Question

identify the point with the given polar coordinates.
\\(\left(-2,\frac{\pi}{4}\
ight)\\)

choose the correct graph below.
\\(\bigcirc\\) a. \\(\bigcirc\\) b. \\(\bigcirc\\) c. \\(\bigcirc\\) d.

Explanation:

Step1: Recall Polar Coordinate Rules

For a polar coordinate \((r, \theta)\), if \(r < 0\), we plot the point by moving in the direction opposite to \(\theta\) (i.e., \(\theta+\pi\)) with \(|r|\) distance. Here, \(r = - 2\), \(\theta=\frac{\pi}{4}\). So the direction is \(\frac{\pi}{4}+\pi=\frac{5\pi}{4}\), and distance from origin is \(| - 2| = 2\)? Wait, no, \(|r| = 2\)? Wait, no, \(r=-2\), so we go in the direction of \(\theta+\pi\) (since \(r\) is negative) with magnitude \(|r|\). So \(\theta+\pi=\frac{\pi}{4}+\pi=\frac{5\pi}{4}\), and \(|r| = 2\)? Wait, no, the distance from the origin is \(|r|\), but when \(r\) is negative, we reflect over the origin. So the point \((-2,\frac{\pi}{4})\) is equivalent to \((2,\frac{\pi}{4}+\pi)=(2,\frac{5\pi}{4})\). Now, \(\frac{5\pi}{4}\) is in the third quadrant (between \(\pi\) and \(\frac{3\pi}{2}\)). Let's check the graphs:

- Graph A: The point is in the first quadrant (direction \(\frac{\pi}{4}\)), \(r = 2\)? No, since our point should be in third quadrant.
- Graph B: The arrow is in third quadrant? Wait, no, the arrow direction: Wait, let's look at the red dots. Wait, the polar coordinate \((-2,\frac{\pi}{4})\): when \(r\) is negative, we go in the direction opposite to \(\frac{\pi}{4}\), i.e., \(\frac{\pi}{4}+\pi=\frac{5\pi}{4}\), and the distance from origin is \(| - 2| = 2\)? Wait, no, the magnitude is \(|r| = 2\), but the direction is \(\theta+\pi\). So the point should be 2 units from origin in the direction of \(\frac{5\pi}{4}\) (third quadrant). Let's check the graphs:

Graph C: The red dot is at \(r=-2\) (since the arrow is in the direction of \(\frac{5\pi}{4}\) (third quadrant) and the distance from origin: the grid lines, each grid is 1 unit? Wait, the red dot in graph C: let's see, the x - axis is from - 5 to 5, y - axis same. The red dot is at \(r=-2\) (since the arrow is in the direction of \(\frac{5\pi}{4}\), and the distance from origin: the dot is at \(r = - 2\) (so 2 units in the opposite direction of \(\frac{\pi}{4}\)). Wait, maybe better: the formula for converting negative \(r\) is \((r,\theta)=(-|r|,\theta)=(|r|,\theta+\pi)\). So \((-2,\frac{\pi}{4})=(2,\frac{5\pi}{4})\). Now, \(\frac{5\pi}{4}\) is 225 degrees, third quadrant. Let's check the graphs:

- Graph A: Point in first quadrant (45 degrees), so no.
- Graph B: The arrow is in third quadrant? Wait, the red dot in B: the arrow is going to third quadrant? No, the arrow in B is going to the left - down, but the red dot is near \(r = 2\)? Wait, no, maybe I messed up. Wait, \(r=-2\) means the point is 2 units from origin but in the direction opposite to \(\theta\). So \(\theta=\frac{\pi}{4}\) (first quadrant), opposite direction is third quadrant (\(\frac{5\pi}{4}\)). So the point should be 2 units from origin in third quadrant. Let's check the red dots:

Graph C: The red dot is at \(r=-2\) (since the arrow is in the direction of \(\frac{5\pi}{4}\), and the dot is 2 units from origin? Wait, the grid: each circle is 1 unit? So the red dot in C is at \(r=-2\) (since it's 2 units from origin in the direction of \(\frac{5\pi}{4}\)). Wait, maybe the correct graph is C? Wait, let's re - express:

For polar coordinates \((r,\theta)\), if \(r<0\), we plot the point by moving \(|r|\) units in the direction of \(\theta+\pi\). So \(r = - 2\), \(\theta=\frac{\pi}{4}\), so direction is \(\frac{\pi}{4}+\pi=\frac{5\pi}{4}\), \(|r| = 2\). So the point is 2 units from origin in the direction of \(\frac{5\pi}{4}\) (third quadrant). Now, looking at the graphs:

- Graph A: Point in first quadrant (direction \(\frac{\pi}{4}\)), \(r = 2\) → wrong.
- Graph B: The arrow is in third quadrant, but the red dot is near \(r = 2\)? No, maybe the red dot is at \(r = 2\) but direction wrong.
- Graph C: The red dot is at \(r=-2\) (since the arrow is in the direction of \(\frac{5\pi}{4}\), and the dot is 2 units from origin in that direction) → correct.
- Graph D: Point in third quadrant but \(r=-2\)? Wait, the red dot in D is at \(r=-2\)? No, the red dot in D is at \(r=-2\)? Wait, the grid: the red dot in D is at \(x=-2,y = 0\)? No, \(\frac{5\pi}{4}\) is not along the negative x - axis. \(\frac{5\pi}{4}\) is 225 degrees, so both x and y negative. So the red dot should be in the third quadrant, with both x and y negative, 2 units from origin (since \(|r| = 2\)). So graph C: the red dot is in the third quadrant, along the line \(\frac{5\pi}{4}\), 2 units from origin? Wait, maybe the correct graph is C.

Step2: Analyze Each Graph

- Graph A: Direction \(\frac{\pi}{4}\) (first quadrant), \(r = 2\) → incorrect (we need direction \(\frac{5\pi}{4}\)).
- Graph B: The arrow direction is not \(\frac{5\pi}{4}\) (maybe \(\frac{3\pi}{4}\) or something else), and the red dot position is not correct.
- Graph C: The red dot is in the third quadrant (direction \(\frac{5\pi}{4}\)), 2 units from origin (since \(|r| = 2\) and \(r=-2\) implies direction \(\theta+\pi\)) → correct.
- Graph D: The red dot is in the third quadrant but along the negative x - axis (direction \(\pi\)), not \(\frac{5\pi}{4}\) → incorrect.

Answer:

C