10.1 Gradient
The gradient ($m$) of a straight line is the ratio of the vertical distance (rise) to the horizontal distance (run) between any two points on the line.
Formulae for Gradient
- Using Two Points $A(x_1, y_1)$ and $B(x_2, y_2)$:
$$m = \frac{\text{Vertical Distance}}{\text{Horizontal Distance}} = \frac{y_2 - y_1}{x_2 - x_1}$$
- Using Intercepts:
$$m = -\frac{y\text{-intercept}}{x\text{-intercept}}$$
Characteristics of Gradient ($m$)
- Positive Gradient ($m > 0$): The line slants upwards from left to right.
- Negative Gradient ($m < 0$): The line slants downwards from left to right.
- Zero Gradient ($m = 0$): Horizontal line (parallel to the $x$-axis).
- Undefined Gradient: Vertical line (parallel to the $y$-axis, as horizontal distance $= 0$).
- Steepness: A larger absolute value of $m$ ($|m|$) indicates a steeper line.