Trigonometric and Polar Functions

Approaches zero

Approaches infinity

Cyclically fluctuates between growth and decline

Stays constant at 200 individuals

0 radians (or 0 degrees)

π/2 radians (or 90 degrees)

3π/2 radians (or 270 degrees)

π radians (or 180 degrees)

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

$0$

$-1$

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

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.

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 $\frac{1}{2}|a-b\cos(\theta)| d\theta$

0 seconds

2.04 seconds

10 seconds

4.08 seconds

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 $\theta$.

$\frac{dy}{dx}$

$\Delta y / \Delta x$

$\frac{d}{dt}$

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

Factor it as a perfect square trinomial.

Apply the quadratic formula.

Expand using FOIL method.

Differentiate using power rule.

There are no breaks or gaps in the values r(theta) takes

Every radian value corresponds to exactly one radius

The derivative r'(theta) exists at every angle

The function's period matches 360 degrees