# Chapter 2: Polynomial, Exponential, and Logarithmic Functions

Because polynomial, exponential, and logarithmic functions have several applications in transport engineering, this chapter will explain the functions. The exponential functions are used to model economic and population growth and to estimate the compound interest formula. Simplifying polynomial expressions and solving corresponding equations is helpful in the design of transportation facilities and understanding the relationship(s) between flow, speed, and density on roadway segments. Logarithmic functions are also sometimes used to model the relationship between traffic speed and density on a roadway segment, where the speed decreases as the density increases. Understanding the properties of these functions also provide a foundation to learn differential and integral calculus for these functions (See Chapter 4).

Learning Objectives

At the end of the chapter, the reader should be able to do the following:

- Generate graphs and charts from exponential and logarithmic functions.
- Solve the exponential and logarithmic equations.
- Visualize polynomial functions on a graph or chart.
- Set up and solve polynomial equations.
- Identify topics in the introductory transportation engineering courses that build on the concepts discussed in this chapter.

## Generate Graphs and Charts from Exponential and Logarithmic Functions

This section will explain exponential growth functions, exponential and logarithmic functions, exponential graphs, graphing exponential growth and decay, and introduce you to logarithms with videos to help your comprehension. Also, short problems to check your understanding are included.

### Introduction to Exponential Functions

Check Your Understanding: Exponential Functions

### Exponential Function Graph

### Graphing Exponential Functions

Check Your Understanding: Graphing Exponential Functions

### Graphs of Exponential Growth

Check Your Understanding: Graphs of Exponential Growth

### Graphing Exponential Growth and Decay

Check Your Understanding: Graphing Exponential Growth and Decay

### Analyzing Graphs of Exponential Functions

Check Your Understanding: Analyzing Graphs of Exponential Functions

### Analyzing Tables of Exponential Functions

Check Your Understanding: Analyzing Tables of Exponential Functions

### Introduction to Logarithms

Check Your Understanding: Logarithms

### Graphs of Logarithmic Functions

Check Your Understanding: Graphs of Logarithmic Functions

### Relationship Between Exponentials and Logarithms: Graphs

### Relationship Between Exponentials and Logarithms: Tables

Check Your Understanding: Relationship Between Exponentials and Logarithms

## Solve the Exponential and Logarithmic Equations

This section will explain how to solve exponential equations using exponent properties in both basic and advanced problems with videos to help your comprehension. Also, short problems to check your understanding are included.

### Solving Exponential Equations Using Exponent Properties

Check Your Understanding: Solving Exponential Equations Using Exponent Properties

### Solving Exponential Equations Using Exponent Properties (Advanced)

Check Your Understanding: Solving Exponential Equations Using Exponent Properties (Advanced)

### Introduction to Logarithm Properties

Please read this link on Introduction to Logarithm Properties.

Check Your Understanding: Introduction to Logarithm Properties

### Justifying the Logarithm Properties

Please read this link on Justifying the Logarithm Properties.

### Logarithm Change of Base Rule Introduction

Please read this link on Logarithm Change of Base Rule Introduction

Check Your Understanding: Justifying the Logarithm Properties & Logarithm Change of Base Rule

### Logarithmic Equations: Variable in the Argument

Check Your Understanding: Logarithmic Equations: Variable in the Argument

### Logarithmic Equations: Variable in the Base

Check Your Understanding: Logarithmic Equations: Variable in the Base

## Visualize Polynomial Functions on a Graph or Chart

This section will explain how to graph polynomial functions with videos to help your comprehension. Also, short problems to check your understanding are included.

### Graphing Polynomial Functions

### Graphing Lab – Quadratics and Polynomials

If you’d like extra practice, please perform this lab.

### Graphs of Polynomials Overview

Please read this link on Graphs of Polynomials Overview

Check Your Understanding: Graphs of Polynomials

## Set Up and Solve Polynomial Equations

This section will explain how to set up and solve polynomials with variables, multiplying binomials with polynomials, and solving polynomial equations by factoring with videos to help your comprehension. Also, short problems to check your understanding are included.

### Adding and Subtracting Polynomials

Please read this link on Adding and Subtracting Polynomials

Check Your Understanding: Adding and Subtracting Polynomials

### Adding and Subtracting Polynomials with Two Variables

Please read this link on Adding and Subtracting Polynomials with Two Variables

Check Your Understanding: Adding and Subtracting Polynomials with Two Variables

### Multiplying Binomials by Polynomials

Please read this link on Multiplying Binomials by Polynomials

Check Your Understanding: Multiplying Binomials by Polynomials

### Solving Polynomial Equations by Factoring

### Factoring Polynomials by Taking a Common Factor

Please read this link on Factoring Polynomials by Taking a Common Factor

Check Your Understanding: Factoring Polynomials by Taking a Common Factor

### Dividing Polynomials: Long Division

Check Your Understanding: Dividing Polynomials: Long Division

### Solving Polynomial Equations by using Synthetic Substitution

Check Your Understanding: Solving Polynomial Equations by using Synthetic Substitution

## Relevance to Transportation Engineering Coursework

This section explains the relevance of engineering economics, traffic assignment, vertical curve design, and traffic flow to transportation engineering coursework.

