches towards the wall.

Exercise3.1.21

An ant is initially 150 inches from a wall, each minute the ant walks 25% of the distance to the wall.

Comparing linear and exponential graphs there are some clear differences.

  • The linear function graphs as a staight line while the exponential function graphs as a concave-up curve.

  • The linear function has a horizontal intercept, but the exponential fucntion appears not to touch the horizontal axis.

Let's explore the graph of the exponential function more closely and return to the example of the ant moving towards a wall by reducing the distance by \(25 \%\) each minute.

Exercise3.1.22Horizontal Asymptotes

An ant initially \(150\) inches from a wall moves towards the wall by \(25 \%\) each minute.

All exponential functions (of this form) have horizontal asymptotes. The equation for a horizontal asymptote is \(y=k\) where \(k\) is the height of the horizontal line the function approaches.

In the last example the function decreases to near zero, so \(y=0\) is the equation of the horizontal asymptote.