Final answer:
The transformation of f(x) = x² to g(x) = -(x-1)² involves a reflection over the x-axis and a horizontal translation of one unit to the right, resulting in an inverted parabola with its vertex moved to (1,0).
Step-by-step explanation:
The student is asking about transforming the function f(x) = x² to g(x) = -(x-1)². To understand this transformation, we should recognize that this involves a reflection and translation of the parent function f(x) = x². Applying the negative sign in front of the function reflects it over the x-axis, and the (x-1) inside the function indicates a horizontal translation one unit to the right.
Firstly, consider the original function f(x) = x², which is a parabola facing upwards. When we introduce the negative sign, as in g(x) = -x², it reflects the parabola about the x-axis, meaning the parabola now faces downwards. Lastly, by replacing x with (x-1), we shift the entire parabola to the right by one unit, which means the vertex of the parabola is now at (1,0) instead of (0,0).
To make this transformation clearer with an example, if we consider the point (0,0) on the graph of f(x), under the transformation to g(x), this point would shift to (1,0) and since it's been reflected, if it originally had a positive value, it would be negative now, hence the point on g(x) is (1,0).