Answer
This equation describes a parabola that is concave upward, and has its vertex at $(0,0)$.
Work Step by Step
- Knowing that $rcos(\theta) = x$ and $rsin(\theta) = y$:
$r = tan(\theta)sec(\theta)$
$r = \frac {sin(\theta)}{cos(\theta)} \frac 1 {cos(\theta)}$
$r = \frac {sin(\theta)}{cos^2(\theta)}$
$rcos^2(\theta) = sin(\theta)$
- Multiply both sides by $r$:
$r^2cos^2(\theta) = rsin(\theta)$
$x^2 = y$
- This is a simple quadratic equation, so it describes a parabola.
- Since the coefficient of $x^2$ is positive (1), the parabola is concave upward.
- The parabola has its vertex at $(0,0)$, since that is the point with the lowest $y$ value.
