In Exercises 27–32, select the representations that do not change the location of the given point. (−5, − π/4) (−5, 7π/4)

Verified step by step guidance

1. Understand that the point is given in polar coordinates as \((r, \theta)\), where \(r\) is the radius (distance from the origin) and \(\theta\) is the angle measured from the positive x-axis.
2. Recall that changing the angle \(\theta\) by adding or subtracting full rotations of \(2\pi\) radians (i.e., \(\theta + 2k\pi\), where \(k\) is an integer) does not change the location of the point because angles are periodic with period \(2\pi\).
3. Note that changing the radius \(r\) to its negative value \(-r\) and adding \(\pi\) to the angle \(\theta\) (i.e., \((-r, \theta + \pi)\)) also represents the same point, because moving in the opposite direction by \(\pi\) radians with a negative radius points to the same location.
4. Apply these principles to the given points: for \(( -5, -\frac{\pi}{4} )\), consider if changing the angle by \(2\pi\) or adding \(\pi\) to the angle and negating the radius results in the same point; similarly, analyze \(( -5, \frac{7\pi}{4} )\) under these transformations.
5. Summarize which representations keep the point's location unchanged by verifying if the transformed coordinates correspond to the same position in the plane using the above rules.

Key Concepts

Polar Coordinates and Their Representation

Polar coordinates represent points in the plane using a radius and an angle, written as (r, θ). The radius r is the distance from the origin, and θ is the angle measured from the positive x-axis. Understanding how points are plotted in polar form is essential to analyze transformations that affect their location.

Equivalent Polar Coordinates

A single point in polar coordinates can have multiple representations due to periodicity of angles and sign changes in radius. For example, (r, θ) is equivalent to (−r, θ + π) and (r, θ + 2πk) for any integer k. Recognizing these equivalences helps identify which transformations preserve the point's location.

Effect of Angle and Radius Transformations on Point Location

Changing the angle by adding multiples of 2π or adjusting the radius sign with a corresponding angle shift can produce the same point. However, other transformations may move the point. Understanding how these changes affect the point's position is key to selecting representations that do not alter its location.