Rose

Definition

A rose is a type of polar graph that appears as a petal-like pattern. It is represented by the equation $r = a \cos(k\theta)$ or $r = a \sin(k\theta)$, where $a$ and $k$ are constants.

5 Must Know Facts For Your Next Test

1. The number of petals in a rose graph depends on the value of $k$. If $k$ is even, the graph will have $2k$ petals; if odd, it will have $k$ petals.
2. The amplitude 'a' determines the length of each petal in the rose graph.
3. Roses are symmetric about the pole (origin) and can be symmetric with respect to both the x-axis and y-axis depending on whether cosine or sine functions are used.
4. For the equation $r = a \cos(k\theta)$, if $k$ is an integer, there will be rotational symmetry every $\pi/k$ radians.
5. The petals of a rose curve intersect at the origin when using polar coordinates.

Review Questions

• How does the value of 'k' in the equation of a rose affect the number of petals?
• What role does 'a' play in determining the appearance of a rose graph?
• Explain why roses exhibit symmetry based on their trigonometric function.

Related terms

Polar Coordinates: A coordinate system where each point is determined by an angle and distance from a reference point (the pole).

Parametric Equations: Equations that express coordinates as continuous functions of one or more parameters, often used to describe curves.

$Lemniscate$: $A figure-eight shaped curve defined by an equation like $(r^2=a^2\cos(2\theta))$, often studied in polar coordinates.$