Slope of a Line

The slope (or gradient) of a line is the single number measuring how steeply it rises or falls, and it is the starting point for everything else in this chapter.

Inclination of a line. The angle of inclination $\theta$ of a line is the angle it makes with the positive direction of the x-axis, measured counter-clockwise, taken so that $0^\circ \le \theta < 180^\circ$. A line parallel to the x-axis has $\theta=0^\circ$; a line parallel to the y-axis has $\theta=90^\circ$.

Definition of slope. The slope of a line, denoted $m$, is defined as

where $\theta$ is its inclination — provided $\theta \ne 90^\circ$, since $\tan 90^\circ$ is undefined. A line parallel to the y-axis is therefore said to have no slope (an undefined slope). A line parallel to the x-axis has $\theta=0^\circ$, so $m=\tan 0^\circ = 0$.

Slope from two points on the line (derivation). Let $P(x_1,y_1)$ and $Q(x_2,y_2)$, with $x_1 \ne x_2$, be two points on a line of inclination $\theta$. Draw a horizontal line through $P$ and a vertical line through $Q$; they meet at $R(x_2,y_1)$, forming a right triangle $PQR$ with $PR = x_2-x_1$ and $QR = y_2-y_1$ (taking $x_2>x_1$ for the picture), and with $\angle RPQ = \theta$ (or its supplement, depending on which side $Q$ lies — the tangent ratio comes out the same value either way). In this right triangle,

Hence

This is the working formula used throughout the chapter. It does not matter which point is labelled $1$ and which $2$: reversing both signs leaves the ratio unchanged, $\dfrac{y_1-y_2}{x_1-x_2}=\dfrac{y_2-y_1}{x_2-x_1}$.

Sign of the slope. If $m>0$, the line rises as $x$ increases (acute inclination, $0^\circ<\theta<90^\circ$). If $m<0$, the line falls as $x$ increases (obtuse inclination, $90^\circ<\theta<180^\circ$). If $m=0$, the line is horizontal. …