The Cartesian Plane and Plotting Points

Overview

The Cartesian plane is a two-dimensional plane defined by a horizontal axis (x-axis) and a vertical axis (y-axis). Points on the plane are represented by coordinates (x, y), where 'x' specifies the position along the horizontal axis and 'y' specifies the position along the vertical axis.

Understanding the Quadrants

The Cartesian plane is divided into four quadrants:

  • Quadrant I: Both x and y values are positive (top-right).
  • Quadrant II: x is negative and y is positive (top-left).
  • Quadrant III: Both x and y values are negative (bottom-left).
  • Quadrant IV: x is positive and y is negative (bottom-right).

Plotting Points

To plot a point, locate the x-coordinate on the x-axis and the y-coordinate on the y-axis. The point is where these two coordinates meet. For example, to plot the point (3, 2):

  • Locate 3 on the x-axis (move 3 units to the right of the origin).
  • Locate 2 on the y-axis (move 2 units up from the origin).
  • The point (3, 2) is where these positions intersect.

Practice Problems

  1. Identify the quadrant for each point:
    • (4, 5):
      Solution

      Quadrant I

    • (-3, 2):
      Solution

      Quadrant II

    • (-1, -4):
      Solution

      Quadrant III

    • (5, -6):
      Solution

      Quadrant IV

  2. Plot the following points on a Cartesian plane:
    • (2, 3)
    • (-4, 1)
    • (-3, -5)
    • (6, -2)

    Sketch a graph and plot these points to practice locating each quadrant.

  3. Determine the coordinates of each point plotted in this diagram:
    Example Solution

    If the point is 3 units to the right of the origin and 4 units up, the coordinates are (3, 4).

  4. If point A is located at (-2, 4) and point B is located at (5, -3), calculate the distance between A and B using the distance formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
    Solution

    Using the coordinates for A (-2, 4) and B (5, -3):

    \[ d = \sqrt{(5 - (-2))^2 + (-3 - 4)^2} = \sqrt{7^2 + (-7)^2} = \sqrt{49 + 49} = \sqrt{98} \approx 9.9 \]

  5. Write the coordinates of a point in Quadrant III that is 5 units away from the origin. (Hint: Remember both coordinates must be negative.)
    Solution

    A point could be (-3, -4) or any other point where the distance formula from (0, 0) equals 5 units.

Additional Practice

  1. Given the point (-4, -3), determine:
    • The x-coordinate: -4
    • The y-coordinate: -3
  2. For the points (1, 2), (3, -2), (-1, -1), and (-3, 3), identify which quadrant each point belongs to:
    • (1, 2):
      Solution

      Quadrant I

    • (3, -2):
      Solution

      Quadrant IV

    • (-1, -1):
      Solution

      Quadrant III

    • (-3, 3):
      Solution

      Quadrant II

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