Coordinate geometry involves working with points on a graph that is known as the Cartesian coordinate plane. This perfectly flat surface has a system that allows you to identify the position of points by using pairs of numbers.
Cartesian Coordinate System
In a coordinate plane, any point can be represented by a pair of numerical coordinates. These pairs of numbers represent the points’ distances from an origin on perpendicular axes. The coordinate of any particular point is the set of numbers that identifies the location of the point, such as (3, 4) or (x, y).
x-axis: The x-axis is the horizontal axis (number line) on a coordinate plane. The values start at the origin, which has a value of 0. Numbers increase in value to the right of the origin and decrease in value to the left. The x value of a point’s coordinate is listed first in its ordered pair.
y-axis: The y-axis is the vertical axis (number line) on a coordinate plane. Its values start at the origin, which has a value of 0. Numbers increase in value going up from the origin and decrease in value going down. The y value of a point’s coordinate is listed second in its ordered pair.
Origin: The origin is the point (0, 0) on the coordinate plane. It is where the x- and y-axes intersect.
Ordered pair: Also known as a coordinate pair, this duo is the set of two values that expresses the distance a point lies from the origin. The horizontal (x) coordinate is always listed first, and the vertical (y) coordinate is listed second.
x-intercept: The value of x where a line, curve, or some other function crosses the x-axis. The value of y is 0 at the x-intercept. The x-intercept is often the solution or root of an equation.
y-intercept: The value of y where a line, curve, or some other function crosses the y-axis. The value of x is 0 at the y-intercept.
Slope: Slope measures how steep a line is and is commonly referred to as the rise over the run.
You can identify any point on the coordinate plane by its coordinates, which designate the point’s location along the x- and y-axes. For example, the ordered pair (2, 3) has a coordinate point located two units to the right of the origin along the horizontal (x) number line and three units up on the vertical (y) number line.
The intersection of the x- and y-axes forms four quadrants on the coordinate plane.
All points in Quadrant I have a positive x value and a positive y value.
All points in Quadrant II have a negative x value and a positive y value.
All points in Quadrant III have a negative x value and a negative y value.
All points in Quadrant IV have a positive x value and a negative y value.
All points along the x-axis have a y value of 0.
All points along the y-axis have an x value of 0.
Assume you have two points, A (x1, y1) and B (x2, y2), on a line. The formula to find the distance between A and B is
For calculating the midpoint coordinates of a line segment on the coordinate plane, you simply apply the midpoint formula:
M stands for midpoint and the x and y variables are the x and y coordinates of the line’s two endpoints.
Slope of a Line
If a line isn’t parallel to one of the coordinate axes, it either rises or falls from the left-hand side of the coordinate plane to the right-hand side. The measure of the steepness of the line’s rising or falling is its slope.
Slope (m) = (y2 - y1) / (x2 - x1)
The x and y values in the equation stand for the coordinates of two points on the line. The formula is just the ratio of the vertical distance between two points and the horizontal distance between those same two points. You subtract the y-coordinate of one point from the y-coordinate of the other point to get the numerator. Then you subtract the x-coordinate of one point from the x-coordinate of the other point to get the denominator.
Types of Slope
- A line with a negative slope falls from left to right (its left side is higher than its right), and its slope is less than 0.
- A line with a positive slope rises from left to right (its right side is higher than its left), and its slope is greater than 0.
- A horizontal line has a slope of 0; it neither rises nor falls and is parallel to the x-axis.
- The slope of a vertical line is undefined because you don’t know whether it’s rising or falling; it has no slope and is parallel to the y-axis.
Slope-intercept Form of Line
The characteristics of a line can be conveyed through a mathematical formula. The equation of a line (also known as the slope-intercept form) generally shows y as a function of x, like this:
y = mx + c
In the slope-intercept form, the coefficient m is a constant that indicates the slope of the line, and the constant b is the y-intercept (that is, the point where the line crosses the y-axis).