We list important relations and functions that will be explored throughout this course and used often in Calculus.

Throughout this course and in Calculus you will study certain relations in depth. These relations are usually functions, and they are the functions that come up often in real world applications. In this section, we will give you a list of some of these functions, including their formula, their graph, and a table of some of their most important values. You might not recognize some of these functions or know what their forumlas mean. That is ok. We will learn more about them throughout this course. Remember, though, that a relation or function can be given by a graph. For now, familiarize yourself with these graphs. They will come up in examples throughout the course.

Linear Functions

Of the most important types of functions is a linear function. The graph of a linear function is a line.

A prototypical example of a linear function

In general, linear functions can be written as where and can be any numbers. You can play with changing the values of and on the graph using Desmos and see how that changes the line.

Quadratic Functions

The graph of a quadratic function is a parabola.

A prototypical example of a quadratic function is

In general, quadratic functions can be written as where , , and can be any numbers. You can play with changing the values of , , and on the graph using Desmos and see how that changes the parabola.

Absolute Value Function

The absolute value function takes all -values and makes them positive. The absolute value function is written as

Square Root Function

The square root function is written as

Exponential Functions

The exponential growth function is written as

Here is the mathematical constant known as Euler’s number. .

In general, we can talk about exponential functions of the form where is a positive number not equal to . You can play with changing the values of on the graph using Desmos and see how that changes the graph. Pay particular attention to the difference between and .

Logarithm Functions

The most famous logarithm function is

Here is the mathematical constant known as Euler’s number. .

You may notice that the table of values for and are similiar. This is becase these two functions are interconnected. We will explore this more later in the course.

In general, we can talk about logarithmic functions of the form where is a positive number not equal to . You can play with changing the values of on the graph using Desmos and see how that changes the graph. Pay particular attention to the difference between and .

Sine

The sine function is written as

This function comes from trigonometry. In the table below we will use another mathematical constant, (“pi” pronounced pie). .

In general, we can consider . You can play with changing the values of and on the graph using Desmos and see how that changes the graph.