Question

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.

Answer 1

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.

& \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{aligned}$$ 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 $\begin{aligned} & \Rightarrow y=-2\left( {{x}^{3}}+2x \right) \\\ & \Rightarrow y=-2{{x}^{3}}-4x \\\ \end{aligned}$ 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.