Revise Linear functions and equations of straight lines for IB Mathematics: Analysis and Approaches (2.1)

The gradient, denoted $m$, measures both the steepness and the direction of a line. It is the ratio of the vertical change ('rise') to the horizontal change ('run') between any two distinct points on the line — and because the line is straight, that ratio is the same wherever you measure it. A positive gradient rises from left to right; a negative gradient falls from left to right. The key habit is to subtract the coordinates in the same order top and bottom.

Gradient $\; m = \dfrac{\text{rise}}{\text{run}} = \dfrac{y_2 - y_1}{x_2 - x_1}$

The same straight line can be written in several forms, each convenient for a different job. The IB often specifies the form your final answer must take, so you must be able to convert freely between them. All three describe identical lines — moving between them is pure rearrangement.

You can tell how two lines are related just by comparing their gradients. Parallel lines have the same steepness, so they never meet. Perpendicular lines cross at a right angle, and turning a line through $90^\circ$ both flips its rise-over-run and reverses its direction — which is why the two gradients multiply to $-1$.

For two non-vertical lines with gradients $m_1$ and $m_2$:

Parallel: $\; m_1 = m_2$

Perpendicular: $\; m_1 \times m_2 = -1 \quad\Longleftrightarrow\quad m_2 = -\dfrac{1}{m_1}$

Two lines cross at the single $(x, y)$ pair that satisfies both equations at once, so finding the intersection means solving the two equations simultaneously. Substitution is quickest when one equation is already in the form $y = \dots$; elimination is neat when the equations are both in general form. If the lines are parallel their gradients are equal and there is no solution — no intersection.

Two more formula-booklet tools finish the toolkit. The midpoint of a segment is simply the average of its endpoints' coordinates, and the distance between two points is the Pythagorean length of the segment joining them. The midpoint is exactly what you need for a perpendicular bisector — the line at right angles to a segment through its middle.

For points $A(x_1, y_1)$ and $B(x_2, y_2)$:

Midpoint $= \left( \dfrac{x_1 + x_2}{2}, \; \dfrac{y_1 + y_2}{2} \right)$

Distance $= \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

In Paper 1 the marks are analytic: each is tied to a specific line of working. A method mark (M) is awarded for a correct approach even if the arithmetic later slips; an accuracy mark (A) is awarded for a correct result and is dependent on the relevant method mark being earned. Follow-through (FT) means that once you have a value, a later part done correctly on THAT value still scores, even if the value itself was wrong. And the engine applies 'accept equivalent forms' — any algebraically correct rearrangement earns the mark. All of this protection exists only if your method is written down. Study how each mark below is pinned to one line of working.

The straight line is the seed of the whole course. Its gradient — a constant rate of change — becomes the derivative, the instantaneous gradient of any curve, so the tangent to a curve is a straight line found by exactly the methods here. The point-slope form reappears every time you write a tangent or normal in calculus (a normal is just a perpendicular line, negative reciprocal and all). Simultaneous solving extends to where curves meet, and the midpoint and distance formulas underpin coordinate geometry throughout. Master the habit — find the gradient, pick a form, substitute a point, show every line — and the geometry and calculus that follow become variations on a method you already own.