I am struggling with number b where my logic is that the asymptote will not change. But the answers say I am wrong. Please help.

Think of the function 4f(x) as stretching the original function f(x) vertically by a factor of 4. All 'y' values become 4 times as large while the 'x' values remain the same. Thus the max point becomes (-2,20) - which I see is what you wrote on the paper. Similarly the asymptote becomes y=4.

But the equation y=1/(x+1) or any similar equation will have asymptote at y=0. And when y=4f(x), y=4/(x+1). Still asymptote at y equal to 0. Why is it different here?

But the equation y=1/(x+1) or any similar equation will have asymptote at y=0. And when y=4f(x), y=4/(x+1). Still asymptote at y equal to 0. Why is it different here?

Because 4 times zero is zero, whereas 4 times 1 is 4. If the asymptote was at y=4, then the asymptote for y=4f(x) would be 4 x 4 = 16.

Right, thanks hahaha ☺