Question

f(x) is transformed from y = x³ by horizontally stretching by a factor of 2, shifting 5 units to the left, reflecting about the x-axis, and shifting 3 units up. Find the equation for f(x).

Answer 1

To obtain f(x), we multiply y = x³ by -1, replace x with x/2 for horizontal stretch, replace x with (x+5) for 5 unit left shift and add 3 for the upward shift resulting in f(x) = -((x+5)/2)³ + 3.

Step-by-step explanation:

When transforming the function y = x³, we need to apply several modifications to obtain the new function f(x). To reflect the function about the x-axis, we multiply the function by -1. This gives us y = -x³. To horizontally stretch the function by a factor of 2, we replace x with x/2, resulting in y = -(x/2)³. Shifting 5 units to the left is achieved by replacing x with (x+5) in this new function, which gives us y = -((x+5)/2)³. Finally, shifting 3 units up is done by adding 3 to our function, resulting in f(x) = -((x+5)/2)³ + 3.