Answer :
Let's analyze the given functions [tex]\( f(x) = 4(x-3)^2 + 6 \)[/tex] and [tex]\( g(x) = -2(x+1)^2 + 4 \)[/tex] step by step to determine the relationships between them.
1. Direction of Opening:
- Function [tex]\( f(x) \)[/tex] has a positive leading coefficient (4), which means it opens upwards.
- Function [tex]\( g(x) \)[/tex] has a negative leading coefficient (-2), which means it opens downwards.
- Therefore, function [tex]\( g \)[/tex] opens in the opposite direction of function [tex]\( f \)[/tex].
2. Vertices of the Functions:
- For [tex]\( f(x) = 4(x-3)^2 + 6 \)[/tex]:
- The vertex form of a parabola [tex]\( a(x-h)^2 + k \)[/tex] gives us the vertex at [tex]\( (h, k) \)[/tex].
- Here, [tex]\( f \)[/tex] has a vertex at [tex]\( (3, 6) \)[/tex].
- For [tex]\( g(x) = -2(x+1)^2 + 4 \)[/tex]:
- Similarly, the vertex form gives us the vertex at [tex]\( (-1, 4) \)[/tex].
3. Relative Positions of the Vertices:
- The vertex of function [tex]\( f \)[/tex] is at [tex]\( (3, 6) \)[/tex].
- The vertex of function [tex]\( g \)[/tex] is at [tex]\( (-1, 4) \)[/tex].
- Now, let's check the relative positions one by one:
- The x-coordinate of [tex]\( g \)[/tex] is [tex]\(-1\)[/tex], and the x-coordinate of [tex]\( f \)[/tex] is [tex]\( 3 \)[/tex]. Thus, [tex]\( -1 \)[/tex] is left of [tex]\( 3 \)[/tex], indicating the vertex of [tex]\( g \)[/tex] is indeed to the left of the vertex of [tex]\( f \)[/tex] by 4 units.
- The y-coordinate of [tex]\( g \)[/tex] is [tex]\( 4 \)[/tex], and the y-coordinate of [tex]\( f \)[/tex] is [tex]\( 6 \)[/tex]. Thus, [tex]\( 4 \)[/tex] is less than [tex]\( 6 \)[/tex], meaning the vertex of [tex]\( g \)[/tex] is 2 units below the vertex of [tex]\( f \)[/tex].
- The vertex of [tex]\( g \)[/tex] is not 4 units to the right of the vertex of [tex]\( f \)[/tex] because [tex]\(-1\)[/tex] to [tex]\(3\)[/tex] is a leftward move.
- The vertex of [tex]\( g \)[/tex] is not 2 units above the vertex of [tex]\( f \)[/tex] because [tex]\( 4 \)[/tex] is less than [tex]\( 6 \)[/tex].
In conclusion, the following statements are true:
1. Function [tex]\( g \)[/tex] opens in the opposite direction of function [tex]\( f \)[/tex].
2. The vertex of function [tex]\( g \)[/tex] is 4 units to the left of the vertex of function [tex]\( f \)[/tex].
3. The vertex of function [tex]\( g \)[/tex] is 2 units below the vertex of function [tex]\( f \)[/tex].
1. Direction of Opening:
- Function [tex]\( f(x) \)[/tex] has a positive leading coefficient (4), which means it opens upwards.
- Function [tex]\( g(x) \)[/tex] has a negative leading coefficient (-2), which means it opens downwards.
- Therefore, function [tex]\( g \)[/tex] opens in the opposite direction of function [tex]\( f \)[/tex].
2. Vertices of the Functions:
- For [tex]\( f(x) = 4(x-3)^2 + 6 \)[/tex]:
- The vertex form of a parabola [tex]\( a(x-h)^2 + k \)[/tex] gives us the vertex at [tex]\( (h, k) \)[/tex].
- Here, [tex]\( f \)[/tex] has a vertex at [tex]\( (3, 6) \)[/tex].
- For [tex]\( g(x) = -2(x+1)^2 + 4 \)[/tex]:
- Similarly, the vertex form gives us the vertex at [tex]\( (-1, 4) \)[/tex].
3. Relative Positions of the Vertices:
- The vertex of function [tex]\( f \)[/tex] is at [tex]\( (3, 6) \)[/tex].
- The vertex of function [tex]\( g \)[/tex] is at [tex]\( (-1, 4) \)[/tex].
- Now, let's check the relative positions one by one:
- The x-coordinate of [tex]\( g \)[/tex] is [tex]\(-1\)[/tex], and the x-coordinate of [tex]\( f \)[/tex] is [tex]\( 3 \)[/tex]. Thus, [tex]\( -1 \)[/tex] is left of [tex]\( 3 \)[/tex], indicating the vertex of [tex]\( g \)[/tex] is indeed to the left of the vertex of [tex]\( f \)[/tex] by 4 units.
- The y-coordinate of [tex]\( g \)[/tex] is [tex]\( 4 \)[/tex], and the y-coordinate of [tex]\( f \)[/tex] is [tex]\( 6 \)[/tex]. Thus, [tex]\( 4 \)[/tex] is less than [tex]\( 6 \)[/tex], meaning the vertex of [tex]\( g \)[/tex] is 2 units below the vertex of [tex]\( f \)[/tex].
- The vertex of [tex]\( g \)[/tex] is not 4 units to the right of the vertex of [tex]\( f \)[/tex] because [tex]\(-1\)[/tex] to [tex]\(3\)[/tex] is a leftward move.
- The vertex of [tex]\( g \)[/tex] is not 2 units above the vertex of [tex]\( f \)[/tex] because [tex]\( 4 \)[/tex] is less than [tex]\( 6 \)[/tex].
In conclusion, the following statements are true:
1. Function [tex]\( g \)[/tex] opens in the opposite direction of function [tex]\( f \)[/tex].
2. The vertex of function [tex]\( g \)[/tex] is 4 units to the left of the vertex of function [tex]\( f \)[/tex].
3. The vertex of function [tex]\( g \)[/tex] is 2 units below the vertex of function [tex]\( f \)[/tex].
