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To write down two different tests for symmetry with respect to the polar axis and find the examples.
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Explanation of Solution
Tests for symmetry:
Symmetry along the polar axis (x- axis)
In a polar equation, replaceby
If an equitant equation results, the graph has a symmetry along the polar axis.
Symmetry along the line (y- axis)
In a polar equation, replaceby
If an equivalent equation results, the graph has a symmetry along the line.
Symmetry along the pole(origin)
In a polar equation, replaceby
or
by
If an equivalent equation results, the graph is symmetric with respect to the pole.
Observe the tests for symmetry, an equation may fail these tests and still have a graph that has a symmetry along the pole origin, the line or the pole.
For example:
The graph of turns out to be symmetric with respect to the polar axis, the line
and the pole, but only the test for symmetry with respect to the pole (replace
by
works.
Chapter 9 Solutions
Precalculus
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