Example 4. The three vertices of a triangle are $(0,0)$, $(2,3)$, and $(3,-2)$. When the triangle is translated, the translation path of $(0,0)$ is $\mathrm{y}^{2}=2 \mathrm{x}$. Find the translation paths of the other two points.
Solve: Write $\mathrm{y}^{2}=2 \mathrm{x}$ as a parametric equation
$$
\left\{\begin{array}{l}
x=-\frac{t^{2}}{2} \\
y=t .
\end{array}\right.
$$
$\left(-\frac{t^{2}}{2}, t\right)$ is the translation increment of $(0,0)$, and also the translation increment of the triangle. Therefore, the translation trajectory of (2,3) is
$$
\left\{\begin{array}{l}
x=2+\frac{t^{2}}{2} \\
y=3+t
\end{array}\right.
$$

Eliminating $t$, we get $(y-3)^{2}=2(x-2)$.
Similarly, the translation trajectory of $(3,-2)$ is
$$
(y+2)^{2}=2(x-3) \text {. }
$$
$$
\text { Let } f(x, y)=0 \text { . }
$$

The translation point of $(x, y)$ with respect to $(h, k)$ is $\left(x^{\prime}, y^{\prime}\right)$, then we have
$$
f\left(x^{\prime}-\dot{n}, y^{\prime}-k\right)=0 \text {. }
$$

Removing the dot
$$
f(x-h, y-k)=0 .
$$

We call (B) the translation curve of (A) with respect to $(\mathrm{h}, \mathrm{k})$.

This result provides us with a method for plotting the translation transformation of function curves and equation curves.