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Understanding the concept of radial distance is crucial for solving polar coordinate problems. In polar coordinates, the radial distance, often denoted as \( r \), is the distance from the origin (also called the pole) to a point in the plane. Think of it as how far you need to travel to get from the center to a specific point.
For example, if \( r = 3 \), you would move three units away from the origin in the direction specified by the angle \( \theta \).
In our exercise, the radial distance is given as 5, but it has a negative sign. So instead of moving 5 units away in the direction of \( \theta \), you move 5 units in the opposite direction. This simple understanding helps you visualize and sketch the graph accurately.

The idea of a negative radius can seem confusing at first, but it is quite simple once understood. A negative radius in polar coordinates means that instead of plotting a point in the direction of the angle \( \theta \), you plot it in the opposite direction.

For example:

• If \( \theta = 0 \) and \( r = -3 \), instead of moving 3 units to the right (along the positive x-axis), you move 3 units to the left.
• If \( \theta = \frac{\pi}{4} \) and \( r = -5 \), plot the point 5 units away in the direction at angle \( \theta + \pi \).

This concept helps in visualizing more complex graphs such as the one in the exercise, where \( r = -5 \) simply tells us to draw a circle of radius 5, but think about the direction opposite to what the angle suggests.