Welcome to Westonci.ca, your ultimate destination for finding answers to a wide range of questions from experts. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.
Sagot :
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].
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].
Thank you for trusting us with your questions. We're here to help you find accurate answers quickly and efficiently. We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Thank you for trusting Westonci.ca. Don't forget to revisit us for more accurate and insightful answers.
