STEP BY STEPS OF COORDINATE PLANE NOTES
I. Introduction
A. Definition of the coordinate plane
1. A coordinate plane is a two-dimensional surface that is formed by the intersection of two number lines - one running vertically (called the y-axis) and the other running horizontally (called the x-axis).
2. Also called the Cartesian plane, after French mathematician René Descartes, who introduced the concept.
II. Basic Terminologies
A. Coordinate system
1. Consists of two intersecting number lines or axes (x-axis and y-axis).
2. The point where the two axes intersect is called the origin.
B. Origin
1. The intersection of the x-axis and y-axis.
2. Represented by the ordered pair (0, 0).
C. Quadrants
1. When the x-axis and y-axis intersect, they divide the coordinate plane into four parts called quadrants.
2. Quadrants are numbered in a counterclockwise direction starting from the top right quadrant (I, II, III, and IV).
D. Ordered pairs/points (x, y)
1. Each point on a coordinate plane corresponds to a unique ordered pair (x, y).
2. The first value (x) represents the horizontal position, and the second value (y) represents the vertical position.
III. Plotting Points
A. To plot a point on a coordinate plane, you must know its ordered pair (x, y).
1. Locate the position along the x-axis (horizontal) equivalent to the x-coordinate value.
2. Locate the position along the y-axis (vertical) equivalent to the y-coordinate value.
3. Place a point where the respective x and y positions intersect.
B. Example
1. Ordered pair: (3, -2)
2. Start at the origin (0,0), move 3 units to the right for the x-coordinate, and 2 units down for the y-coordinate.
3. Place a dot at the intersection point, representing the point (3, -2).
IV. Distance and Midpoints
A. Distance Formula
1. To find the distance between two points (x1, y1) and (x2, y2), use the following formula:
2. Distance = √((x2 - x1)² + (y2 - y1)²)
B. Midpoint Formula
1. To find the midpoint of a line segment with endpoints (x1, y1) and (x2, y2), use the following formula:
2. Midpoint = ((x1 + x2)/2, (y1 + y2)/2)
V. Slope and Graphing Equations
A. Slope
1. A measure of how steep a line is on a coordinate plane.
2. Can be calculated as the ratio of the change in y-coordinates (the rise) to the change in x-coordinates (the run) between two points, (x1, y1) and (x2, y2).
3. Slope (m) = (y2 - y1) / (x2 - x1)
B. Graphing linear equations
1. Linear equations can be graphed on a coordinate plane in the form y = mx + b, where m is the slope and b is the y-intercept (the y-coordinate of the point where the line intercepts the y-axis).
2. To graph the equation, find the y-intercept point (0, b) and plot it on the coordinate plane.
3. Use the slope (m) to find another point by moving along the line either to the right or the left.
4. Connect the points with a straight line.
VI. Conclusion
A. The coordinate plane plays a crucial role in understanding various mathematical concepts, such as graphing linear equations, calculating distances, and finding midpoints.
B. It is also a valuable tool for visualizing relationships between different sets of data in various fields, such as science, economics, and engineering.