How does both shifting and scaling a function affect the Fourier Transform?
Your logic is correct. There are two ways to determine the answer. Yours is one of them:

$$\begin{align}v(t)=x(at)&amp;\Longleftrightarrow \frac{1}{|a|}X(\omega/a)\\v(t-t_0/a)=x(at-t_0)&amp;\Longleftrightarrow\frac{1}{|a|}X(\omega/a)e^{-j\omega t_0/a}\end{align}$$

The other option is

$$\begin{align}w(t)=x(t-t_0)&amp;\Longleftrightarrow X(\omega)e^{-j\omega t_0}\\w(at)=x(at-t_0)&amp;\Longleftrightarrow \frac{1}{|a|}X(\omega/a)e^{-j(\omega/a) t_0}\end{align}$$

Of course, both options lead to the same result.