Answer
$$ r = \frac 8 {1 - sin(\theta)}$$
Work Step by Step
1. Determine the equation that we are going to use:
The vertex is at $(4, \frac{3\pi}{2})$
The angle $(\frac {3\pi} 2)$ indicates that the directrix is a horizontal line below the x-axis. $(y = -c)$
If we multiply 4 by 2, we will get the absolute value of $c$, which is 8.
Directrix: $y = -8$
Thus, we are going to use $sin(\theta)$ in the formula.
Since $-8$ is negative, the equation will have "$-sin(\theta)$":
$$r = \frac{ed}{1 - esin(\theta)}$$
2. Substitute the given values for $d$ (directrix) and e (eccentricity):
** Parabolas: e = 1
** d = 8 (absolute value)
$$r = \frac{(1)(8)}{1 - (1)sin(\theta)} = \frac 8 {1 - sin(\theta)}$$
