Correct way to describe this geometrical transformation

Question

I am trying to describe to geometrical transformation that maps the graph of y = sinx onto the graph y = 2sinx. I know what the two separate graphs look like, and I could draw them, the only difference is 2sinx has it's peaks and troughs as 2 instead of one, but how do I describe the geometrical transformation, if there is a more mathematical way to put it than I have already stated?

Answer 1

You can see the graph of $f(x) = sin(x)$ as a subset of $\mathbb{R}^2$, thus and precisely $\{ (x, \sin(x) )\}$.

Thus, you can see the transformation that maps one graph to the other as the following matrix,

\begin{equation*} A = \begin{pmatrix} 1 & 0 \\ 0 & 2 \end{pmatrix} \end{equation*}

Indeed

\begin{equation*} \begin{pmatrix} x \\ 2 \sin(x) \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 2 \end{pmatrix} \begin{pmatrix} x \\ \sin(x) \end{pmatrix} \end{equation*}