### Engineering Economics

Engineering economics is the application of economic analysis methods to quantify benefits and costs to assist in the decision-making process. The decisions are related to transportation engineering alternative designs (e.g., constructing a stop-controlled intersection or a roundabout), prioritization of transportation projects, and maintenance budgets, among others. In engineering economics, the time value of money (present value of money and future value of money) and interest rate, also known as the discount rate, need to be considered to provide meaningful Benefit-Cost B/C analysis. The analysis includes the compound interest formula and the exponential growth of money. Exponential functions are also used to model the depreciation of assets. Relevant exponential functions are described in the section titled “Generate Graphs and Charts from Exponential and Logarithmic Functions” of this chapter.

### Traffic Assignment

In the 4-step travel demand modeling process, traffic assignment is typically the last step, which is used to estimate the number of trips on each route of the roadway network. A critical part of traffic assignment is the Link Performance Function for each route, which defines travel time as a function of the number of users on a route. Travel time increases exponentially as traffic volume increases and is defined based on functional forms discussed in “Generate Graphs and Charts from Exponential and Logarithmic Functions”.

### Vertical Curve Design

In highway engineering, one of the fundamental design elements is the vertical profile (i.e., the upgrades, the downgrades, and the curves that connect consecutive roadway sections with different grades, also known as the vertical curves). The vertical curve’s elevation is defined as a quadratic equation, a special type of polynomial function (see “Solve the Exponential and Logarithmic Equations” above).

### Relationship among Traffic Flow, Speed, and Density

In traffic flow theory for uninterrupted flow facilities (such as freeways), the fundamental relationship relates speed and density on the segment with the flow rate past a point on the road segment. The decrease in speed as segment density increases may be modeled using linear, polynomial, or logarithmic functions. The nature of speed-density functions may be used to model relationships between the flow rate and density and/or the flow rate and speed. Polynomial and Logarithmic functions discussed in this chapter are critical to this analysis.

Key Takeaways

- Exponential functions are useful for transportation practitioners in applying the compound interest formula and modeling asset depreciation and population growth rate in engineering economics.
- Exponential functions are also used in defining link performance functions in traffic assignment to model travel time as a function of road users on a network route
- Logarithmic functions are sometimes used to define speed as a function of roadway segment density.
- Polynomial functions are used in several applications, including the equation of parabolic curves connecting two consecutive grades on a roadway segment.
- Speed-flow and flow-density relationships on roadway segments are often also defined as polynomial functions.

## Glossary: Key Terms

**Binomial[1]** – a mathematical expression consisting of two terms connected by a plus sign or minus sign

**Common Factor[1]** – (also called common divisor) a number or expression that divides two or more numbers or expressions without remainder

**Exponent[1]** – a symbol written above and to the right of a mathematical expression to indicate the operation of raising to a power

**Exponential Function[1]** – a mathematical function in which an independent variable appears in one of the exponents

**Exponential Growth[2]** – growth whose rate becomes ever more rapid in proportion to the growing total number or size.

**Logarithm[1]** – the exponent that indicates the power to which a base number is raised to produce a given number

**Logarithmic Function[1]** – a function (such as y =loga x or y = ln x) that is the inverse of an exponential function (such as y = ax or y = ex) so that the independent variable appears in a logarithm

**Polynomial[1]** – a mathematical expression of one or more algebraic terms each of which consists of a constant multiplied by one or more variables raised to a nonnegative integral power (such as a + bx + cx2)

## Media Attributions

Note: All Khan Academy content is available for free at (www.khanacademy.org).

