Rose graphs and other polar graphs

A rose graph is a type of polar graph that represents equations of the form $$r = a \times \text{sin}(n\theta)$$ or $$r = a \times \text{cos}(n\theta)$$ , where 'a' determines the length of the petals and 'n' determines the number of petals.

For the equation $$r = a \times \text{sin}(n\theta)$$ or $$r = a \times \text{cos}(n\theta)$$ , if 'n' is odd, the graph has 'n' petals. If 'n' is even, the graph has '2n' petals.

The parameter 'a' in the equation $$r = a \times \text{sin}(n\theta)$$ or $$r = a \times \text{cos}(n\theta)$$ determines the maximum length of each petal.

The rose graph $$r = a \times \text{sin}(n\theta)$$ is symmetric about the y-axis if 'n' is even and symmetric about the origin if 'n' is odd.

The rose graph $$r = a \times \text{cos}(n\theta)$$ is symmetric about the x-axis if 'n' is even and symmetric about the origin if 'n' is odd.

A polar coordinate system is a two-dimensional coordinate system where each point is determined by a distance from a reference point (the pole) and an angle from a reference direction.

The equation of a circle centered at the pole with radius 'r' is given by $$r = a$$ , where 'a' is the radius.

The equation of a spiral of Archimedes is given by $$r = a + b\theta$$ , where 'a' and 'b' are constants.

The equation of a logarithmic spiral is given by $$r = ae^{b\theta}$$ , where 'a' and 'b' are constants.

Sine rose graphs (e.g., $$r = a \times \text{sin}(n\theta)$$ ) start at the origin, while cosine rose graphs (e.g., $$r = a \times \text{cos}(n\theta)$$ ) start at the maximum value along the x-axis.