TMUA Graph Sketching and Transformations

Graph questions on TMUA Paper 1 reward candidates who can move between algebraic and visual representations without losing detail. The transformations themselves are small in number; the test is whether you apply them in the right order on the right axis.

• The four single transformations. $y = f(x) + a$ shifts up by $a$; $y = f(x - a)$ shifts right by $a$; $y = af(x)$ stretches vertically by factor $a$; $y = f(ax)$ stretches horizontally by factor $1/a$. Negative scalars reflect.
• Composing transformations. Inside the function (the $x$ side) versus outside (the $y$ side) work in opposite directions, and the sign on inside transformations is the reverse of what reads natural. Factor before you transform: $f(2x + 4) = f\big(2(x + 2)\big)$, which is the graph of $f$ scaled horizontally by $1/2$ and then shifted left by $2$ — not right.
• Asymptotic behaviour. For a rational function $p(x)/q(x)$, horizontal asymptote comes from comparing degrees; vertical asymptote is where $q(x) = 0$ and $p(x) \neq 0$; oblique asymptote when $\deg p = \deg q + 1$.
• Roots and turning points. Real roots are $x$-axis crossings; repeated roots are touches not crossings; the sign of the leading coefficient determines end behaviour for polynomials.

The move. Before sketching, find the roots, the $y$-intercept, the asymptotes, the end behaviour. The picture builds itself from those four pieces; you almost never need to plot points.