Coefficients in cost function in A-star
For A* to get the optimal path it requires that $f(n) \leq g(goal)$. In other words that the heuristic underestimates the cost from the node to the goal.

Multiplying a valid hueristic with $0 \lt\alpha\lt 1$ will not violate this requirement.

Multiplying $g(n)$ is not allowed because you can end up with $f(goal) = \alpha g(goal) &lt; f(n)$ which would violate the requirement for getting the optimal path.