Trigonometric and Polar Functions

Approaches zero

Approaches infinity

Cyclically fluctuates between growth and decline

Stays constant at 200 individuals

0 seconds

2.04 seconds

10 seconds

4.08 seconds

It would shift up by 1 unit.

It would shift to the right by $\pi/4$.

It would reflect over the polar axis.

It would stretch vertically by a factor of 2.

0 radians (or 0 degrees)

π/2 radians (or 90 degrees)

3π/2 radians (or 270 degrees)

π radians (or 180 degrees)

$-\frac{\sqrt{3}}{2}$

$0$

$-1$

$\frac{\sqrt{2}}{2}$

$\frac{dy}{dx}$

$\Delta y / \Delta x$

$\frac{d}{dt}$

$\frac{D^2 y}{dx^2}$

Integral from $0$ to $\pi$ of $\frac{1}{2}(a-b\cos(\theta)) d\theta$

Integral from $-\pi/2$ to $\pi/2$ of $|a-b\cos(\theta)| d\theta$

Integral from $\pi/20$ to $-\pi/20$ of $\frac{1}{2}(a-b\cos(\theta)) d\theta$

Integral from $-\pi/20$ to $\pi/20$ of $\frac{1}{2}|a-b\cos(\theta)| d\theta$

Continuous straight lines radiating out from the pole.

Isolated points scattered throughout all quadrants.

Perfect circles around each plotted point.

Loops or petals extending into quadrants opposite those indicated by $\theta$.

Absolute values eliminate all negative inputs forcing the graph to be entirely contained in the first quadrant.

Odd-values leads to a non-symmetrical appearance while even-valued angles display mirrored symmetry relative to the pole axis.

Even-valued angles will produce a positive feedback loop creating an infinite expansion radius.

The graph remains unchanged because absolute valuations do not affect linear relationships.

A vertical asymptote present on the graph.

An unbroken curve on the graph.

A sharp corner on the graph.

A gap between parts of the curve.