Lab
Parametric Grapher
The first graph is the path in the plane: the point (x(t), y(t)) for every t in the interval, joined in order of increasing t, with arrows for the direction of motion, a square at the start and a filled dot at the end. The two graphs under it are x against t and y against t, the two functions that build the path. Move t and watch the point move on all three. Before you move it into a new stretch, say from the graphs against t whether the point will move left or right and up or down, then check. These topics are in Unit 4, which is not assessed on the AP Exam.
Challenges
- Choose the circle or ellipse with h = 0, k = 0, a = 1, b = 1 and n = 1, and set the interval from 0 to 2π. Where does the point start and end, and which way does it go round?
- Keep that circle and change a to −1. The path is the same circle. Where does it start now, and which way is it traced?
- Put a back to 1 and set n = 2 on the same interval. How many times is the circle traced, and what interval of t traces it exactly once?
- Choose the parabola y = a(x − h)² + k as (t, f(t)). On its starting interval, 0 to 3, find the lowest point of the path and the value of t where the path crosses the x-axis. Find each one on the graph of y against t too.
- With the starting ellipse, x(t) = 3 cos t and y(t) = 2 sin t from 0 to π/2, find the average rates of change of x and of y by hand. Divide them to get the slope between the start and end points, then compare with the lab.
- Choose the path that crosses itself. The point is at the origin when t = 0 and again when t = π. Which way is it moving each time?
- Choose the segment from (−2, −1) to (3, 2). What are the rates of change of x and y with respect to t, and at what t is the point halfway along?
- Choose the hyperbola opening left and right, with a = 2 and b = 1. Move t and watch the readout with x(t) and y(t) substituted: why is the left side always 1? Then stretch the interval to include π/2 and say what happens to the path there.
- Choose the cubic, note its start and end points, then choose the inverse of the cubic. How are the start and end points of the two paths related, and why?