Straight Lines - Various Forms of the Equation of a Line: Horizontal/Vertical, Point-Slope, Two-Point, Slope-Intercept, Intercept Form

Horizontal and Vertical Lines: A horizontal line is parallel to the x-axis and has a constant y-coordinate for every point; it appears as a flat line crossing the y-axis at $(0, k)$. A vertical line is parallel to the y-axis with a constant x-coordinate, appearing as a straight upright line crossing the x-axis at $(h, 0)$.

Slope-Intercept Form: This form is used when the slope $m$ and the y-intercept $c$ are known. Visually, $c$ is the point where the line crosses the vertical axis $(0, c)$, and $m$ dictates the 'steepness' or 'gradient' of the line as it moves across the coordinate plane.

Point-Slope Form: This equation is used when we know one specific point $(x_1, y_1)$ on the line and its slope $m$. It is based on the principle that the slope between any arbitrary point $(x, y)$ and the fixed point $(x_1, y_1)$ must be constant.

Two-Point Form: This is used when the coordinates of two distinct points $(x_1, y_1)$ and $(x_2, y_2)$ are given. Geometrically, this defines a unique straight line passing through both points, where the slope is first calculated as the vertical change divided by the horizontal change.

Intercept Form: This form relates the line to its intercepts on both the x-axis (at $a$) and the y-axis (at $b$). Visually, the line cuts the x-axis at $(a, 0)$ and the y-axis at $(0, b)$, creating a triangle with the origin.

Geometric Significance of Slope ($m$): The slope represents $\tan \theta$, where $\theta$ is the angle of inclination with the positive direction of the x-axis. If $m > 0$, the line rises from left to right; if $m < 0$, it falls from left to right.