IDNLearn.com makes it easy to find the right answers to your questions. Our experts provide timely, comprehensive responses to ensure you have the information you need.
Sagot :
To determine which functions have a maximum and are transformed to the left and down of the parent function [tex]\( f(x) = x^2 \)[/tex], we need to analyze each function's behavior in terms of the direction of the parabola (whether it opens upward or downward) and its vertex transformations relative to [tex]\( f(x) = x^2 \)[/tex].
1. Function: [tex]\( p(x) = 14(x+7)^2 + 1 \)[/tex]
- The coefficient of the quadratic term (14) is positive, indicating that the parabola opens upwards, which means it has a minimum, not a maximum.
- The vertex of this parabola is at [tex]\((-7, 1)\)[/tex], which indicates it is translated to the left by 7 units and up by 1 unit.
- Since this function does not have a maximum, it is not a candidate.
2. Function: [tex]\( q(x) = -5(x+10)^2 - 1 \)[/tex]
- The coefficient of the quadratic term (-5) is negative, indicating that the parabola opens downwards, which means it has a maximum.
- The vertex of this parabola is at [tex]\((-10, -1)\)[/tex], which indicates it is translated to the left by 10 units and down by 1 unit.
- This function has a maximum and is indeed transformed to the left and down of the parent function [tex]\( f(x) = x^2 \)[/tex].
3. Function: [tex]\( s(x) = -(x-1)^2 + 0.5 \)[/tex]
- The coefficient of the quadratic term (-1) is negative, indicating that the parabola opens downwards, which means it has a maximum.
- The vertex of this parabola is at [tex]\((1, 0.5)\)[/tex], which indicates it is translated to the right by 1 unit and up by 0.5 units.
- Although this function has a maximum, it is transformed to the right, not to the left and down.
4. Function: [tex]\( g(x) = 2x^2 + 10x - 35 \)[/tex]
- To analyze this function, let's rewrite it in vertex form by completing the square:
[tex]\[ g(x) = 2(x+2.5)^2 - 47.5 \][/tex]
- The coefficient of the quadratic term (2) is positive, indicating that the parabola opens upwards, which means it has a minimum, not a maximum.
- Therefore, this function does not have a maximum and is not a candidate.
5. Function: [tex]\( t(x) = -2x^2 - 4x - 3 \)[/tex]
- To analyze this function, let's rewrite it in vertex form by completing the square:
[tex]\[ t(x) = -2(x+1)^2 - 1 \][/tex]
- The coefficient of the quadratic term (-2) is negative, indicating that the parabola opens downwards, which means it has a maximum.
- The vertex of this parabola is at [tex]\((-1, -1)\)[/tex], which indicates it is translated to the left by 1 unit and down by 1 unit.
- This function has a maximum and is transformed to the left and down of the parent function [tex]\( f(x) = x^2 \)[/tex].
Based on this analysis, the functions that have a maximum and are transformed to the left and down of the parent function [tex]\( f(x) = x^2 \)[/tex] are:
- [tex]\( q(x) = -5(x+10)^2 - 1 \)[/tex]
- [tex]\( t(x) = -2x^2 - 4x - 3 \)[/tex]
Therefore, the correct answers are functions 2 and 5: [tex]\( q(x) \)[/tex] and [tex]\( t(x) \)[/tex].
1. Function: [tex]\( p(x) = 14(x+7)^2 + 1 \)[/tex]
- The coefficient of the quadratic term (14) is positive, indicating that the parabola opens upwards, which means it has a minimum, not a maximum.
- The vertex of this parabola is at [tex]\((-7, 1)\)[/tex], which indicates it is translated to the left by 7 units and up by 1 unit.
- Since this function does not have a maximum, it is not a candidate.
2. Function: [tex]\( q(x) = -5(x+10)^2 - 1 \)[/tex]
- The coefficient of the quadratic term (-5) is negative, indicating that the parabola opens downwards, which means it has a maximum.
- The vertex of this parabola is at [tex]\((-10, -1)\)[/tex], which indicates it is translated to the left by 10 units and down by 1 unit.
- This function has a maximum and is indeed transformed to the left and down of the parent function [tex]\( f(x) = x^2 \)[/tex].
3. Function: [tex]\( s(x) = -(x-1)^2 + 0.5 \)[/tex]
- The coefficient of the quadratic term (-1) is negative, indicating that the parabola opens downwards, which means it has a maximum.
- The vertex of this parabola is at [tex]\((1, 0.5)\)[/tex], which indicates it is translated to the right by 1 unit and up by 0.5 units.
- Although this function has a maximum, it is transformed to the right, not to the left and down.
4. Function: [tex]\( g(x) = 2x^2 + 10x - 35 \)[/tex]
- To analyze this function, let's rewrite it in vertex form by completing the square:
[tex]\[ g(x) = 2(x+2.5)^2 - 47.5 \][/tex]
- The coefficient of the quadratic term (2) is positive, indicating that the parabola opens upwards, which means it has a minimum, not a maximum.
- Therefore, this function does not have a maximum and is not a candidate.
5. Function: [tex]\( t(x) = -2x^2 - 4x - 3 \)[/tex]
- To analyze this function, let's rewrite it in vertex form by completing the square:
[tex]\[ t(x) = -2(x+1)^2 - 1 \][/tex]
- The coefficient of the quadratic term (-2) is negative, indicating that the parabola opens downwards, which means it has a maximum.
- The vertex of this parabola is at [tex]\((-1, -1)\)[/tex], which indicates it is translated to the left by 1 unit and down by 1 unit.
- This function has a maximum and is transformed to the left and down of the parent function [tex]\( f(x) = x^2 \)[/tex].
Based on this analysis, the functions that have a maximum and are transformed to the left and down of the parent function [tex]\( f(x) = x^2 \)[/tex] are:
- [tex]\( q(x) = -5(x+10)^2 - 1 \)[/tex]
- [tex]\( t(x) = -2x^2 - 4x - 3 \)[/tex]
Therefore, the correct answers are functions 2 and 5: [tex]\( q(x) \)[/tex] and [tex]\( t(x) \)[/tex].
We value your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. Find reliable answers at IDNLearn.com. Thanks for stopping by, and come back for more trustworthy solutions.
