9–13. Graph the points with the following polar coordinates. Give two alternative representations of the points in polar coordinates.


(-4, 3π/2)

Verified step by step guidance

1. Recall that a point in polar coordinates is given as \((r, \theta)\), where \(r\) is the radius (distance from the origin) and \(\theta\) is the angle measured from the positive x-axis.
2. Plot the point \((-4, \frac{3\pi}{2})\) by first considering the angle \(\frac{3\pi}{2}\), which corresponds to the downward direction along the negative y-axis.
3. Since \(r\) is negative, move in the opposite direction of the angle \(\frac{3\pi}{2}\). This means you move 4 units upward (opposite to downward) along the positive y-axis.
4. To find two alternative representations, use the fact that adding \(2\pi\) to the angle does not change the point, and changing the sign of \(r\) while adding \(\pi\) to the angle gives the same point. So, the alternatives are:
5. 1) \((r, \theta + 2\pi) = (-4, \frac{3\pi}{2} + 2\pi)\) and 2) \((-r, \theta + \pi) = (4, \frac{3\pi}{2} + \pi)\). Write these explicitly as your alternative polar coordinates.

Key Concepts

Polar Coordinates System

Polar coordinates represent points in a plane using a radius and an angle, denoted as (r, θ). The radius r is the distance from the origin, and θ is the angle measured from the positive x-axis. This system is useful for describing locations in circular or rotational contexts.

Graphing Points in Polar Coordinates

To graph a point (r, θ), start at the origin, rotate counterclockwise by angle θ, then move outward (or inward if r is negative) by distance r. Negative radius values mean moving in the opposite direction of the angle, which affects the point's position on the plane.

Alternative Representations of Polar Coordinates

A single point can have multiple polar coordinate representations by adding or subtracting full rotations (2π) to the angle or by changing the sign of the radius and adjusting the angle by π. For example, (r, θ) is equivalent to (-r, θ + π) and (r, θ + 2πk) for any integer k.