This lesson invites students to consider the need for a new type of function in which the output values of the function repeat at regular intervals. None of the function types students have studied up to this point have this characteristic, outside of functions whose value never changes. The terms periodic function and period are introduced and linked to circular motion using the idea of the motion of the hands of a clock, a context that students will revisit throughout the unit (MP1). In a future lesson, students will expand their understanding of these terms as they learn about cosine, sine, and tangent as functions.
Students reason abstractly and quantitatively as they assign coordinate values to points on a clock hand (MP2). In the following lessons, students build on the foundational work of this lesson as they use trigonometry to identify coordinates of points on a circle and establish the Pythagorean Identity and the unit circle.
- Comprehend that periodic functions are ones with outputs that repeat at regular intervals.
- Use the Pythagorean Theorem to calculate coordinates of points on a circle centered on the origin.
Let’s think about moving in circles.
- I can use the Pythagorean Theorem to find coordinates of points on a circle centered at the origin.
- I understand that a periodic function is one with outputs that repeat at regular intervals.
The length of an interval at which a periodic function repeats. A function \(f\) has a period, \(p\), if \(f(x+p) = f(x)\) for all inputs \(x\).
A function whose values repeat at regular intervals. If \(f\) is a periodic function then there is a number \(p\), called the period, so that \(f(x + p) = f(x)\) for all inputs \(x\).