Coordinate geometry, graphs and functions

📌 Key points

1. Gradient = change in ychange in x = y₂ − y₁x₂ − x₁. Lines going up from left to right have positive gradients; lines going down have negative gradients.
2. In y = mx + c, m is the gradient and c is the y-intercept. To find a line’s equation, find m first, then substitute a point to find c.
3. Parallel lines have equal gradients. If two lines are perpendicular, their gradients multiply to −1, e.g. 2 and −12.
4. The midpoint of (x₁, y₁) and (x₂, y₂) is (x₁ + x₂2, y₁ + y₂2); the length between them is √((x₂ − x₁)² + (y₂ − y₁)²).
5. Graph shapes: y = ax² + bx + c is a parabola; y = x³ is a cubic; y = kx is a reciprocal graph with the axes as asymptotes; y = a^x is exponential and passes through (0, 1).
6. Solve equations graphically by reading the x-coordinates where two graphs intersect; the roots of f(x) = 0 are where y = f(x) crosses the x-axis.
7. Functions: fg(x) means do g first, then f. The inverse f⁻¹(x) undoes f: write y = f(x), swap x and y, and rearrange for y.
8. The gradient of a curve at a point is the gradient of the tangent there. On a distance–time graph the gradient is speed; on a speed–time graph it is acceleration and the area underneath is distance.
9. Differentiation: if y = axⁿ then dydx = naxⁿ⁻¹. At a turning point dydx = 0.
10. Inequalities on a graph: use a broken line for a strict inequality and a solid line for ≤ or ≥, then shade as the question instructs.

📖 Key terms

Gradient

The steepness of a line: change in y ÷ change in x.

y-intercept

Where a graph crosses the y-axis (x = 0).

Perpendicular

At right angles (90°) to each other.

Asymptote

A line that a curve gets closer and closer to but never meets.

Composite function

One function applied after another, e.g. fg(x) = f(g(x)).

Inverse function

The function that reverses f, written f⁻¹(x).

Tangent

A straight line that touches a curve at one point.

Turning point

A point where the gradient is zero (a maximum or a minimum).