Why does the ratio of rise and run in a linear function always the same?

Answer 1

The ratio of rise and run in a linear function (which its graph is a nonvertical line) is called the slope of the line. We can calculate the slope of the line by using any two points on the line and always the ratio of changes on their y-coordinates and the changes on x-coordinates (rise and run) is constant (unchanged). The reason is that two points determine a line and its direction, so the line is fixed and extends without end in that direction. Any line intersects the x-axis only at one point (we call it the x-intercept point) by forming an angle α (the inclination angle), where 0˚ ≤ α < 180˚, that is measured from the positive x-axis to the line. There is a theorem that state that the slope of a nonvertical line is the tangent of its inclination: m = tan α if α ≠ 90˚ (if α = 90˚, then the line is vertical and has no slope).