$Y = \alpha X + \beta$
Mean = $\mu$
Standard Deviation = $\sigma$
Variance = $\sigma ^2$
Let Z be a random variable such that
$Z = \alpha X$
So Z is scaled by some $\alpha > 0$
$\mu _ Z = \alpha \mu$
$\sigma _Z = \alpha \sigma$
$Z \sim$ $N(\mu _ Z , \sigma _Z ^2 ) \sim N(\alpha \mu, (\alpha \sigma)^2)$
Now,
$Y = Z + \beta$
So the distribution will be shifted depending upon the magnitude and sign of $\beta$,
But it doesn't change the standard deviation but it does change the mean.
$\mu_y = \mu_z + \beta = \alpha \mu + \beta$
Hence,
$Y \sim N(\alpha \mu + \beta, (\alpha \sigma)^2)$
https://www.khanacademy.org/math/ap-statistics/random-variables-ap/transforming-random-variables/v/impact-of-scaling-and-shifting-random-variables