To plot the point in polar coordinates, we are given $r = 4$ and $\theta = \frac{5\pi}{4}$.

1. Plot the point in polar coordinates:

• The point is at a distance of 4 from the origin and at an angle of $\frac{5\pi}{4}$.
• The angle $\frac{5\pi}{4}$ is in the third quadrant (because $\frac{5\pi}{4}$ is between $\pi$ and $\frac{3\pi}{2}$).

2. Find three additional polar representations for the point, keeping $-2\pi < \theta < 2\pi$:

To find multiple polar representations of the same point, we use the following properties:

• A point in polar coordinates $(r, \theta)$ can also be represented by $(-r, \theta + \pi)$, because the point will be diametrically opposite to the original point.
• You can add or subtract $2\pi$ from $\theta$ to find other representations without changing the location of the point.

Step-by-step transformations:

1. Original point:

   • $(r, \theta) = (4, \frac{5\pi}{4})$.

2. First alternative:

   • Add $2\pi$ to $\theta$: $\theta = \frac{5\pi}{4} + 2\pi = \frac{5\pi}{4} + \frac{8\pi}{4} = \frac{13\pi}{4}.$ But this doesn't satisfy $-2\pi < \theta < 2\pi$. To get a valid representation, subtract $2\pi$ from $\frac{13\pi}{4}$: $\theta = \frac{13\pi}{4} - 2\pi = \frac{13\pi}{4} - \frac{8\pi}{4} = \frac{5\pi}{4}.$ So, this is the same as the original point.

3. Second alternative:

   • Using $(-r, \theta + \pi)$, the new point is: $(-4, \frac{5\pi}{4} + \pi) = (-4, \frac{5\pi}{4} + \frac{4\pi}{4}) = (-4, \frac{9\pi}{4}).$ We need to reduce $\frac{9\pi}{4}$ within the range $-2\pi < \theta < 2\pi$: $\frac{9\pi}{4} - 2\pi = \frac{9\pi}{4} - \frac{8\pi}{4} = \frac{\pi}{4}.$ So, one alternative is $(-4, \frac{\pi}{4})$.

4. Third alternative:

   • Another way is to subtract $2\pi$ from $\theta$: $\theta = \frac{5\pi}{4} - 2\pi = \frac{5\pi}{4} - \frac{8\pi}{4} = -\frac{3\pi}{4}.$ So, another representation is $(4, -\frac{3\pi}{4})$.

5. Fourth alternative:

   • You can subtract $\pi$ to obtain another valid representation: $\theta = -\frac{3\pi}{4} - \pi = -\frac{3\pi}{4} - \frac{4\pi}{4} = -\frac{7\pi}{4}.$ And so, add $2\pi$ to get back to the valid range: $-\frac{7\pi}{4} + 2\pi = -\frac{7\pi}{4} + \frac{8\pi}{4} = \frac{\pi}{4}.$

The final representations (from smallest to largest by r-value, then by θ-value):

1. $(-4, \frac{\pi}{4})$
2. $(4, -\frac{3\pi}{4})$
3. $(4, \frac{5\pi}{4})$
4. $(4, \frac{9\pi}{4})$

Let me know if you need more details or have any questions!