Let's analyze the function \( g(x) = -(x-4)^2 + 6 \). This function is a transformed version of the parent quadratic function \( f(x) = x^2 \).
1. Reflection across the \( x \)-axis:
The function \( g(x) \) has a negative sign in front of the squared term, which means that the graph of the function is reflected across the \( x \)-axis. This statement is true.
2. Translation of the vertex:
The general form of a vertex-translated quadratic function is \( a(x-h)^2 + k \), where \((h, k)\) is the vertex.
In this case, \( g(x) = -(x-4)^2 + 6 \), so the vertex is \((4, 6)\).
However, the statement indicates the translation is to \((-4, 6)\), which is incorrect. Thus, this statement is false.
3. Behavior as \( x \) approaches negative infinity:
As \( x \) approaches both positive and negative infinity, the dominating term is the squared term multiplied by the negative coefficient. This means \( g(x) \) will approach negative infinity in both directions. Therefore, as \( x \) approaches negative infinity, \( g(x) \) also approaches negative infinity. This statement is true.
4. Function being always decreasing:
The function \( g(x) \) represents a downward opening parabola. It decreases before the vertex and increases after the vertex. Therefore, it is not always decreasing. This statement is false.
5. Function being always negative:
The vertex of the function \( g(x) \) is \((4, 6)\), and since the parabola opens downwards, the maximum value of \( g(x) \) is 6 at the vertex. Hence, the function takes positive values around the vertex and is not always negative. This statement is false.
6. Symmetrical about the point \((4, 6)\):
Any quadratic function in the form \( a(x-h)^2 + k \) is symmetrical about its vertex. Since the vertex of \( g(x) \) is \((4, 6)\), the function is symmetrical about this point. This statement is true.
So, the correct statements about the function \( g(x) = -(x-4)^2 + 6 \) are:
- Function \( g \) reflected function \( f \) across the \( x \)-axis.
- For function \( g \), as \( x \) approaches negative infinity, \( g(x) \) approaches negative infinity.
- Function [tex]\( g \)[/tex] is symmetrical about the point [tex]\((4, 6)\)[/tex].
