Lab
Polar Grapher
The first graph is the polar graph of r = f(θ). The second is the same function on rectangular axes, with θ across and r up, so where it dips below the axis r is negative and the point sits on the ray opposite θ's terminal ray. Move θ and watch the point move on both. Before you move it into a new stretch, say whether the point will move toward the pole or away from it, then check. The two interval sliders pick the part of the graph that is drawn bold and the interval for the average rate of change.
Challenges
- With r = 1 + 2 cos θ, set θ to 5π/6. Is r positive or negative? Which quadrant is the point in, and why is it not on the terminal ray of 5π/6?
- With r = 1 + 2 cos θ, find the two angles between 0 and 2π where the curve passes through the pole. What does the rectangular graph do at those angles?
- Set θ to π/3 with r = 4 cos θ. Write the point as (r, θ), as (x, y), and as another polar pair with r negative.
- Choose r = 3 cos(3θ) and set θ to π/3. r is negative, so where is the point? Find the angle between π and 2π where the curve reaches the same point with r positive, and say why the dashed stretches of the polar graph lie on top of the solid ones.
- Choose r = 3 sin(2θ) and set the interval from 0 to π/2. On which part does the point move away from the pole, and at which angle is it farthest from the pole?
- With r = 3 sin(2θ), set the interval from π/2 to π, where r is negative. On which part is r decreasing, and does the point move toward the pole or away from it there?
- With r = 2 + 2 cos θ, find the average rate of change of r from 0 to π/2. Use it to estimate r at π/6, then compare the estimate with the value of r the lab gives.