Part A
Option B (The graph of g of x \( g(x) \) can be drawn by translating the graph of f of x \( f(x) \) 1 unit right and 2 units up) should be selected.
Part B
Option B should be selected:h of x equals the opposite of x squared plus 1
$$ h(x) = -x^2+1 $$
Part A Explanation
The function g of x \( g(x) \) is a transformation of the function f of x. \( f(x). \) Let's break down the transformations.
(1) Horizontal translation. When adding or subtracting a constant term inside the square function (that is, on the input variable x before any computation), every point on the parabola shifts to the left or right by that amount. The subtraction open parenthesis x minus 1 close parenthesis squared \( (x-1)^2 \) causes a translation to the right. Adding a positive number would cause a translation to the left. Thus the graph is shifted to the right by 1 unit.
(2) Vertical translation. Outside the square function (that is, after the operation of squaring on the input variable x), a constant term added causes a shift upward, while a constant number being subtracted would cause a shift downward. In this case, every point on the graph is shifted two units upward. The addition of 2 outside the square function is a vertical shift.
The combination of these transformations results in a parabola that is shifted 1 unit to the right and 2 units upward compared to the original parabola. The vertex of the f of x \( f(x) \) is at (0, 0), and the vertex of g of x \( g(x) \) is at (1, 2).
Part B Explanation
The graph of f of x \( f(x) \) is a parabola that opens upward with a vertex at the origin. To reflect the parabola across the horizontal x-axis, you need to have every output y \( (y) \) be its opposite. This can be accomplished with the following transformation:f of x equals x squared is transformed to h of x equals the opposite of x squared
$$ f(x)=x^2 \quad\quad \Longrightarrow \quad\quad h(x)=-x^2 $$
You can then translate this graph 1 unit upward by simply adding 1.h of x equals the opposite of x squared is transformed to h of x equals the opposite of x squared plus 1
$$ h(x) = -x^2 \quad\quad \Longrightarrow \quad\quad h(x) = -x^2+1 $$
