Answer:
B) g(x) = -(4x - 4)² - 9
Explanation:
f(x) = (x - 4)² + 3
to shrink horizontally by ¼, we need to multiply the x-variable by 4 wherever it appears in the original function definition.
in other words : g(x) = f(4x) = (4x - 4)² + 3
why ? because g(x) is still working with the same x-timing as f(x) before, but f(4x) makes the original f(x)- values appear much "sooner" (by factor 4), and so the graph is contracted to 1/4 width.
then we translate this 6 units up. that means we are not chasing the general behavior of the function at all. we are just adding 6 to every function value.
so,
g(x) = f(4x) + 6 = (4x - 4)² + 3 + 6 = (4x - 4)² + 9.
and finally we reflect this around the x-axis. that means the absolute function values stay the same, but positive turns to negative, and negative turns to positive.
this is done by multiplying the function by -1 :
g(x) = -(f(4x) + 6) = -f(4x) - 6 = -(4x - 4)² - 9