A student graphs r = 2cos(θ) and reaches θ = π, where r = 2cos(π) = −2. The student concludes this point doesn't exist since r can't be negative. What actually happens?
AThe point doesn't exist; negative r values indicate the curve terminates at that angle.
BThe point is plotted at distance 2 in the direction opposite to θ = π, which lands at the same location as (r = 2, θ = 0) on the positive x-axis.
CThe point is plotted at distance 2 in the direction of θ = π, on the negative x-axis.
DNegative r means the point is reflected across the y-axis to θ = 0 with r = −2.
In polar coordinates, a negative r means plot in the *opposite* direction from θ. At θ = π, the direction points to the left. A negative r flips you to the opposite direction — to the right — so (r = −2, θ = π) lands at the same point as (r = 2, θ = 0). This is why r = 2cos(θ) traces a complete circle as θ goes from 0 to π: the second half of the range (where cos is negative) retraces the same points already plotted in the first half, from the opposite direction.
Question 2 Multiple Choice
How many petals does the rose curve r = cos(4θ) have?
A4 petals, because the formula gives n petals when n is even.
B8 petals, because the formula gives 2n petals when n is even.
C2 petals, because the formula gives n petals when n is odd and 4 is close to 3.
D16 petals, because n is squared.
For rose curves r = cos(nθ), the rule is: n petals if n is odd, 2n petals if n is even. Since n = 4 is even, the curve has 2 × 4 = 8 petals. The extra petals appear because negative r values map the oscillations on one half of θ's range to the opposite direction, filling in petals that would otherwise be missing. For odd n, the negative-r petals coincide with positive-r petals already plotted, so you get n petals instead of 2n.
Question 3 True / False
The polar graph of r = 3 is a circle of radius 3 centered at the origin.
TTrue
FFalse
Answer: True
For every angle θ, the equation r = 3 specifies a point exactly 3 units from the origin. Rotating through all angles traces out all points at distance 3 — a circle of radius 3. This is the simplest polar curve and directly illustrates the radial nature of polar coordinates: holding r constant while θ varies produces a circle, whereas in Cartesian coordinates a circle requires the more complex equation x² + y² = 9.
Question 4 True / False
If replacing θ with π − θ leaves a polar equation unchanged, the graph is symmetric about the polar axis (the positive x-axis).
TTrue
FFalse
Answer: False
Symmetry about the polar axis (x-axis) corresponds to replacing θ with −θ and getting the same equation — this tests whether the graph is its own mirror image across the x-axis. Replacing θ with π − θ tests symmetry about the vertical line through the origin (the y-axis). For example, r = sin(θ) satisfies sin(π − θ) = sin(θ), so it is symmetric about the y-axis, not the x-axis.
Question 5 Short Answer
Explain what happens geometrically when r is negative in a polar equation. Where does the point (r, θ) get plotted when r < 0?
Think about your answer, then reveal below.
Model answer: When r is negative, the point is plotted in the direction exactly opposite to θ — that is, at angle θ + π — at a distance |r| from the origin. Geometrically, you face the direction of θ and then walk backwards by |r| units. For example, (r = −3, θ = π/4) lands at the same point as (r = 3, θ = π/4 + π) = (r = 3, θ = 5π/4), in the third quadrant.
This matters enormously for graphing polar curves. Many standard curves — including circles like r = 2cos(θ) and rose curves — rely on negative r values to complete their shape. Students who skip negative-r points or treat them as 'missing' will draw incomplete or incorrect curves. Understanding that negative r reverses the direction (equivalent to flipping by π radians) is the key to correctly interpreting what polar equations like r = cos(2θ) are actually tracing.