Let's analyze the given functions step-by-step to understand their relationships.
1. Determine the vertices of the functions:
- For [tex]\( f(x) = 4(x-3)^2 + 6 \)[/tex], the vertex form [tex]\( a(x-h)^2 + k \)[/tex] gives the vertex [tex]\((h, k)\)[/tex]. Here, [tex]\( h = 3 \)[/tex] and [tex]\( k = 6 \)[/tex]. Thus, the vertex of [tex]\( f \)[/tex] is [tex]\((3, 6)\)[/tex].
- For [tex]\( g(x) = -2(x+1)^2 + 4 \)[/tex], in the vertex form [tex]\( a(x-h)^2 + k \)[/tex], we have [tex]\( x+1 = x - (-1) \)[/tex], so [tex]\( h = -1 \)[/tex] and [tex]\( k = 4 \)[/tex]. Therefore, the vertex of [tex]\( g \)[/tex] is [tex]\((-1, 4)\)[/tex].
2. Determine the direction in which each function opens:
- For [tex]\( f(x) = 4(x-3)^2 + 6 \)[/tex], the coefficient of the squared term is 4, which is positive. Therefore, [tex]\( f(x) \)[/tex] opens upwards.
- For [tex]\( g(x) = -2(x+1)^2 + 4 \)[/tex], the coefficient of the squared term is -2, which is negative. Thus, [tex]\( g(x) \)[/tex] opens downwards.
3. Check the direction relationship between the functions:
- Since [tex]\( f(x) \)[/tex] opens upwards and [tex]\( g(x) \)[/tex] opens downwards, [tex]\( g \)[/tex] opens in the opposite direction of [tex]\( f \)[/tex].
4. Determine the vertical distance between the vertices:
- The y-coordinate of [tex]\( f \)[/tex]'s vertex is 6, and the y-coordinate of [tex]\( g \)[/tex]'s vertex is 4.
- The vertical distance is [tex]\( |6 - 4| = 2 \)[/tex].
- Since [tex]\( 6 > 4 \)[/tex], the vertex of [tex]\( g \)[/tex] is 2 units below the vertex of [tex]\( f \)[/tex].
5. Determine the horizontal distance between the vertices:
- The x-coordinate of [tex]\( f \)[/tex]'s vertex is 3, and the x-coordinate of [tex]\( g \)[/tex]'s vertex is -1.
- The horizontal distance is [tex]\( |3 - (-1)| = |3 + 1| = 4 \)[/tex].
- Since [tex]\( -1 < 3 \)[/tex], the vertex of [tex]\( g \)[/tex] is 4 units to the left of the vertex of [tex]\( f \)[/tex].
Conclusion of the statements:
- Function [tex]\( g \)[/tex] opens in the opposite direction of function [tex]\( f \)[/tex]. (True)
- The vertex of function [tex]\( g \)[/tex] is 2 units above the vertex of function [tex]\( f \)[/tex]. (False)
- The vertex of function [tex]\( g \)[/tex] is 2 units below the vertex of function [tex]\( f \)[/tex]. (True)
- Function [tex]\( g \)[/tex] opens in the same direction as function [tex]\( f \)[/tex]. (False)
- The vertex of function [tex]\( g \)[/tex] is 4 units to the right of the vertex of function [tex]\( f \)[/tex]. (False)
- The vertex of function [tex]\( g \)[/tex] is 4 units to the left of the vertex of function [tex]\( f \)[/tex]. (True)
Thus, the correct answers are:
- Function [tex]\( g \)[/tex] opens in the opposite direction of function [tex]\( f \)[/tex].
- The vertex of function [tex]\( g \)[/tex] is 2 units below the vertex of function [tex]\( f \)[/tex].
- The vertex of function [tex]\( g \)[/tex] is 4 units to the left of the vertex of function [tex]\( f \)[/tex].
