Question

What is the transformation rule to go from ( f(x) ) to ( g(x) = 3f(-x-2) + 5 )?

a. g(x) = 3f(x+2) + 5
b. g(x) = 3f(-x-2) - 5
c. g(x) = f(-3x+2) + 5
d. g(x) = 3f(-x-2) + 10

Answer 1

The correct transformation rule for getting from f(x) to g(x) = 3f(-x-2) + 5 is a horizontal reflection, a shift to the right by 2 units, a vertical stretch by a factor of 3, and an upward shift by 5 units, which corresponds to option (d).

Step-by-step explanation:

The transformation rule to go from f(x) to g(x) = 3f(-x-2) + 5 involves several steps. First, there's a horizontal reflection because of the '-x' inside the function. The function is reflected across the y-axis. Next, there's a horizontal translation; since it's '-x-2', the function is shifted right by 2 units. This can sometimes be confusing because the '-2' inside the function indicates a movement in the positive x-direction. Finally, the function is vertically scaled by a factor of 3 and then translated upwards by 5 units. The correct transformation rule, based on the given options, would be choice (d) g(x) = 3f(-x-2) + 5.