Question about convolution of shifted function-simple
Considering an LTI continuous-time system with impulse response $h(t)$, its response to any valid input $x(t)$ can be computed with a convolution integral as:
$$y(t) = x(t) \star h(t) = \int_{-\infty}^{\infty} x(\tau) h(t-\tau) d\tau$$ and equivalently
$$y(t) = h(t) \star x(t) = \int_{-\infty}^{\infty} h(\tau) x(t-\tau) d\tau$$
due to the commutativity of the convolution operator.
Hence, shifting the output $y(t)$ by $k$ could be shown as
$$y(t-k) = \int_{-\infty}^{\infty} x(\tau) h(t-k-\tau) d\tau$$ or equivalently as
$$y(t-k) = \int_{-\infty}^{\infty} h(\tau) x(t-k-\tau) d\tau$$
Where, the first one equals $y(t-k) = x(t) \star h(t-k)$ and the second one equals $y(t-k) = h(t) \star x(t-k)$.