You may have seen the term 'polar functions' on various webpages in the context of graphing, but what does it actually mean? The simple answer is that it's a *misnomer* and **in mathematics, there is no such thing as a 'polar function'**. Functions are NOT categorized as "Cartesian" or "polar". The term ** polar** is used to refer to coordinate systems and graphs, NOT

The 'misnomer' arises from misunderstanding the terms **polar function grapher** or **polar function plotter**. When we use these terms, we are NOT referring to graphing or plotting 'polar functions'. We are simply referring to **graphing** or **plotting functions** using the ** polar coordinate system**.

The word ** polar** in these phrases modifies the word "grapher" and more precisely, it modifies "function grapher", not "function". Therefore, the more explicit term is

It should be noted that given any *real valued* **function** of *real numbers*, say **y = f(x)**, can be graphed in *either* the **Cartesian coordinate system** or the **polar coordinate system**. For example, the graph of **f(x) = x** is a **line** in the Cartesian coordinate system and a **spiral** in the polar coordinate system (try graphing it using the **function grapher** in both coordinate systems).

Note that the choice of variables used to describe a function is a matter of convention. You can use **θ** and **r** instead of **x** and **y**, and write **r = f(θ)**. This doesn't change anything; the graph of **r = f(θ) = θ** is the same **line** in the Cartesian coordinate systems and the same **spiral** in the polar coordinate system as they were for **y = f(x) = x**. You just need to rename the axes correspondingly. For example, in the **Cartesian** case you can call the *independent axis* as either the **x-axis** or **θ-axis**, and the *dependent axis* as either the **y-axis** or **r-axis**, depending on your choice of variables.

So, we know that any given **function** has both a **Cartesian graph** and a **polar graph** when plotted in their respective **Cartesian** and **polar coordinate systems**. However, it is a common *misconception* to categorize functions as either "Cartesian" or "polar". This categorization is erroneous and undermines the fundamental concept of a function. It can lead to confusion and difficulty in understanding the significance of graphically representing the same function in different coordinate systems.

To summarize, it is important to remember that the terms **Cartesian** and **polar** only refer to the type of **graph** or the **coordinate system** used to represent the function graphically, not the function itself.
In other words, a function is not Cartesian or polar; it is simply a function. The term "polar functions" is a *misnomer* and should be avoided.