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End First, let’s take a look at…. End 2 A little history End 3 A little history • René Descartes (1596-1650) • philosopher • mathematician • joined algebra and geometry • credited with--Cartesian plane End 4 The year is 1630. Lying on his back, French mathematician René Descartes, watches a fly crawl across the ceiling. Suddenly, an idea comes to him. He visualizes two number lines, intersecting at a 90° angle. He realizes that he can graph the fly's location on a piece of paper. Descartes called the main horizontal line the x-axis and the main vertical line the y-axis. He named the point where they intersect the origin. End Descartes represented the fly's location as an ordered pair of numbers. The first number, the x-value, is the horizontal distance along the x-axis, measured from the origin. The second number, the y-value, is the vertical distance along the y-axis, also measured from the origin. End Now, let’s take a look at… End 7 Cartesian plane Formed by intersecting two real number lines at right angles End 8 Cartesian plane Horizontal axis is usually called the x-axis End 9 Cartesian plane Vertical axis is usually called the y-axis End 10 Cartesian plane Also called: • x-y plane • rectangular coordinate system End 11 Cartesian plane Divides into Four Quadrants and… II I III IV End 12 Cartesian plane The intersection of the two axes is called the origin End 13 Cartesian plane Math Alert The quadrants do not include the axes II I III IV End 14 Cartesian plane Math Alert A point on the x or y axis is not in a quadrant II I III IV End 15 Cartesian plane Each point in the x-y plane is associated with an ordered pair, (x,y) (x,y) (x,y) (x,y) (x,y) End 16 Cartesian plane (x,y) The x and y of the ordered pair, (x,y), are called its coordinates (x,y) (x,y) (x,y) End 17 Cartesian plane Math Alert There is an infinite amount of points in the Cartesian plane End 18 COORDINATE PLANE ORIGIN The plane determined by a horizontal number line, called the x-axis, and a vertical number line, called the y-axis, intersecting at a point called the origin. Each point in the coordinate plane can be specified by an ordered pair of numbers. The point (0, 0) on a coordinate plane, where the x-axis and the y-axis intersect. End Take note of these graphing basics End 20 Cartesian plane • Always start at (0,0)---every point “originates” at the origin End 21 Cartesian plane y • In plotting (x,y) ---remember the directions of both the x and y axis x End 22 Cartesian plane • (x,---) x-axis goes left and right End 23 Cartesian plane • (---,y) y-axis goes up and down End 24 Now, let’s look at graphing… (2,1) End 25 Cartesian plane (2,1) • Start at (0,0) + •( , ---) • Move right 2 (2,1) End 26 Cartesian plane (2,1) • (---, +) • (---, 1) • Move up 1 (2,1) End 27 Now, let’s look at graphing… (4, 2) End 28 Cartesian plane (4, 2) • Start at (0,0) • ( +, ---) • Move right 4 (4, 2) End 29 Cartesian plane (4, 2) • (---, - ) • (---, -2) • Move down 2 (4, 2) End 30 Cartesian plane (3,5) • Start at (0,0) (, ---) • Move left 3 (3,5) End 31 Cartesian plane (3,5) (3,5) • (---, +) • (---, 5) • Move up 5 End 32 Cartesian plane (0, 4) • Start at (0,0) • (none,---) • No move right or left (0, 4) End 33 Cartesian plane (0, 4) (0, 4) • (0, + ) • (---, 4) • Move up 4 End 34 Now, let’s look at graphing… (5,0) End 35 Cartesian plane (5,0) • Start at (0,0) • ( ,---) • Move left 5 (5,0) End 36 Cartesian plane (5,0) • ( ---, 0) • No move up or down (5,0) End 37 Points in To make it easy Quadrant 2 have to talk about negative x but where on the positive y coordinates. coordinate plane a point is, we divide the coordinate plane into four sections called quadrants. Points in Quadrant 3 have negative x and negative y coordinates. End Points in Quadrant 1 have positive x and positive y coordinates. Points in Quadrant 4 have positive x but negative y coordinates End Cartesian plane Directions: Approximate the coordinates of the point--Or what is the ‘(x,y)’of the point? End 40 Cartesian plane Directions: Approximate the coordinates of the point (2, 4) End 41 Cartesian plane Directions: Approximate the coordinates of the point End 42 Cartesian plane Directions: Approximate the coordinates of the point ( 4, 2) End 43 Cartesian plane Directions: Approximate the coordinates of the point End 44 Cartesian plane Directions: Approximate the coordinates of the point (0,3) End 45 Cartesian plane Directions: Approximate the coordinates of the point End 46 Cartesian plane Directions: Approximate the coordinates of the point (3, 3) End 47 Cartesian plane Directions: Approximate the coordinates of the point End 48 Cartesian plane Directions: Approximate the coordinates of the point (1, 6) End 49 Cartesian plane Directions: Approximate the coordinates of the point End 50 Cartesian plane Directions: Approximate the coordinates of the point (5, 0) End 51 Cartesian plane Directions: Find the coordinates of the point two units to the left of the y-axis and five units above the x-axis End 52 Cartesian plane Directions: Find the (2,5) coordinates of the point two units to the left of the y-axis and five units above the x-axis End 53 Cartesian plane Directions: Find the coordinates of the point on the x-axis and three units to the left of the y-axis End 54 Cartesian plane Directions: Find the coordinates of a point on the xaxis and three units to the left of the y-axis (3, 0) End 55 End 56