Example 15 Given the graph of the function $y=f(x)$ as shown in Figure 1-2.
(1) Write the analytical expression for $y=f(x)$;
(2) Find the analytical expression for $y=f(2 x)$, and draw the graph of $y=f(2 x)$;
(3) Find the analytical expression for $y=f(2 x-1)$, and draw the graph of $y=f(2 x-1)$.
(1) From Figure $1-2$, we can see that $f(x)=|x|(-1 \leqslant x \leqslant 1)$.
(2) From the first part, we know $f(2 x)=|2 x|=2|x|$, noting that $-1 \leqslant 2 x \leqslant 1$, so
$$
f(2 x)=2|x|\left(-\frac{1}{2} \leqslant x \leqslant \frac{1}{2}\right) .
$$

Its graph is shown in Figure $1-3$.
(3) Similarly, we get $f(2 x-1)=|2 x-1|(0 \leqslant x \leqslant 1)$, its graph is shown in Figure $1-4$.