Sketching the Graph of a Rose Curve (n Even)

Testing for symmetry, we find again that the symmetry tests do not tell the whole story. The graph is not only symmetric with respect to the polar axis, but also with respect to the line $\theta =\frac{\pi }{2}$ and the pole.

Now we will find the zeros. First make the substitution $u=4\theta .$

The zero is $\theta =\frac{\pi }{8}.$ The point $\left(0,\frac{\pi }{8}\right)$ is on the curve.

Next, we find the maximum $\left|r\right|.$ We know that the maximum value of $\cos u=1$ when $\theta =0.$ Thus,

The point $\left(2,0\right)$ is on the curve.

The graph of the rose curve has unique properties, which are revealed in Table 8.

As $r=0$ when $\theta =\frac{\pi }{8},$ it makes sense to divide values in the table by $\frac{\pi }{8}$ units. A definite pattern emerges. Look at the range of r-values: 2, 0, −2, 0, 2, 0, −2, and so on. This represents the development of the curve one petal at a time. Starting at $r=0,$ each petal extends out a distance of $r=2,$ and then turns back to zero $2n$ times for a total of eight petals. See the graph in Figure 16.