Before we begin: A road rises 10 metres for every 100 metres of horizontal distance. Another road rises 15 metres for every 200 metres of horizontal distance. Which road is steeper? Explain your reasoning.
Come back to this after you have worked through the lesson.
Gradient measures how much a line rises or falls for every unit it runs horizontally. It is the single most important property of a straight line. A positive gradient means the line goes up from left to right; a negative gradient means it goes down. A horizontal line has zero gradient, and a vertical line has an undefined gradient because you cannot divide by zero.
- Calculate the gradient of a line given two points.
- Identify whether a gradient is positive, negative, zero, or undefined.
- Find the gradient from an equation in the form $y = mx + c$.
- Interpret the meaning of gradient in practical situations.
Wrong: Reversing rise and run. Calculating $\dfrac{x_2 - x_1}{y_2 - y_1}$ instead of $\dfrac{y_2 - y_1}{x_2 - x_1}$.
Right: Gradient = $\dfrac{\text{rise}}{\text{run}} = \dfrac{\Delta y}{\Delta x} = \dfrac{y_2 - y_1}{x_2 - x_1}$.
Wrong: Confusing horizontal and vertical lines. Horizontal has $m = 0$; vertical has $m$ undefined.
Right: Horizontal: $m = 0$ (no rise). Vertical: $m$ is undefined (run = 0, cannot divide by zero).
For two points $A(x_1, y_1)$ and $B(x_2, y_2)$ on a line, the gradient $m$ measures how much the line rises (or falls) for every unit it runs horizontally. It is calculated as rise over run.
$m = \dfrac{y_2 - y_1}{x_2 - x_1}$
Gradient tells you both steepness and direction. A positive gradient means the line slopes upward from left to right. A negative gradient means it slopes downward. A horizontal line has zero gradient because there is no rise at all. A vertical line has an undefined gradient because the run is zero and division by zero is impossible.
$m > 0$ up. $m < 0$ down. $m = 0$ flat. undefined vertical.
When a linear equation is written in gradient-intercept form $y = mx + c$, the gradient appears as the coefficient of $x$. If an equation is not in this form, rearrange it by isolating $y$. The number in front of $x$ is always the gradient.
$y = mx + c$. $m$ = gradient. $c$ = y-intercept.
Australian road signs express steepness as a percentage. A 10% grade means the road rises 10 m for every 100 m of horizontal distance, giving a gradient of $m = 0.10$. Wheelchair ramps must not exceed a 1:8 gradient (about 12.5%) to be safe. Roof pitches, ski slopes, and railway lines all use gradient to describe steepness.
10% grade = $m = 0.10$. 1:8 ramp = $m = 0.125$.
Identify: $x_1 = 2, y_1 = 5, x_2 = 6, y_2 = 11$.
Substitute: $m = \dfrac{11 - 5}{6 - 2} = \dfrac{6}{4} = \dfrac{3}{2} = 1.5$
Since $m > 0$, the line slopes upward from left to right.
Substitute: $m = \dfrac{3 - 7}{5 - (-3)} = \dfrac{-4}{8}$
$m = -\dfrac{1}{2} = -0.5$
Since $m < 0$, the line slopes downward from left to right.
Rearrange to $y = mx + c$ form:
$4x - 2y + 8 = 0$
$-2y = -4x - 8$
$y = 2x + 4$
The coefficient of $x$ is $2$.
Brain Trainer
4 quick-fire drills. Beat the clock.
5 MCQs and 3 short-answer questions. Target: 80% accuracy.
Your answer:
$m = \dfrac{-9 - 3}{2 - (-4)} = \dfrac{-12}{6} =$ $-2$
The line slopes downward from left to right because $m < 0$.
Your answer:
(a) $2y = -4x + 6$ → $y = -2x + 3$
(b) Gradient $m = -2$, y-intercept $c = 3$
(c) When $x = 0$, $y = 3$. When $x = 1$, $y = 1$. Gradient $= \dfrac{1-3}{1-0} = \dfrac{-2}{1} = -2$ ✓
Your answer:
(a) $m_{AB} = \dfrac{8-2}{4-1} = \dfrac{6}{3} =$ $2$
$m_{BC} = \dfrac{5-8}{7-4} = \dfrac{-3}{3} =$ $-1$
(b) The gradients are different ($2 \neq -1$), so $AB$ and $BC$ have different directions. Therefore $A$, $B$, and $C$ cannot be collinear.
(c) Any point on the line through $A$ with gradient 2 works. For example, from $B(4, 8)$ with gradient 2: $D = (5, 10)$. Check: $m_{AD} = \dfrac{10-2}{5-1} = \dfrac{8}{4} = 2$ ✓
Consolidate and reflect before moving on.
A line passes through $(2, -3)$ and has gradient $\dfrac{4}{3}$. Find two other points on this line that are exactly 5 units away from $(2, -3)$. (Hint: use the 3-4-5 triangle relationship. If run = 3 and rise = 4, the distance is 5.)
Gradient = $\dfrac{\text{rise}}{\text{run}} = \dfrac{y_2 - y_1}{x_2 - x_1}$. Positive = up, negative = down, zero = flat, undefined = vertical. From $y = mx + c$, the coefficient of $x$ is the gradient.
Reversing rise and run. Remember: y-difference goes on top. Also confusing horizontal ($m = 0$) with vertical (undefined).
Gradient is the foundation of linear equations. In the next lessons, you will use gradient to find the equation of a line and to determine if lines are parallel or perpendicular.
Find the gradient of the line through $(-2, 4)$ and $(3, -6)$, then find the gradient of $3x + 6y = 12$. Are they the same? Time yourself, can you do both in under 60 seconds?
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