CARTESIAN COORDINATE SYSTEM - Week 1 Notes

CARTESIAN COORDINATE PLANE: PARTS, TERMINOLOGY, AND FUNDAMENTALS

• What the Cartesian (rectangular) coordinate plane is
  • Imagine two straight number lines crossing each other at their zero points. One goes left-to-right (horizontal), and the other goes up-and-down (vertical).
  • This creates a flat surface (a "plane") that acts like a map, where you can find any spot using two numbers.
  • It's named after a smart French guy named René Descartes, who came up with this idea.
  • This map is divided into four main sections called "quadrants." They are numbered 1, 2, 3, and 4, starting from the top-right and going around counterclockwise.
• Axes and origin
  • X-axis: This is the horizontal (left-to-right) number line. It tells you how far left or right a point is.
  • Y-axis: This is the vertical (up-and-down) number line. It tells you how far up or down a point is.
  • Origin: This is the exact middle where the two lines cross. It's like the starting point of your map and is always written as $(0,0)$.
• Quadrants
  • Quadrant I: This is the top-right section. Here, both numbers for a point are positive (like $(+, ext{+})$).
  • Quadrant II: This is the top-left section. Here, the first number (x) is negative, and the second number (y) is positive (like $(-, ext{+})$).
  • Quadrant III: This is the bottom-left section. Here, both numbers (x and y) are negative (like $(-,-)$).
  • Quadrant IV: This is the bottom-right section. Here, the first number (x) is positive, and the second number (y) is negative (like $(+,-)$).
  • Important: If a point is directly on one of the number lines (axes), it's not in any quadrant.
• Key vocabulary (important words)
  • Quadrant: One of the four sections of the coordinate plane.
  • Ordered pair or coordinates: A set of two numbers, like $(x, y)$, that tells you the exact location of a point on the plane. The order matters!
  • Abscissa: This is just another name for the x-coordinate. It tells you the horizontal distance from the Y-axis.
  • Ordinate: This is another name for the y-coordinate. It tells you the vertical distance from the X-axis.
  • Cartesian Plane or Rectangular Coordinate Plane: The whole grid system made by the two crossing number lines.
  • X-axis: The horizontal number line.
  • Y-axis: The vertical number line.
  • Origin: The point $(0,0)$ where the X-axis and Y-axis meet.
• How to show a point
  • Every point on the plane is shown by an ordered pair $(x, y)$. The first number, $x$, tells you to move left or right. The second number, $y$, tells you to move up or down.
  • You always start at the origin $(0,0)$.
  • Example: To find $(1, 3)$, start at $(0,0)$. Move 1 step to the right (because 1 is positive). Then, move 3 steps up (because 3 is positive). This point is in Quadrant I.
  • Example 2: To find $(-2, 4)$, start at $(0,0)$. Move 2 steps to the left (because -2 is negative). Then, move 4 steps up (because 4 is positive). This point is in Quadrant II.
• Quick checks for special spots
  • If the $x$ number is $0$ (like $(0, 5)$), the point is on the Y-axis. It's not in a quadrant.
  • If the $y$ number is $0$ (like $(3, 0)$), the point is on the X-axis. It's not in a quadrant.
  • The origin $(0,0)$ is special; it's right where the axes cross and isn't in any quadrant.
• History note
  • René Descartes came up with this brilliant idea to connect math about shapes (geometry) with math about numbers and