A transformation is defined by the matrix \[\left[ \begin{matrix}
   1 & 0 \\
   0 & -2 \\
\end{matrix} \right]\]. Find the equation of the iimage of the graph of the cubic equation $y={{x}^{3}}+2x$ under this transformation.

Hint: In the above question, we have been given a transformation in the form of a matrix which is given as \[\left[ \begin{matrix}
   1 & 0 \\
   0 & -2 \\
\end{matrix} \right]\]. We need to multiply the given matrix \[\left[ \begin{matrix}
   1 & 0 \\
   0 & -2 \\
\end{matrix} \right]\] with the coordinate matrix $\left[ \begin{matrix}
   x \\
   y \\
\end{matrix} \right]$ to obtain the new transformed coordinate matrix. The transformed coordinates have to be then substituted in place of the respective coordinates in the given cubic equation, which is $y={{x}^{3}}+2x$. Then we need to manipulate the obtained equation in terms of the standard equation $y=f\left( x \right)$.

Complete step-by-step answer:
The cubic equation given in the above question is
$\Rightarrow y={{x}^{3}}+2x$
Its graph is given as

In the question, we have been given a transformation defined by the matrix \[\left[ \begin{matrix}
   1 & 0 \\
   0 & -2 \\
\end{matrix} \right]\]. Before applying this transformation onto the given equation, we need to multiply the given matrix with the coordinate matrix which is $\left[ \begin{matrix}
   x \\
   y \\
\end{matrix} \right]$, as shown below.
\[\begin{align}
  & \Rightarrow \left[ \begin{matrix}
   1 & 0 \\
   0 & -2 \\
\end{matrix} \right]\left[ \begin{matrix}
   x \\
   y \\
\end{matrix} \right] \\
 & \Rightarrow \left[ \begin{matrix}
   1\cdot x+0\cdot y \\
   0\cdot x-2\cdot y \\
\end{matrix} \right] \\
 & \Rightarrow \left[ \begin{matrix}
   x \\
   -2y \\
\end{matrix} \right] \\
\end{align}\]
So we have obtained a new transformed coordinate matrix. According to the transformed matrix, the x coordinate is not changed. But the y coordinate is changed to $-2y$. This means that in the graph of $y={{x}^{3}}+2x$, the x coordinate will remain the same, while the y coordinate of each of the points will get doubled and reversed. Therefore, the new transformed equation will be given as
$\begin{align}
  & \Rightarrow y=-2\left( {{x}^{3}}+2x \right) \\
 & \Rightarrow y=-2{{x}^{3}}-4x \\
\end{align}$
We can see this change in the graph below.

Note: According to the given transformation, the y coordinate is changed to $-2y$. However, we must note that, it does not means that we can replace $y$ with $-2y$. We have to transform the given equation such that the y coordinate gets doubled along with the change in its sign. Therefore, we have multiplied $-2$ with the RHS.