AHL 2.16 (HL)—Further graph transformations

Apply modulus and reciprocal transformations in the correct order.

For y=|f(x)|, negative parts reflect above the x-axis; y=f(|x|) mirrors the right-hand graph into x<0; y=1/f(x) keeps zeros as excluded inputs and swaps large/small values.

Worked example

If f(x)=x−1, then |f(x)| has a corner at x=1, while f(|x|)=|x|−1 has a V-shaped graph with a corner at x=0.

Transform the graph in stages and preserve domain restrictions; the two modulus forms are not interchangeable.

Absolute value outside changes outputs; absolute value inside changes which inputs are used.

Further transformations: $y=[f(x)]^2$ makes outputs non-negative and retains the zeros of $f$; $y=f(ax+b)$ applies horizontal scaling and translation through the input. For example, $f(2x-4)=f(2(x-2))$ is horizontally compressed by factor $1/2$ and shifted right $2$. Modulus example: $|2x-3|\le5$ is equivalent to $-5\le2x-3\le5$, giving $-1\le x\le4$.