Final answer:
The graph of g(x) = 4(x + 3) + 1 is a horizontal shift to the left by 3 units of the graph of f(x) = 4x + 1, with the same slope and y-intercept.
Step-by-step explanation:
When comparing two linear functions such as f(x) = 4x + 1 and g(x) = 4(x + 3) + 1, we can analyze the effects of the transformations applied to the function 'f' to obtain 'g'. In this case, 'g(x)' is derived from 'f(x)' by replacing 'x' with 'x + 3', which translates to a horizontal shift to the left by 3 units, since we are adding 3 inside the function.
The coefficient of x (which is 4 in both equations) represents the slope of the line, and since it remains unchanged, both graphs have the same slope. The '+1' is the y-intercept, which also remains the same in both equations, meaning the point where each line intersects the y-axis is at y=1. Therefore, the graph of 'g' is simply a left-shifted version of the 'f' graph by 3 units.