I face this rather-insightful problem in the BB where it shows how f(x) has a y-intercept of 1, then a 2f(x) has a y-intercept of 2.
So, assuming there are no phase or vertical shifts, what role does a coefficient play???
The coefficient of 2 scales the function vertically by a factor of 2.
This is correct, but more specifically, the coordinates of the y-intercept of f(x) are (0, f(0)).
But as to your original question, no, the lead coefficient does NOT have to equal the y intercept. It just happens to be the case this time because the original function had an intercept equal to one.
In general, if (x,y) is a point on the graph of y=f(x) then (x, ky) will be a point on the graph of y=k f(x)
And since that is true for any x value, it is also automatically true when x = 0, at the y-intercept. But for example, if we start with
f(x) = 2^(x+1) and then ask for the y-intercept of y=3f(x), since the original function’s intercept was at (0,2) the new function will have its intercept at (0,6).