Trigonometric and Polar Functions

Undefined coordinates

(1, 0)

(0, $\theta$) for any angle $\theta$

(r, $\pi/2$) where $r > 0$

Symmetry with respect to the horizontal axis

No symmetry at all

Symmetry with respect to the pole

Symmetry with respect to the line y = x

A straight line sloped downward.

A horizontal line crossing the y-axis.

A parabola that opens upward.

A parabola opening towards the right side of the polar grid.

An ellipse aligned along lines $\theta = \frac{\pi}{4}$ and $\theta = -\frac{3\pi}{4}$.

A circle centered at $\left( \frac{\pi}{4}, \frac{\pi}{4} \right)$.

A straight line through the pole making a 45-degree angle with the positive x-axis.

Plotting the graph and looking for the point where $r = 0$.

Performing polar-to-cartesian conversion to find coordinates at the origin.

Utilizing conic section techniques to determine the point of intersection.

Plotting the graph and locating the peak value of the r function.

Number of distinctive locations peak curve occurs multiplied by a factor of four

As many local maxima as there are integer multiples between the interval [0, m] inclusive

Sum total angles divisible by the number of petals minus one

Twice the number of points where the derivative function equals zero given the interval [0, m]

It has a vertical reflection across the x-axis.

It is symmetric with respect to the line $y = x$.

It has a horizontal reflection across the y-axis.

It is symmetric with respect to the origin.

The maximum value of a function.

The distance from the origin to a point.

The angle in radians from the positive x-axis.

The coordinate along the horizontal axis.

(0, $\pi/4$)

(2, $\pi/2$)

(1, $\pi$)

(-2, $-\pi/6$)

Plot rotates by $\pi$ radians, mirroring over initial petal distribution line through pole.

Each loop size decreases by half keeping total number constant.

Graph compresses radially toward pole while preserving angular symmetry.

The number of loops doubles without affecting their overall size or shape.