plot the point given in polar coordinates. $\\left(-5,-\\frac{4\\pi}{3}\\right)$ select the graph that…

Explanation:

Step1: Analyze the polar coordinate ((r,\theta)=(-5,-\frac{4\pi}{3}))

In polar coordinates, if (r < 0), we add (\pi) to the angle (\theta) and take the absolute - value of (r). So, (r'=\vert - 5\vert=5) and (\theta'=-\frac{4\pi}{3}+\pi=-\frac{\pi}{3})

Step2: Locate the angle (-\frac{\pi}{3})

The angle (-\frac{\pi}{3}) is measured clock - wise from the positive (x) - axis.

Step3: Locate the point with (r = 5) along the ray of (\theta=-\frac{\pi}{3})

Since (r = 5>0), we move 5 units from the origin along the ray corresponding to (\theta =-\frac{\pi}{3})

Based on the above analysis of polar - coordinate transformation ((r=-5,\theta =-\frac{4\pi}{3}) is equivalent to (r = 5,\theta=-\frac{\pi}{3})), we can match the graph. (Since we don't have the actual visual content of options A - D, but if we assume the standard polar - coordinate graphing rules: for a positive (r) and negative angle (-\frac{\pi}{3}), we move 5 units along the ray (\theta=-\frac{\pi}{3}))

Answer:

Assuming the graph that has a point 5 units from the origin along the ray (\theta =-\frac{\pi}{3}) (clock - wise (\frac{\pi}{3}) from the positive (x) - axis) is the correct one. If we assume option D is such a graph (based on standard polar - coordinate plotting after transformation), then the answer is D.