11.1: THE ORIGIN OF THE CARTESIAN COORDINATE SYSTEM
The year is 1630. Lying on his back, French mathematician René Descartes(pronounced "day-cart") watched a fly crawl across the ceiling. Suddenly, an idea came to him.
Descartes visualized two number lines intersecting at a 90° angle. He realized that he could determine the fly's location on a piece of paper. Descartes called the horizontal line the x-axis and the vertical line the y-axis. He named the point where the axes intersect at 90 degrees the origin. He then placed "tic marks" evenly spaced along the vertical and horizontal axes, and assigned positive and negative numbers to those tic marks.
Descartes decided to represent the fly's location as an ordered pair of numbers. The first number of the ordered pair, the x-value, is the horizontal distance along the x-axis, measured from the origin. The second number of the ordered pair, the y-value, is the vertical distance along the y-axis, also measured from the origin. The locations in the plane where the x and y values intersect are called coordinates. There is an x-coordinate and there is a y-coordinate which are grouped together in parentheses and seperated by a comma. The x-coordinate always comes first in this grouping and the y-coordinate always follows the x-coordinate in this grouping. This grouping of two numbers seperated by a comma and paced in parentheses is called an "ordered pair."
The Cartesian Coordinate System
The fly's position above is represented by the ordered pair of numbers (5, -4) made up of the x-coordinate, 5, and the y-coordinate, -4. You can say the fly is at a point represented by the ordered pair (5, -4). The plane above containing the ordered pair (5, -4) is called the "Cartesian Coordinate System," or the "Coordinate Plane."
The Cartesian Coordinate System is divided in 4 regions called "quadrants." The fly above is in quadrant 4. It is important that you are able to tell someone what quadrant in which a point sits. The graph below contains 9 points labeled with their respective ordered pairs. Once you read the rules on determing what quadrant a point sits in you will easily be able to determine what a quadrant a point is in.
|RULES FOR DETERMINING WHICH QUADRANT A POINT IS IN
RULE 1) Any point on the x or y axis is said to "not be in a quadrant."
ex. In the graph below, the points defined by the ordered pairs (0,4), (-6,0), (0,0), (4,0) and (0,-3) are not in any quadrant.
RULE 2) QUADRANT 1- Where the x and y coordinate are both positive
ex. In the graph below, the point defined by the ordered pair (3,2) is in quadrant 1
RULE 3) QUADRANT 2 - Where the x coordinate is negative and the y coordinate is positive
ex. In the graph below, the point defined by the ordered pair (-2, 3) is in quadrant 2
RULE 4) QUADRANT 3 - Where the x coordinate is negative and the y coordinate is negative
ex. In the graph below, the point defined by the ordered pair (-4, -2) is in quadrant 3
RULE 5) QUADRANT 4 - Where the x coordinate is positive and the y coordinate is negative
ex. In the graph below, the point defined by the ordered pair (5, -4) is in quadrant 4
|YOU MUST KNOW HOW TO USE THE FOLLOWING LANGUAGE
1) Cartesian Coordinate System or Coordinate Plane
2) vertical axis and horizontal axis
3) tic marks, also called hash marks
5) x-axis, positive x-axis, and negative x-axis
6) y-axis, positive y-axis, and negative y-axis
7) quadrant I, quadrant II, quadrant III, and quadrant IV
8) tic and hash marks
9) ordered pair, x-value, y-value, coordinate(s), x-coordinate, y-coordinate, and point
In the next chapter we will look more fully at the use of the above language and the Cartesian Coordinate System.
THE SUMMER STUDY GUIDE
RETURN TO THE SUMMER STUDY GUIDE MAIN PAGE
- CHAPTER 1: THE NUMBER SYSTEM
- CHAPTER 2: ORDER OF OPERATIONS
- CHAPTER 3: VARIABLES, MONOMIALS,
BINOMIALS, TRINOMIALS, POLYNOMIALS,
COEFFICIENTS, TERMS AND LIKE TERMS
- CHAPTER 4: SIGNED NUMBERS,
ABSOLUTE VALUE, AND INEQUALITY SYMBOLS
- CHAPTER 5: FACTORS, COMMON
FACTORS, LEAST COMMON FACTORS AND GREATEST COMMON FACTORS
- CHAPTER 6: PROPERTIES OF NUMBERS
- CHAPTER 7: THE WORLD OF FRACTIONS
- CHAPTER 8: EXPONENTS
- CHAPTER 9: ROOTS
- CHAPTER 10: ALGEBRAIC EXPRESSIONS
- CHAPTER 11: CARTESIAN COORDINATE SYSTEM
- CHAPTER 12: SETS, RELATIONS AND FUNCTIONS
- CHAPTER 13: AVERAGE RATE OF CHANGE OF Y WITH RESPECT TO X, SLOPE, PYTHAGOREAN THEOREM, AND DISTANCE FORMULA BETWEEN TWO POINTS
- CHAPTER 14: X-INTERCEPT(ZERO) AND Y INTERCEPT(B)
- CHAPTER 15: LINES
- CHAPTER 16: FUNCTIONS
- CHAPTER 17: MULTIPLYING POLYNOMIALS
- CHAPTER 18: FACTORING
- CHAPTER 19: RATIONAL EXPRESSIONS
- CHAPTER 20: SOLVING EQUATIONS
- CHAPTER 21:SOLVING INEQUALITIES
- CHAPTER 22: SOLVING A SYSTEM OF EQUATIONS
- CHAPTER 23: QUADRATICS
- CHAPTER 24: CIRCLES
- CHAPTER 25: AREAS AND PERIMETERS OF PLANE FIGURES
- CHAPTER 26: VOLUMES