Answer :
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].
