BackFundamental Concepts in the Rectangular Coordinate System
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Rectangular Coordinate System
Graphing Equations in the Rectangular Coordinate System
The rectangular coordinate system, also known as the Cartesian plane, is a two-dimensional plane defined by a horizontal axis (x-axis) and a vertical axis (y-axis). Graphing equations in this system allows us to visually represent relationships between variables.
Coordinate Axes: The x-axis (horizontal) and y-axis (vertical) intersect at the origin (0, 0).
Ordered Pair: Each point is represented by an ordered pair (x, y).
Graphing Equations: To graph an equation, plot points that satisfy the equation and connect them appropriately.
Example: To graph $y = 2x + 1$, choose values for x, calculate corresponding y values, plot the points, and draw the line.
Finding the Distance Between Two Points
The distance between two points in the plane can be found using the distance formula, which is derived from the Pythagorean Theorem.
Distance Formula: For points $(x_1, y_1)$ and $(x_2, y_2)$, the distance $d$ is given by:
$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
Example: The distance between (1, 2) and (4, 6) is $d = \sqrt{(4-1)^2 + (6-2)^2} = \sqrt{9 + 16} = \sqrt{25} = 5$.
Finding the Midpoint of a Line Segment
The midpoint of a line segment connecting two points is the point exactly halfway between them. The midpoint formula calculates the average of the x-coordinates and the y-coordinates.
Midpoint Formula: For points $(x_1, y_1)$ and $(x_2, y_2)$, the midpoint $M$ is:
$M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$
Example: The midpoint between (1, 2) and (4, 6) is $M = \left( \frac{1+4}{2}, \frac{2+6}{2} \right) = (2.5, 4)$.
