102–104. Spirals Graph the following spirals. Indicate the direction in which the spiral is generated as θ increases, where θ>0. Let a=1 and a=−1.


Spiral of Archimedes: r = aθ

Verified step by step guidance

1. Understand the given spiral equation: the Spiral of Archimedes is defined by the polar equation \(r = a\theta\), where \(r\) is the radius (distance from the origin) and \(\theta\) is the angle in radians.
2. Set the parameter \(a\) to the given values, first \(a = 1\) and then \(a = -1\), to see how the spiral changes with positive and negative values of \(a\).
3. For each value of \(a\), plot points by choosing several values of \(\theta > 0\) (for example, \(\theta = 0, \frac{\pi}{4}, \frac{\pi}{2}, \pi, 2\pi\), etc.), then calculate the corresponding \(r\) using \(r = a\theta\).
4. Convert each polar coordinate \((r, \theta)\) to Cartesian coordinates using the formulas \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\) to help with graphing on the Cartesian plane.
5. Determine the direction of the spiral as \(\theta\) increases: since \(r\) increases linearly with \(\theta\) for \(a=1\), the spiral moves outward counterclockwise; for \(a=-1\), \(r\) becomes negative, which reflects the point across the origin, causing the spiral to move outward in the opposite direction. Indicate these directions on your graph.

Key Concepts

Polar Coordinates

Polar coordinates represent points in the plane using a radius and an angle (r, θ), where r is the distance from the origin and θ is the angle from the positive x-axis. Understanding how to plot points in this system is essential for graphing spirals like the Spiral of Archimedes.

Spiral of Archimedes

The Spiral of Archimedes is defined by the equation r = aθ, where r increases linearly with θ. This spiral moves outward at a constant rate as θ increases, and the sign of a determines the direction in which the spiral expands.

Direction of Spiral Generation

As θ increases, the spiral is traced out in a specific direction, typically counterclockwise for positive θ. The sign of the parameter a affects whether the spiral expands outward or inward and influences the orientation of the curve.