How a polar curve moves as the angle grows

Find the sign and size of $r$ on the interval. For $0<\theta <\frac{\pi }{2}$, $\sin \theta$ runs from 0 up to 1, so $$r=-1+\sin \theta$$ runs from $-1$ up to 0. It is negative the whole time.

Where are the points? A negative $r$ puts the point on the opposite side of the pole from the angle. The angles are in the first quadrant, so the points are in the third quadrant, below the horizontal axis. For example, at $\theta =\frac{\pi }{6}$, $r=-\frac{1}{2}$, so $x=r\cos \theta \approx -0.43$ and $y=r\sin \theta =-\frac{1}{4}$.

Toward or away from the pole? Distance from the origin is $|r|$, which shrinks from 1 to 0 as $\theta$ increases. So the points get closer to the origin.

Below the horizontal axis and getting closer: choice C.

Each choice

• A. The points do get closer, yet they sit below the axis, not above it. Plotting each point in the direction of $\theta$ without accounting for the negative $r$ puts them in quadrant I.
• B. Both parts go wrong here: the negative $r$ flips the points below the axis, and $r$ increasing from $-1$ to 0 means the distance $|r|$ is shrinking. Treating "r is increasing" as "farther away" produces this choice.
• C (correct). With $r$ negative the points sit opposite the first-quadrant angles, below the axis, and $|r|$ falls from 1 to 0.
• D. The location below the axis is right, but $r$ climbing from $-1$ toward 0 means $|r|$ is decreasing. The points approach the origin rather than moving away.