Example QF.3.1

Example QF.3.1 : $h =2; q(x) = (x-2)^2$
This is best understood as the composition of the linear function $g(x)= x-2$ followed by the core quadratic function $f(x) = x^2$, so $q(x) = f(g(x))$.
Draw a mapping diagram yourself or use the diagram created with GeoGebra to explore the diagram further.
Compare the mapping diagram with the graphs of $g(x)$ and $f(x)$

Graphs of $g(x)$ and $f(x)$

Mapping Diagram Showing Composition.

Given a point / number, $x$, on the source line, there is a blue arrow meeting the target line at the point / number, $(x-2)^2$
This point corresponds to the quadratic function's value for $x$.
The values for the core mapping diagram for $x^2$ in red are applied to the values $g(x)=x-2$ (green) .

As $x$ increases, $q(x)$ decreases to value $q(2)=0$ and then increases.