Question

102–104. Spirals Graph the following spirals. Indicate the direction in which the spiral is generated as θ increases, where θ\u003e0. Let a=1 and a=−1. Spiral of Archimedes: r = aθ

Accepted Answer

Understand the given spiral equation: the Spiral of Archimedes is defined by the polar equation $$r = a	heta$$, where $$r is $$the radius (distance from the origin) and $$	heta is $$the angle in radians. 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 $$a. $$For each value of $$a$$, plot points by choosing several values of $$	heta \u003e 0 ($$for example, $$	heta = 0, \frac{\pi}{4}, \frac{\pi}{2}, \pi, 2\pi$$, etc.), then calculate the corresponding $$r$$ using $$r = a	heta. $$Convert each polar coordinate $$(r, 	heta) to $$Cartesian coordinates using the formulas $$x = r \cos(	heta)$$ and $$y = r \sin(	heta) to $$help with graphing on the Cartesian plane. Determine the direction of the spiral as $$	heta$$ increases: since $$r$$ increases linearly with $$	heta$$ 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.

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

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Key Concepts

Polar Coordinates

Spiral of Archimedes

Direction of Spiral Generation

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