Question

The shape of f(x)=x^3, but shifted six units to the left, nine units down, and then reflected in the y-axis (in that order).

What is the equation for the transformed function?

a) f(x) = -(x + 6)^3 - 9
b) f(x) = -(x - 6)^3 + 9
c) f(x) = (x - 6)^3 - 9
d) f(x) = (x + 6)^3 + 9

Answer 1

The transformed function after shifting to the left, shifting down, and reflecting in the y-axis is f(x) = -(x + 6)^3 - 9, which is option (a).

Step-by-step explanation:

When transforming the shape of the function f(x) = x^3, the following steps need to be taken:

1. Shift six units to the left: we apply f(x) = (x + 6)^3, because moving left is equivalent to adding to the x-value.
2. Shift nine units down: modify the function to f(x) = (x + 6)^3 - 9.
3. Reflect in the y-axis: we use y(x) = −y(−x), which changes the function to f(x) = -((x + 6)^3) - 9.

Therefore, the correct equation for the transformed function is f(x) = -(x + 6)^3 - 9, which corresponds to option (a) in the given choices.