Final answer:
To shift the function f(x) = eˣ to the right by 2, up by 4, and vertically stretch it by 3, while reflecting it about the x-axis, we apply each transformation step-by-step. The resulting function is f(x) = -3eˣ₋₂ - 4.
Step-by-step explanation:
To analyze how the function f(x) = eˣ changes with a shift to the right by 2, up by 4, and a vertical stretch of 3, reflected about the x-axis, we can apply the transformations to the original function step-by-step.
To shift the function to the right by 2, we replace 'x' with '(x - 2)' in the original function: f(x) = eˣ becomes f(x) = eˣ₋₂.
To shift the function up by 4, we add 4 to the original function: f(x) = eˣ₋₂ becomes f(x) = eˣ₋₂ + 4.
To vertically stretch the function by 3, we multiply the original function by 3: f(x) = eˣ₋₂ + 4 becomes f(x) = 3eˣ₋₂ + 4.
To reflect the function about the x-axis, we multiply the entire function by -1: f(x) = 3eˣ₋₂ + 4 becomes f(x) = -3eˣ₋₂ - 4.