Dave4Math / Courses / Functions and Their Graphs

Functions and Their Graphs

Learn functions and their graphs. Master function notation, graphing functions, the transformations of functions, and inverse functions. With this course, you'll be experienced at using functions and understanding their graphs.


January 11, 2021

Created by Dave

This course begins Summer 2021.
Current Status
Not Enrolled
Get Started

Are you terrible at memorization? Do you find it distasteful? Introducing Dave’s online course: Functions and Their Graphs. With Dave, he always ensures you know the course material rather than regurgitate it for an exam. Your success is his success!

Who This Course is For

  • Students in college who want to learn more mathematics.
  • Anyone interested in functions and their graphs.
  • Students who are enrolled in precalculus and want to improve their grade.
  • Anyone interested in getting prepared for calculus 1.
  • Anyone wanting to learn any of the following: graphs of equations, lines in the plane, graphs of functions, transforming functions, combinations of functions, and inverse functions.


There is no required textbook, though you will need an up-to-date web browser, paper, and pen. A hand-held scientific calculator of your choice is required.

What You’ll Learn in Functions and Their Graphs

  • The Graph of an Equation
  • Using a Graphing Utility
  • Applications
  • The Slope of a Line
  • The Point-Slope Form
  • Sketching Graphs of Lines
  • Parallel and Perpendicular Lines
  • Introduction to Functions
  • Function Notation
  • The Domain of a Function
  • Applications
  • The Graph of a Function
  • Increasing and Decreasing Functions
  • Relative Minimum and Maximum Values
  • Graphing Step Functions
  • Graphing Piecewise-Defined Functions
  • Even and Odd Functions
  • Summary of Graphs of Parent Functions
  • Vertical and Horizontal Shifts
  • Reflecting Graphs
  • Arithmetic Combinations of Functions
  • Compositions of Functions
  • Inverse Functions
  • The Graph of an Inverse Function
  • The Existence of an Inverse Function
  • Finding Inverse Functions Algebraically

Course Description

This course begins with an explanation of the Cartesian plane. After that, I discuss plotting points, the distance formula, the midpoint formula, and the Pythagorean Theorem.

Next, we study graphing equations. For example, we are sketching the graph of an equation by plotting points. Then I show that finding the intercepts of a graph and graphical (and algebraic) tests for symmetry are important.

Now we specialize in linear equations in two variables. First, we examine the slope-intercept form of the equation of a line and finding the slope of a line. Then, the slope of a line passing through two points, and of course, the point-slope form of an equation of a line. While working through many examples, we also consider parallel and perpendicular lines.

After these beginnings, we now introduce our main topic: functions. We first discuss domain and range and then drive home the concept of a function by examining the characteristics of a function from one set to another. Importantly, we discuss four ways to represent a function, verbally, numerically, graphically, and algebraically. We discuss each way at length while also explaining function notation.

With a thorough understanding of what a function is, we now being to analyze their graphs. In particular, we next study the zeros of a function, where the graph of a function is increasing (decreasing), and the relative extrema.

Next, we switch gears and move to study a library of parent functions. We will use these parent functions to build more complex functions useful in a large number of applications. For example, we consider parent functions such as linear, squaring, cubic, square root, and reciprocal functions. After that, we investigate certain transformations of functions such as shifting graphs, reflecting graphs, and non-rigid transformations.

Now that we understand some parent functions and how to transform them, we consider combining them into more complex functions. So we consider the sum, difference, product, and quotients of functions. In the end, the result is to have a useful, more extensive array of scenarios, where we can understand a function from its basic building blocks.

When we began our study of functions early in the course, we discussed relations. So it is easy to define an inverse relation, but now we focus on inverse functions. We define inverse functions, learn how to verify inverse functions, and how to graph inverse functions. After that, we characterize one-to-one functions and work algebraically to find the inverse of a function.

Throughout this course, we present many applications by working through examples. Towards the end, though, applications take center stage. First, we investigate mathematical modeling and then discuss least square regression. Finally, we discuss direct variation, inverse variation, and joint variation.

I recommend the prerequisite course Fundamental Concepts of Algebra. If you’re not sure if this course is for you, checkout the course contents below or find out more by checking out my free precalculus articles.

Course Content

Expand All

David A. Smith

Mathematics Educator

I sincerely believe that the potential in every student can be unlocked. As an accomplished and dedicated instructor, I have firm confidence that I can provide significant value to your studies.

I relish teaching and use it as a channel for my creativity as I seek to deliver real-life and relevant connections between the mathematics that I teach and my students. Promoting an active learning environment and excitement about mathematics is my dream come true.

David Smith (Dave) has a B.S. and M.S. in Mathematics and has enjoyed teaching precalculus, calculus, linear algebra, and number theory at both the junior college and university levels for over 20 years. David is the founder and CEO of Dave4Math.