图中显示了方程为 $y = f(x)$ 和 $y = g(x)$ 的图像。

完整描述一组由两个变换组成的过程，使得 $y = f(x)$ 的图像变换为 $y = g(x)$ 的图像。请明确说明这些变换应按什么顺序进行。

两条折线的转折点横坐标相同。比较纵坐标差的变化，再找出整体的竖直位移。

The points $(0,6)$ and $(3,3)$ on $f$ correspond to $(0,-2)$ and $(3,-8)$ on $g$. The change in height doubles, so first stretch by scale factor $2$ in the $y$-direction. This sends the height $6$ to $12$, so next translate by $\begin{pmatrix}0\\-14\end{pmatrix}$. The resulting map is $(x,y)\mapsto(x,2y-14)$.

如果先平移再伸缩，原先的竖直位移也会翻倍。因此只需先下移一半的距离。

First translate by $\begin{pmatrix}0\\-7\end{pmatrix}$, then stretch by scale factor $2$ in the $y$-direction. The resulting height is $2(y-7)=2y-14$, as required.