Use the equation $y = mx + c$ to find the gradient and intercept; find the equation of a line through given points; and identify parallel and perpendicular lines (perpendicular at Higher tier).

What this dot point is asking

OCR references A9 and A10 cover straight line graphs: using $y = mx + c$ to read the gradient and intercept, finding the equation of a line through given points, and identifying parallel and (at Higher tier) perpendicular lines. The straight line is the simplest graph and the gateway to coordinate geometry, real-life linear graphs and the graphical solution of simultaneous equations. It is tested on every tier and frequently on the non-calculator paper.

The equation y = mx + c

Every non-vertical straight line can be written in this form.

So $y = 3x - 2$ has gradient $3$ and crosses the $y$-axis at $-2$. To plot it, mark $(0, -2)$ and step up $3$ for every $1$ across. If an equation is given as $2y = 4x + 6$, rearrange to $y = 2x + 3$ before reading off $m$ and $c$. A line written as $x + y = 5$ also needs rearranging, to $y = -x + 5$, which has gradient $-1$ and intercept $5$.

A horizontal line has the form $y = k$ (gradient $0$, no $x$ term), and a vertical line has the form $x = k$ (an undefined gradient, so it cannot be written as $y = mx + c$). Recognising these special cases stops you trying to force a vertical line into the standard form. The gradient itself is a rate: in a real-life graph it might be a cost per item or a speed, so reading $m$ correctly is the key to interpreting context, which OCR rewards under AO3.

Gradient between two points

The gradient measures steepness as a ratio.

For $(2, 3)$ and $(6, 11)$, the gradient is $\dfrac{11 - 3}{6 - 2} = \dfrac{8}{4} = 2$. The order of subtraction must be consistent: subtract the coordinates in the same order on top and bottom, or the sign of the gradient flips.

Finding the equation of a line

Two pieces of information fix a line: a gradient and a point, or two points.

Parallel and perpendicular lines

Gradients reveal how two lines relate.

Parallel lines never meet and have equal gradients, so any line parallel to $y = 4x - 1$ has gradient $4$; only the intercept changes. Perpendicular lines cross at a right angle, and their gradients are negative reciprocals: their product is $-1$. So a line perpendicular to one with gradient $2$ has gradient $-\tfrac{1}{2}$, and a line perpendicular to gradient $-\tfrac{3}{4}$ has gradient $\tfrac{4}{3}$. To find the negative reciprocal, flip the fraction and change the sign; for a whole-number gradient $m$, the perpendicular gradient is $-\tfrac{1}{m}$. Perpendicular lines are a Higher-tier focus and a frequent multi-mark question.

A typical Higher question gives you a line and a point and asks for the perpendicular (or parallel) line through that point. The method is always the same: decide the new gradient from the parallel or perpendicular rule, then substitute the point into $y = mx + c$ to find the intercept. Coordinate geometry questions may also ask for the midpoint of a line segment, found by averaging the coordinates, or the length of a segment, found with Pythagoras on the horizontal and vertical differences, so straight-line work links directly to the geometry strand.