### Videos

- Video 1: Introduction to Exponential Functions by Khan Academy is licensed by Creative Commons Attribution-NonCommercial-ShareAlike 3.0 United States (CC BY-NC-SA 3.0 US)
- Video 2: Exponential Function Graph by Khan Academy is licensed by Creative Commons Attribution-NonCommercial-ShareAlike 3.0 United States (CC BY-NC-SA 3.0 US)
- Video 3: Graphing Exponential Functions by Khan Academy is licensed by Creative Commons Attribution-NonCommercial-ShareAlike 3.0 United States (CC BY-NC-SA 3.0 US)
- Video 4: Graphs of Exponential Growth by Khan Academy is licensed by Creative Commons Attribution-NonCommercial-ShareAlike 3.0 United States (CC BY-NC-SA 3.0 US)
- Video 5: Graphing Exponential Growth and Decay by Khan Academy is licensed by Creative Commons Attribution-NonCommercial-ShareAlike 3.0 United States (CC BY-NC-SA 3.0 US)
- Video 6: Analyzing Graphs of Exponential Functions by Khan Academy is licensed by Creative Commons Attribution-NonCommercial-ShareAlike 3.0 United States (CC BY-NC-SA 3.0 US)
- Video 7: Analyzing Tables of Exponential Functions by Khan Academy is licensed by Creative Commons Attribution-NonCommercial-ShareAlike 3.0 United States (CC BY-NC-SA 3.0 US)
- Video 8: Introduction to Logarithms by Khan Academy is licensed by Creative Commons Attribution-NonCommercial-ShareAlike 3.0 United States (CC BY-NC-SA 3.0 US)
- Video 9: Graphs of Logarithmic Functions by Khan Academy is licensed by Creative Commons Attribution-NonCommercial-ShareAlike 3.0 United States (CC BY-NC-SA 3.0 US)
- Video 10: Relationship Between Exponentials and Logarithms: Graphs by Khan Academy is licensed by Creative Commons Attribution-NonCommercial-ShareAlike 3.0 United States (CC BY-NC-SA 3.0 US)
- Video 11: Relationship Between Exponentials and Logarithms: Tables by Khan Academy is licensed by Creative Commons Attribution-NonCommercial-ShareAlike 3.0 United States (CC BY-NC-SA 3.0 US)
- Video 12: Solving Exponential Equations Using Exponent Properties by Khan Academy is licensed by Creative Commons Attribution-NonCommercial-ShareAlike 3.0 United States (CC BY-NC-SA 3.0 US)
- Video 13: Solving Exponential Equations Using Exponent Properties (Advanced) by Khan Academy is licensed by Creative Commons Attribution-NonCommercial-ShareAlike 3.0 United States (CC BY-NC-SA 3.0 US)
- Video 14: Logarithmic Equations: Variable in the Argument by Khan Academy is licensed by Creative Commons Attribution-NonCommercial-ShareAlike 3.0 United States (CC BY-NC-SA 3.0 US)
- Video 15: Logarithmic Equations: Variable in the Base by Khan Academy is licensed by Creative Commons Attribution-NonCommercial-ShareAlike 3.0 United States (CC BY-NC-SA 3.0 US)
- Video 16: Graphing Polynomial Functions by Michael Pemberton is licensed by Creative Commons Attribution 3.0 Unported (CC BY 3.0)
- Video 17: Solving Polynomial Equations by Factoring by Keith Mann is licensed by Creative Commons Attribution 3.0 Unported (CC BY 3.0)
- Video 18: Dividing Polynomials: Long Division by Khan Academy is licensed by Creative Commons Attribution-NonCommercial-ShareAlike 3.0 United States (CC BY-NC-SA 3.0 US)
- Video 19: Solving Polynomial Equations by using Synthetic Substitution by Keith Mann is licensed by Creative Commons Attribution 3.0 Unported (CC BY 3.0)

## References

- Farid, A. (2022).
*Engineering Economics.*Personal Collection of Ahmed Farid, California Polytechnic State University, San Luis Obispo, CA. - Mannering, F., and Washburn, S. (2013). Chapter 5: Fundamentals of Traffic Flow and Queuing Theory. In:
*Principles of Highway Engineering and Traffic Analysis 5th Edition*. John Wiley & Sons, Inc., Hoboken, NJ. pp. 135-174. - Farid, A. (2022).
*Transportation Planning 1.*Personal Collection of Ahmed Farid, California Polytechnic State University, San Luis Obispo, CA. - Farid, A. (2022).
*Transportation Planning 2.*Personal Collection of Ahmed Farid, California Polytechnic State University, San Luis Obispo, CA. - Farid, A. (2022).
*Highway Design.*Personal Collection of Ahmed Farid, California Polytechnic State University, San Luis Obispo, CA.

a function (such as y =loga x or y = ln x) that is the inverse of an exponential function (such as y = ax or y = ex) so that the independent variable appears in a logarithm

a mathematical expression of one or more algebraic terms each of which consists of a constant multiplied by one or more variables raised to a nonnegative integral power (such as a + bx + cx2)

growth whose rate becomes ever more rapid in proportion to the growing total number or size.

a mathematical function in which an independent variable appears in one of the exponents

a symbol written above and to the right of a mathematical expression to indicate the operation of raising to a power

the exponent that indicates the power to which a base number is raised to produce a given number

a mathematical expression consisting of two terms connected by a plus sign or minus sign

(also called common divisor) a number or expression that divides two or more numbers or expressions without remainder