Foundational Math for Water Engineering
Welcome
Learning goals
Software
1
The Basics
1.1
Real numbers and their properties
1.1.1
Translating signs carefully
1.1.2
Practice Problems
1.2
BEDMAS and fractions
1.2.1
Complex numerical fractions
1.2.2
Units, conversion factors, and precision
1.2.3
Practice Problems
1.3
Sets, inequalities, and interval notation
1.3.1
Converting among representations
1.3.2
Practice Problems
1.4
Absolute value and distance
1.4.1
Practice Problems
1.5
Exponents, roots, and rational exponents
1.5.1
Simplifying roots
1.5.2
Practice Problems
1.6
Algebraic expressions and polynomials
1.6.1
Polynomials in two variables
1.6.2
Domains of algebraic expressions
1.6.3
Practice Problems
1.7
Expanding and factoring
1.7.1
A factoring decision process
1.7.2
Practice Problems
1.8
Polynomial division and rational zeros
1.8.1
Long and synthetic division
1.8.2
Finding all rational zeros
1.8.3
Practice Problems
1.9
Equations and an initial look at inequalities
1.9.1
Checking and classifying results
1.9.2
Practice Problems
1.10
Lines and linear applications
1.10.1
Intercepts and related lines
1.10.2
Practice Problems
1.11
Fractional and rational expressions
1.11.1
A complete simplification routine
1.11.2
Practice Problems
1.12
Solving linear and quadratic equations
1.12.1
Solving by factoring
1.12.2
Using the quadratic formula
1.12.3
Interpreting the discriminant
1.12.4
Practice Problems
1.13
Rearranging formulas and radical equations
1.13.1
Formula and radical checks
1.13.2
Practice Problems
1.14
Solving inequalities
1.14.1
Sign charts and tolerances
1.14.2
Practice Problems
1.15
Chapter Review
1.15.1
Chapter summary
1.15.2
Common mistakes
1.15.3
Exercises
1.15.4
Questions for discussion
2
Functions
2.1
Function language, evaluation, domain, and range
2.1.1
Evaluating functions and recognizing graphs
2.1.2
Domain, codomain, and range
2.1.3
Practice Problems
2.2
Parent functions, piecewise models, and rates of change
2.2.1
Parent functions and piecewise definitions
2.2.2
Net change, average rate of change, and difference quotients
2.2.3
Practice Problems
2.3
Graphical behaviour and transformations
2.3.1
Graphical equations, inequalities, and extrema
2.3.2
Transforming parent graphs
2.3.3
Practice Problems
2.4
Function operations, composition, and inverses
2.4.1
Arithmetic with functions
2.4.2
Composition of functions
2.4.3
One-to-one and onto functions
2.4.4
Inverse functions
2.4.5
Practice Problems
2.5
Chapter Review
2.5.1
Chapter summary
2.5.2
Common mistakes
2.5.3
Exercises
2.5.4
Questions for discussion
3
Polynomial and Rational Functions
3.1
Quadratic functions and completing the square
3.1.1
Vertex form and intercepts
3.1.2
Completing the square
3.1.3
Practice Problems
3.2
Polynomial behaviour and graphing
3.2.1
End behaviour and local extrema
3.2.2
Zeros, multiplicity, and a systematic sketch
3.2.3
Practice Problems
3.3
Rational functions, holes, and asymptotes
3.3.1
Domains, holes, and vertical asymptotes
3.3.2
Horizontal asymptotes
3.3.3
Practice Problems
3.4
Sketching and interpreting rational functions
3.4.1
A systematic graphing process
3.4.2
Domain and range from a rational graph
3.4.3
Practice Problems
3.5
Chapter Review
3.5.1
Chapter summary
3.5.2
Common mistakes
3.5.3
Exercises
3.5.4
Questions for discussion
4
Exponential and Logarithmic Functions
4.1
Exponential functions and transformations
4.1.1
Basic graphs and properties
4.1.2
Transformations of exponential graphs
4.1.3
Practice Problems
4.2
Exponential models in applications
4.2.1
Discrete and continuous growth and decay
4.2.2
Logistic growth
4.2.3
Practice Problems
4.3
Logarithmic functions and laws
4.3.1
Definition, graph, and domain
4.3.2
Product, quotient, and power laws
4.3.3
Practice Problems
4.4
Solving exponential and logarithmic equations
4.4.1
Exponential equations
4.4.2
Logarithmic equations and domain checks
4.4.3
Practice Problems
4.5
Change of base, logarithmic scales, and flow duration
4.5.1
Change of base and orders of magnitude
4.5.2
Semi-log, log-log, and flow-duration graphs
4.5.3
Practice Problems
4.6
Chapter Review
4.6.1
Chapter summary
4.6.2
Common mistakes
4.6.3
Exercises
4.6.4
Questions for discussion
5
Trigonometric Functions
5.1
Angles, radians, and right triangles
5.1.1
Degree and radian measure
5.1.2
Right-triangle ratios
5.1.3
Practice Problems
5.2
Special angles, reference angles, and the unit circle
5.2.1
Special triangles and reference angles
5.2.2
Trigonometric functions of any angle
5.2.3
Practice Problems
5.3
Trigonometric graphs and transformations
5.3.1
Parent graphs and domains
5.3.2
Amplitude, period, and phase shift
5.3.3
Practice Problems
5.4
Inverse trigonometric functions
5.4.1
Principal values and solving angles
5.4.2
Practice Problems
5.5
Trigonometric identities and angle formulas
5.5.1
Fundamental identities
5.5.2
Even, odd, addition, and subtraction formulas
5.5.3
Practice Problems
5.6
Chapter Review
5.6.1
Chapter summary
5.6.2
Common mistakes
5.6.3
Exercises
5.6.4
Questions for discussion
6
Statistics
6.1
Data, variables, populations, and samples
6.1.1
Data matrices and variable types
6.1.2
Populations, samples, parameters, and statistics
6.1.3
Practice Problems
6.2
Sampling, observational studies, and experiments
6.2.1
Bias and sampling methods
6.2.2
Observational studies and experiments
6.2.3
Practice Problems
6.3
Measures of centre and variability
6.3.1
Mean, median, and mode
6.3.2
Variance and standard deviation
6.3.3
Practice Problems
6.4
Quartiles, outliers, and distribution shape
6.4.1
Five-number summaries and boxplots
6.4.2
Skewness and choosing summaries
6.4.3
Practice Problems
6.5
Chapter Review
6.5.1
Chapter summary
6.5.2
Common mistakes
6.5.3
Exercises
6.5.4
Questions for discussion
Glossary
Appendices
A
Formula Reference
A.1
Algebra and geometry
A.2
Functions and rates
A.3
Exponential and logarithmic models
A.4
Trigonometry
A.5
Descriptive statistics
References
Foundational Mathematics for Water Engineering
Fall 2026 Edition
Joya Danyluk and Joe Hobart
Foundational Mathematics for Water Engineering
References
Diez, David M., Christopher D. Barr, and Mine Çetinkaya-Rundel. 2019.
OpenIntro Statistics
. 4th ed. OpenIntro.