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Chapter 11: Problem 7

Explain three symmetries in polar graphs and how they are detected in equations.

Short Answer

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Short answer: Three symmetries in polar graphs are symmetry about the polar axis (horizontal symmetry), symmetry about the line 胃 = 蟺/2 (vertical symmetry), and symmetry about the origin (rotational symmetry). To detect each symmetry type: 1. For symmetry about the polar axis, replace 胃 with -胃 in the equation of the polar graph. If the equation remains the same, it has horizontal symmetry. 2. For symmetry about the vertical line of the form 胃 = 蟺/2, substitute 胃 with 蟺 - 胃 in the equation. If the resulting equation is the same as the original equation, the graph has vertical symmetry. 3. For symmetry about the origin, replace 胃 with 胃 + 蟺 in the equation. If the resulting equation is the same as the original equation after simplification, the polar graph has rotational symmetry.

Step by step solution

01

Explain symmetry about the polar axis (horizontal symmetry)

A polar graph has symmetry about the polar axis if it remains unchanged when reflected across the horizontal x-axis. This means that the graph will look the same when the angle 胃 is replaced with -胃. This type of symmetry can be detected in the equation if replacing 胃 with -胃 yields the same equation. For example, the equation r = 2cos(胃) has symmetry about the polar axis because if we replace 胃 with -胃, the equation still remains r = 2cos(胃).
02

Explain symmetry about the line 胃 = 蟺/2 (vertical symmetry)

A polar graph has symmetry about the line 胃 = 蟺/2 if it remains unchanged when the angle 胃 is replaced with 蟺 - 胃. This means that the graph will look the same if it is reflected across the vertical y-axis. To detect this symmetry in the equation of a polar graph, replace 胃 with 蟺 - 胃 and check if the resulting equation is the same as the original equation. For example, the equation r = 2sin(胃) has symmetry about the line 胃 = 蟺/2 because r = 2sin(蟺 - 胃) still gives us r = 2sin(胃).
03

Explain symmetry about the origin (rotational symmetry)

A polar graph has symmetry about the origin if it remains unchanged when the curve is rotated by 180 degrees or when 胃 is replaced by 胃 + 蟺. This type of symmetry requires that the graph appears identical when reflected across both the x-axis and the y-axis. To detect this symmetry in the equation of a polar graph, replace 胃 with 胃 + 蟺 and check if the resulting equation is the same as the original equation after some simplification. For example, the equation r = 1 - cos(胃) has symmetry about the origin because r = 1 - cos(胃 + 蟺) gives us r = 1 + cos(胃), which simplifies to the original equation. By understanding these three types of polar symmetries and knowing how to detect them in equations, you can determine the overall shape and behavior of a polar graph without having to plot every point.

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