Let's analyze the polar coordinates and the corresponding graphs step-by-step to determine the correct one.

The two polar coordinates given are:

• $\left(3, \frac{\pi}{3}\right)$
• $\left(-2, -\frac{\pi}{2}\right)$

1. First point: $\left(3, \frac{\pi}{3}\right)$

• r = 3: The point is at a distance of 3 units from the origin.
• $\theta = \frac{\pi}{3}$: This angle is $60^\circ$ counterclockwise from the positive x-axis.
  • Therefore, the point should be located in the first quadrant, with a distance of 3 units along the direction of this angle.

2. Second point: $\left(-2, -\frac{\pi}{2}\right)$

• r = -2: The negative sign means the point is in the opposite direction of the angle.
• $\theta = -\frac{\pi}{2}$: This angle is $-90^\circ$, which points downward along the negative y-axis.
  • Since the radius is negative, the point will be 2 units upward along the positive y-axis, opposite to the given angle's direction.

Now, examining the options:

• Graph A: This graph places the second point on the negative x-axis, which is incorrect.
• Graph B: The first point is in the first quadrant at an angle of $\frac{\pi}{3}$ and 3 units away, and the second point is on the positive y-axis, 2 units up. This matches the description.
• Graph C: The placement of the second point is incorrect, as it's shown on the negative y-axis.
• Graph D: The positions of both points seem to be swapped, which is incorrect.

Conclusion:

The correct graph is B.

Would you like to explore alternative coordinate pairs or dive deeper into the topic?

Here are 5 related questions:

1. How can polar coordinates be converted to Cartesian coordinates?
2. What are the characteristics of points with negative radius values in polar coordinates?
3. How can angles greater than $2\pi$ or negative angles be normalized in polar coordinates?
4. What are the typical applications of polar coordinates in physics or engineering?
5. How would you convert Cartesian coordinates back to polar coordinates?

Tip: Remember that a negative radius reverses the direction of the point from the given angle.