Points are written (x, y): across first, then up or down. The axes cross at the origin (0, 0).
The gradient measures steepness: m = \frac{\text{change in } y}{\text{change in } x} = \frac{y2 - y1}{x2 - x1}.
A line sloping up from left to right has positive gradient; sloping down, negative; horizontal, zero.
Gradient from two points
The equation \(y = mx + c\)
Every non-vertical straight line can be written as y = mx + c, where m is the gradient and c is the y-intercept. A vertical line has equation x = k and undefined gradient.
y = 2x + 3 has gradient 2 and crosses the y-axis at (0, 3).
Horizontal lines are y = k; vertical lines are x = k.
Reading y = 2x + 3
Drawing a line
Make a table of values: substitute at least three x-values and plot the points.
Or use the gradient and intercept: start at c on the y-axis, then move across 1 and up m.
Finding the equation of a line
Through two points: work out the gradient m, then substitute one point into y = mx + c to find c.
Through (1, 5) and (3, 9): m = \frac{9 - 5}{3 - 1} = 2; 5 = 2(1) + c gives c = 3, so y = 2x + 3.
Line through two points
Midpoints and lengths
The midpoint of (x1, y1) and (x2, y2) is \left(\frac{x1 + x2}{2}, \frac{y1 + y2}{2}\right).
The length of a line segment comes from Pythagoras' theorem.
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