Straight Lines: Slopes, Intercepts, Normal Form A straight line is a plane curve that passes through two distinct points in the plane. We can represent a...
A straight line is a plane curve that passes through two distinct points in the plane. We can represent a straight line using different equations depending on its characteristics.
Slope:
The slope of a line is a measure of how steep it is. It is defined as the rise divided by the run, which is the horizontal distance between any two points on the line.
For a line with slope m, the rise is m units and the run is 1 unit.
The slope can also be calculated as the change in y divided by the change in x.
Intercept:
A line can intersect with the coordinate plane at exactly one point, which is called an intercept.
The x-intercept occurs when the line intersects the x-axis, and the y-intercept occurs when it intersects the y-axis.
The coordinates of the intercept point can be found by substituting the values of x and y into the equation of the line.
Normal Form:
The normal form of a straight line is y = mx + b, where:
m is the slope
b is the y-intercept
This equation represents a line with slope m and y-intercept b.
The normal form is particularly useful because it can be used to find the equation of a line given its slope and intercept.
Examples:
A line with slope 2 passing through the points (3, 5) and (7, 13) has a slope of 2.
The line y = 3x + 1 intersects the x-axis at the point (0, 1).
The equation y = 2x - 3 is in normal form, with m = 2 and b = -3