Slope:
For a straight line, slope is equal to rise over run, or
Slope = rise/run = change in y/change in x = Δy/Δx = y2-y1/x2 - x1
Let consider the curve display in above figure. Points P1 and P2 are any two different points on the line, and a right triangle is drawn whose legs are parallel to the coordinate axes. The length of the leg parallel to the x-axis is the difference among the x-coordinates of the two points and is known as "Δx," read "delta x," or "the change in x." The leg parallel to the y-axis has length Δy, that is the difference among the y-coordinates. For instance, consider the line holding points (1,3) and (3,7) in the second part of the figure. The difference among the x-coordinates is Δx = 3-1 = 2. The difference between the y-coordinates is Δy = 7-3 = 4. The ratio of the differences, Δy/ Δx, is the slope, that in the preceding example is 4/2 or 2. It is significant to remember that if other points had been chosen on the similar line, the ratio Δy/ Δ x would be the same, since the triangles are clearly similar. If the points (2,5) and (4,9) had been selected, then Δy/ Δx = (9-5)/(4-2) = 2, which is the same number as before. Thus, the ratio Δy/ Δx depends on the inclination of the line, m = rise [vertical (y-axis) change] ÷ run [horizontal (x-axis) change].
Figure: Slope
Since slope m is a measure of the steepness of a line, a slope has the subsequent features:
1. A horizontal line has zero slope.
2. A line that rises to the right has positive slope.
3. A line rising to the left has negative slope.
4. A vertical line has undefined slope because the calculation of the slope would involve division by zero. (Δy/Δx approaches infinity as the slope approaches vertical.)