How to Graph Polar Coordinates with Negative Values

• When the angle is negative: Negative angles move in a clockwise direction.

  Visualizing simple and complex polar coordinates

  This figure shows an example point, D. To locate the polar coordinate point D at

  

  first locate the angle

  

  and then find the location of the radius, 1, on that line.

• When the radius is negative: When graphing a polar coordinate with a negative radius, you move from the pole in the direction opposite the given positive angle (on the same line as the given angle but in the direction opposite to the angle from the pole). For example, check out point F at

  

  in the figure.

  Some teachers prefer to teach their students to move right along the x- (polar) axis for positive numbers (radii) and left for negative. Then you do the rotation for the angle in a positive direction. You’ll get to the same spot with that method.

  For example, take a look at point F

  

  in the figure. Because the radius is negative, move along the left x-axis 1/2 of a unit. Then rotate the angle in the positive direction (counterclockwise) pi/3 radians. You should arrive at your destination, point F.

• When both the angle and radius are negative: To express a polar coordinate with a negative radius and a negative angle, locate the terminal side of the negative angle first and then move in the opposite direction to locate the radius. For example, point G in the figure has these characteristics at

  

• Positive radius, positive angle

• Positive radius, negative angle

• Negative radius, positive angle

• Negative radius, negative angle

can have three other polar coordinate representations with different combinations of signs for the radius and angle:

When polar graphing, you can change the coordinate of any point you’re given into polar coordinates that are easy to deal with (such as positive radius, positive angle).