Gradients and intercepts

#### Tier: #Foundation #Higher

Description

Every straight-line graph can be written in the form $y = mx + c$, where $m$ is the gradient and $c$ is the $y$-intercept.

Gradient ($m$): measures the steepness. It is the change in $y$ per unit increase in $x$. $$m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1}$$

• Positive gradient: line goes uphill left to right
• Negative gradient: line goes downhill
• Zero gradient: horizontal line

$y$-intercept ($c$): the value of $y$ where the line crosses the $y$-axis ($x = 0$).

Example: Line through $(0, 3)$ and $(4, 11)$: $$m = \frac{11 - 3}{4 - 0} = \frac{8}{4} = 2, \quad c = 3$$ $$\text{Equation: } y = 2x + 3$$

$x$-intercept: set $y = 0$ and solve for $x$. For $y = 2x + 3$: $x = -1.5$.

Common error: confusing the gradient with the $y$-intercept when reading an equation, or calculating gradient with $\frac{\Delta x}{\Delta y}$ instead of $\frac{\Delta y}{\Delta x}$.

Links

Equation of a straight line Parallel and perpendicular lines