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Magazine Chapter 10: Q. C (page 777)
When is an integer, the polar graph of is a rose. (a) How many petals does the rose have when k is an integer? (b) What can you say about the symmetries of either when k is rational? Use a graphing calculator or a computer algebra system to graph several cases before answering the question. (c) What can you say about the polar graph ofrole="math" localid="1663157118656" when k is irrational?
Short Answer
(a). There are 2k and k number of rose petals when k is an even and odd integer.
(b). The graph is symmetric when k is an integer.
(c). The polar graph has infinite number of petals when k is irrational
Step by step solution
Given information
Part (a) Step 1: Number of rose petals when k is an integer.
The graph of the given curves is,
It is observed that when k is even, there are 2k petals on the rose, and when k is odd, there are only k petals. This pattern emerges if k is any integer, including if k was negative.
Part (b) Step 1: Symmetry
For the graph is,
When is odd there will be symmetry.
For
When is even there are symmetry
Graph when k is an irrational number.
When is an irrational number, then the graph is as below
The graph for different irrational numbers:
There are infinite numbers of petals when k is an irrational numbers.
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