N6: Functions and graphs

An ordered pair $(x,y)$ identifies a point in the Cartesian plane. A set of ordered pairs can represent a relationship even when no formula is supplied. Plot points against the stated axes and retain the order of coordinates.

A linear function $y=ax+b$ has constant gradient $a$ and vertical intercept $b$. Gradient is vertical change divided by horizontal change and may be positive, zero or negative.

A quadratic graph is a parabola. The sign of the $x^2$ coefficient determines whether it opens upward or downward. Completed-square form reveals the turning point, while factorised form reveals the roots and axis of symmetry.

Power functions $y=ax^n$ have characteristic shapes determined by the exponent and sign of $a$. Exponential functions $y=ka^x$, for positive integer $a$, change by a constant multiplicative factor across equal increases in $x$.

The gradient changes along a curve. At the required point, draw a tangent that just touches the local curve, choose two distant points on the tangent, and calculate rise over run.