5 min read•june 18, 2024
Kashvi Panjolia
Kashvi Panjolia
Similar to graphing functions in the Cartesian plane, you can on the polar plane. Remember that the polar plane represents points in the plane based on their distance from the origin (0,0) and the angle they make with the positive x-axis.
Polar functions are equations written in the form r = f(θ), where r is the from the origin and θ is the angle. Their graphs consist of input-output pairs of values where the input values are and the output values are radial distances.
In the , the equation y = x creates a line that passes through the origin (0, 0) and has a of 1. This means that for every unit increase in x, there is a corresponding increase of the same amount in y.
In the , however, the equation r = θ creates a that starts at the origin and spirals outward. In polar coordinates, r is the distance from the origin and θ is the angle relative to the positive x-axis. As θ increases, r also increases, causing the points to spiral outward away from the origin.
It is possible to shift or scale this function, but you would still be graphing a spiral. That's why, in order to graph different kinds of functions, are used inside polar functions. For example, the graph of creates a circle touching the origin.
The process for is very similar to the process for graphing any function. It may seem a little tedious, but it is essential you follow this process for accurate and thorough graphs. These are the steps for graphing a polar function:
Here is an example question to help you understand how to graph a polar function:
Graph the polar function r = 2cosθ from θ=0 to θ=𝛑.
We will make a table of values for θ and r from θ=0 to θ=𝛑, incrementing θ by 𝛑/6 radians for accuracy. To do this, we substitute our θ values into r = 2cosθ and simplify to find r for each value of θ:
θ (radians) | r = 2cosθ |
0 | 2 |
𝛑/6 | √3 |
𝛑/3 | 1 |
𝛑/2 | 0 |
2𝛑/3 | -1 |
5𝛑/6 | -√3 |
𝛑 | -2 |
Using our table, we will plot these coordinate points on the polar plane and connect our points to obtain a smooth curve. The graph of the function r=2cosθ from θ=0 to θ=𝛑 is shown below.
This graph is a circle touching the origin because of an important property of the polar plane. One point can be represented by multiple sets of coordinates, so (2, 0) and (-2, 𝛑) are referencing the same point (remember that in the polar plane, coordinates are written using the form (r, θ)).
The polar plane is different from the Cartesian plane because the changes in input values (θ) correspond to changes in the angle measure from the positive x-axis, and changes in output values (r) correspond to changes in distance from the origin. In the Cartesian plane, the changes in the input values (x) led to changes in the horizontal distance from the origin, and the changes in output values (y) led to changes in the vertical distance from the origin. Be sure to practice graphing more polar functions so you can wrap your head around this concept.
One of the key features of polar function graphs is symmetry. A polar function is said to be symmetric about the origin if its graph is unchanged when reflected about the origin. This means that the graph of a polar function is symmetric if it looks the same when rotated by 180°.
Another important feature of polar function graphs is periodicity. A polar function is said to be periodic if its graph repeats after a fixed interval of θ. This means that if you start at a point on the graph and rotate the graph by a fixed angle, you will end up back at the same point on the graph. This feature was demonstrated in the example above. After 𝛑 radians, the function mapped to the same point on the polar plane as it did when 0 radians was substituted as the angle.