Symmetry test

Definition

A symmetry test determines whether a polar graph is symmetric with respect to the polar axis, the line $\theta = \frac{\pi}{2}$, or the origin. Symmetry helps simplify graphing and understanding of polar equations.

5 Must Know Facts For Your Next Test

1. To test for symmetry with respect to the polar axis, replace $\theta$ with $-\theta$ in the equation and see if it remains unchanged.
2. For symmetry with respect to the line $\theta = \frac{\pi}{2}$, replace $r \cdot e^{i\theta}$ with $r \cdot e^{i(\pi - \theta)}$ and check if it holds true.
3. To check for symmetry about the origin, replace $(r, \theta)$ with $(-r, \theta + \pi)$ in the equation.
4. Symmetry tests can reduce the amount of computation needed by identifying repeated patterns in polar graphs.
5. Identifying symmetry helps predict graph behavior without plotting all points.

Review Questions

• What substitution would you use to test for symmetry about the polar axis?
• How do you determine if a polar graph is symmetric with respect to the origin?
• Why is identifying symmetry useful when analyzing polar graphs?

Related terms

Polar Coordinates: A coordinate system where each point on a plane is determined by an angle and a distance from a reference point.

Polar Equation: An equation that expresses a relationship between the radius and angle in polar coordinates.

$r = f(\theta)$: $A common form of a polar equation where r is expressed as a function of theta.$