Polar Functions and Graphs
Lessons in this chapter
About Polar Functions and Graphs
Polar Functions and Graphs introduces an entirely new coordinate system built from a distance and an angle instead of x and y.
Polar Coordinates and Conversions introduces the polar coordinate system and the formulas for converting between polar and rectangular coordinates.
Graphing Polar Functions and Families of Polar Curves cover graphing individual polar equations and recognizing entire families of curves (like roses and cardioids) from their equation’s form.
Rates of Change in Polar Functions closes the chapter by measuring how a polar curve changes, extending the rate-of-change ideas from earlier chapters to this new coordinate system.
The four lessons, Polar Coordinates and Conversions, Graphing Polar Functions, Families of Polar Curves, and Rates of Change in Polar Functions, each open as a worksheet worked in the page. The conversion problems are scored as you answer them, which is how those formulas stop needing to be looked up. Free to try, with no login needed for your first 3.
Frequently asked questions
Polar Coordinates and Conversions locates a point by its distance from the origin and its angle instead of by x and y, and converts between the two systems. Graphing Polar Functions plots curves point by point as the angle sweeps round, Families of Polar Curves covers the rose, cardioid, and limaçon shapes and where each one comes from, and Rates of Change in Polar Functions asks how the distance changes as the angle does. Polar work goes beyond the high school Common Core, which reaches only as far as Functions (F) and the complex number work in Number and Quantity (N); it sits in Unit 3 of the AP Precalculus course framework, Trigonometric and Polar Functions.
Polar coordinates describe a point by its distance from the origin (r) and the angle from the positive x-axis (θ), instead of horizontal and vertical distances.
Use x = r·cos(θ) and y = r·sin(θ) to convert from polar to rectangular, and r² = x² + y² with tan(θ) = y/x to convert from rectangular to polar.
Plot points by testing several angle values, calculating r for each, and tracing the resulting curve: special attention is needed wherever r becomes zero or negative.
A rose curve has petal-shaped loops, a cardioid is heart-shaped, and a limaçon is a more general looped or dimpled curve: each family's shape comes directly from its equation's specific form.
The rate of change of a polar curve measures how the distance r changes as the angle θ changes, extending the average-rate-of-change idea to this new coordinate system.