Question

If y=3x+1 were changed to y=x+1 how would the graph of the new function compare with the original?

Answer 1

F(x)=3x+1  (preimage)
g(x)=x+1  (image)
it is undergoing a reduction/compression with translation.
In general, a linear transformation is
g(x) = a*f(bx-h)+k
h=horizontal translation (right if h>0, left if h<0, note formula has minus sign)
k=vertical translation (up if k>0, down if k<0)
a=vertical stretching, (stretching if |a|>1, compression if |a|<1, also, if a<0, a reflection across the x-axis is performed)
b=horizontal stretching (|b|>0 compression, |b|<0 stretching, also, if b<0, a reflection across the y-axis is performed)

In this case, 
g(x)=f(x/3), so it is a horizontal stretching.
Note that the y-intercept remains unchanged.